ECON 350 - CLASSICAL
ECONOMICS - Descriptive statistics
Question Bank - Set 3
Liberty University
Question 1
Question
Let Xbe a random variable with probability density function given by
f(x) = (2xfor 0 <x<1
0 otherwise
Calculate the mean and standard deviation of X.
Solution
Step 1: To find the mean of X, we use the formula for the expected value of a
continuous random variable:
E(X) = Z∞
−∞
xf(x)dx
Step 2: Substituting the given probability density function f(x) into the
formula, we have:
E(X) = Z1
0
2x·x dx
Step 3: Simplifying the integrand, we get:
E(X) = Z1
0
2x2dx
Step 4: Solving the integral, we find:
E(X) = 2x3
31
0
=2
3
Therefore, the mean of Xis 2
3.
Step 5: To find the standard deviation of X, we use the formula:
Var(X) = E(X2)−[E(X)]2
Step 6: We already know E(X) = 2
3. Now, we need to find E(X2).
Step 7: Using the definition of expectation, we find:
E(X2) = Z1
0
2x·x2dx =Z1
0
2x3dx =1
2
Step 8: Now, we calculate the variance:
Var(X) = 1
2−2
32
=1
2−4
9=1
18
Step 9: Finally, the standard deviation of Xis the square root of the variance,
so:
SD(X) = r1
18 =1
√18 =√2
6
Therefore, the standard deviation of Xis √2
6.
Question 2
Question
Let’s consider the following dataset of exam scores:
8,12,15,17,20,22,26,28,31,35
Calculate the mean, median, mode, range, variance, and standard deviation of
the dataset.
Solution
To find the mean, median, mode, range, variance, and standard deviation of the
dataset, we will follow these steps:
Step 1: Calculate the Mean The mean of a dataset is calculated by
summing all values and dividing by the number of values.
Mean =8 + 12 + 15 + 17 + 20 + 22 + 26 + 28 + 31 + 35
10
Step 2: Calculate the Median The median is the middle value when the
data is ordered from least to greatest. To find the median, we first order the
2
dataset:
8,12,15,17,20,22,26,28,31,35
Since we have 10 values, the median is the average of the 5th and 6th values.
So, the median is 20+22
2= 21.
Step 3: Calculate the Mode The mode is the value that appears most
frequently in the dataset. In this dataset, there is no repeating value, so there
is no mode.
Step 4: Calculate the Range The range is the difference between the
maximum and minimum values in the dataset. Range = Maximum value -
Minimum value = 35 - 8 = 27.
Step 5: Calculate the Variance The variance measures how spread out
the values are from the mean.
V ariance =Pn
i=1(xi−¯x)2
n−1
where xiis each value, ¯xis the mean, and nis the number of values. After
calculating, the variance is found to be 94.6.
Step 6: Calculate the Standard Deviation Finally, the standard devi-
ation is the square root of the variance. Standard Deviation = √V ariance =
√94.6 = 9.73.
Question 3
Question
The following data represents the test scores of a sample of 15 students:
64,72,75,80,82,85,88,90,92,94,97,99,100,100,105
Calculate the coefficient of variation for the given data. Round your answer
to two decimal places.
Solution
Step 1: Find the mean of the data.
Mean = 1
n
n
X
i=1
xi=1
15(64+72+75+80+82+85+88+90+92+94+97+99+100+100+105) = 87.27
Step 2: Find the standard deviation of the data.
Standard Deviation = v
u
u
t
1
n
n
X
i=1
(xi−Mean)2=r1
15((64 −87.27)2+ (72 −87.27)2+... + (105 −87.27)2)
3
Standard Deviation ≈11.55
Step 3: Calculate the coefficient of variation.
Coefficient of Variation = Standard Deviation
Mean ×100% = 11.55
87.27×100% ≈13.22%
Therefore, the coefficient of variation for the given data is approximately
13.22
Question 4
Question
Let’s consider a dataset of 100 students’ exam scores from a certain university
course. The mean exam score is 75 with a standard deviation of 10. If the scores
are normally distributed, what percentage of students scored above an 85 on
the exam?
Solution
Step 1: Calculate the z-score corresponding to an exam score of 85 using the
formula:
z=x−µ
σ
where xis the exam score, µis the mean, and σis the standard deviation.
Plugging in the values:
z=85 −75
10 = 1
Step 2: Use a standard normal distribution table or calculator to find the
percentage of students scoring above a z-score of 1. From the standard nor-
mal distribution table, we know that the area to the right of a z-score of 1 is
approximately 0.1587.
Step 3: Convert this probability to a percentage by multiplying by 100.
Therefore, the percentage of students who scored above an 85 on the exam is:
0.1587 ×100% = 15.87%
So, approximately 15.87
Question 5
Question
Let’s say we have a data set with the following values:
5, 6, 8, 11, 14, 17, 19, 24, 26, 29
Calculate the mean, median, mode, variance, and standard deviation for this
data set.
4
Solution
To find the mean, median, mode, variance, and standard deviation, we will
perform the following calculations step by step:
Step 1: Find the Mean The mean (x) is calculated by summing up all
the values in the data set and dividing by the total number of values.
x=5+6+8+11+14+17+19+24+26+29
10
Step 2: Find the Median To find the median, we first need to arrange
the data set in ascending order: 5, 6, 8, 11, 14, 17, 19, 24, 26, 29. Since we have
10 values, the median is the average of the 5th and 6th values. Median = 14+17
2
Step 3: Find the Mode The mode is the value that appears most fre-
quently in the data set. In this case, there is no value that appears more than
once, so the data set has no mode.
Step 4: Find the Variance The variance (s2) is calculated by finding the
average of the squared differences between each value and the mean.
s2=(5 −x)2+ (6 −x)2+. . . + (29 −x)2
9
Step 5: Find the Standard Deviation The standard deviation (s) is the
square root of the variance. It gives a measure of how spread out the values in
the data set are.
s=√s2
Now, we can substitute the mean (x) from Step 1 into the formulas for
variance and standard deviation to find their values.
Question 6
Question
A researcher collected data on the number of hours students studied per week
and their corresponding final exam scores. The data set has a mean study time
of 10 hours with a standard deviation of 3 hours, and a mean exam score of 75
with a standard deviation of 10. Calculate the coefficient of variation for both
the study time and exam scores, and interpret the results in the context of the
study.
Solution
Step 1: Calculate the coefficient of variation for study time. The coefficient
of variation is calculated as the ratio of the standard deviation to the mean,
multiplied by 100%.
Coefficient of Variation for study time = Standard Deviation of study time
Mean of study time ×100%
5
Coefficient of Variation for study time = 3
10 ×100% = 30%
Step 2: Calculate the coefficient of variation for exam scores. Similarly, the
coefficient of variation for exam scores is:
Coefficient of Variation for exam scores = Standard Deviation of exam scores
Mean of exam scores ×100%
Coefficient of Variation for exam scores = 10
75 ×100% ≈13.33%
Step 3: Interpretation - The coefficient of variation for study time is 30%,
indicating that the variability in study time is relatively high compared to the
mean. - The coefficient of variation for exam scores is approximately 13.33%,
indicating that the variability in exam scores is lower compared to the mean.
In the context of the study, this suggests that the students have a wider
range of study times compared to their exam scores.
Question 7
Question
A researcher collected data on the heights (in centimeters) of 50 university
students. The data had a mean of 170 cm and a standard deviation of 10 cm.
If the researcher decides to remove the top 10
Solution
Step 1: Find the z-score corresponding to the top 10
Let z= z-score corresponding to the top 10%
From the standard normal distribution table,
P(Z≤z)=0.9
z≈1.28
Step 2: Use the z-score formula to find the corresponding height value.
z=X−µ
σ
1.28 = X−170
10
X−170 = 12.8
X= 182.8
Step 3: The minimum height a student must have to remain in the dataset
is 182.8 cm.
6
Question 8
Question
A researcher collected the following data on the number of hours students stud-
ied for an exam: 3, 5, 7, 6, 4, 8, 10, 6, 5, 7. Compute the mean, median, mode,
range, variance, and standard deviation of the data.
Solution
Step 1: Calculate the mean.
Mean = 3+5+7+6+4+8+10+6+5+7
10
=61
10
= 6.1
Step 2: Calculate the median. First, we need to arrange the data in ascend-
ing order: 3, 4, 5, 5, 6, 6, 7, 7, 8, 10
Since we have an even number of data points, the median will be the average
of the two middle numbers: Median = 6+6
2= 6.
Step 3: Calculate the mode. The mode is the number that appears most
frequently in the dataset. In this case, both 5 and 7 appear twice, so the data
is bimodal.
Step 4: Calculate the range. Range = Max value - Min value = 10 - 3 = 7.
Step 5: Calculate the variance.
Variance = (3 −6.1)2+ (5 −6.1)2+ (7 −6.1)2+ (6 −6.1)2+ (4 −6.1)2+ (8 −6.1)2+ (10 −6.1)2+ (6 −6.1)2+ (5 −6.1)2+ (7 −6.1)2
10 −1
=9.61 + 1.21 + 0.81 + 0.01 + 4.41 + 4.41 + 13.69 + 0.01 + 1.21 + 0.81
9
=35.57
9
≈3.95
Step 6: Calculate the standard deviation. Standard deviation = √Variance ≈
√3.95 ≈1.98.
Question 9
Question
Suppose you have a dataset with the following values: 1, 2, 3, 4, 5, 6, 7, 8, 9, 100.
Calculate the mean, median, mode, range, variance, and standard deviation for
this dataset.
7
Solution
Step 1: Begin by arranging the dataset in ascending order: 1, 2, 3, 4, 5, 6, 7, 8,
9, 100.
Step 2: Calculate the mean: The mean is calculated by summing all values
and dividing by the number of values. Mean = 1+2+3+4+5+6+7+8+9+100
10 =145
10 =
14.5.
Step 3: Calculate the median: Since there are 10 values, the median is the
average of the 5th and 6th values. Median = 5+6
2=11
2= 5.5.
Step 4: Calculate the mode: In this dataset, there is no mode as each value
appears only once.
Step 5: Calculate the range: Range = Maximum value - Minimum value =
100 - 1 = 99.
Step 6: Calculate the variance: The variance is the average of the squared dif-
ferences from the mean. Variance = (1−14.5)2+(2−14.5)2+...+(100−14.5)2
10 =(198.25+161.29+...+2087.25)
10 =
3004.85
10 = 300.485.
Step 7: Calculate the standard deviation: The standard deviation is the
square root of the variance. Standard deviation = √300.485 ≈17.34.
Question 10
Question
The following data represents the test scores of students in two different classes:
Class A: 85, 90, 88, 92, 84
Class B: 78, 82, 88, 85, 90
Calculate the mean, median, mode, range, variance, and standard deviation
for each class. Compare the central tendency and dispersion measures between
the two classes.
Solution
Let’s calculate the mean, median, mode, range, variance, and standard deviation
for each class step by step.
Class A:
Mean: ¯x=85+90+88+92+84
5=439
5= 87.8
Median: Arrange the data in ascending order: 84, 85, 88, 90, 92. The
median is 88.
Mode: There is no mode since all values are unique.
Range: 92 −84 = 8
Variance: s2=1
nPn
i=1(xi−¯x)2
s2=(85−87.8)2+(90−87.8)2+(88−87.8)2+(92−87.8)2+(84−87.8)2
5=47.2
5= 9.44
8
Standard Deviation: s=√9.44 ≈3.08
Class B:
Mean: ¯x=78+82+88+85+90
5=423
5= 84.6
Median: Arrange the data in ascending order: 78, 82, 85, 88, 90. The
median is 85.
Mode: There is no mode since all values are unique.
Range: 90 −78 = 12
Variance: s2=1
nPn
i=1(xi−¯x)2
s2=(78−84.6)2+(82−84.6)2+(88−84.6)2+(85−84.6)2+(90−84.6)2
5=56.4
5= 11.28
Standard Deviation: s=√11.28 ≈3.36
Comparing the two classes: - Class A has a higher mean, median, and range
compared to Class B. - Class A has a lower variance and standard deviation
compared to Class B, indicating less dispersion in the data set.
Question 11
Question
Suppose a researcher collected data on the heights (in centimeters) of 100 adult
individuals. The mean height was found to be 170 cm with a standard devi-
ation of 10 cm. If the researcher wants to identify the interval within which
approximately 95
Solution
Step 1: Calculate the margin of error using the formula Z×σ
√n, where Zis
the Z-score corresponding to the desired level of confidence, σis the standard
deviation, and nis the sample size.
Given: - Z-score for 95- Standard deviation, σ= 10 cm. - Sample size,
n= 100.
Plugging in the values: Margin of error = 1.96 ×10
√100 = 1.96. Thus, the
margin of error is 1.96 cm.
Step 2: Find the lower and upper bounds of the interval by adding and
subtracting the margin of error from the mean.
Lower bound: 170 −1.96 = 168.04. Upper bound: 170 + 1.96 = 171.96.
Therefore, the interval within which approximately 95
9
Question 12
Question
A researcher collects data on the heights (in inches) of students in a mathe-
matics class. The following descriptive statistics were computed for the data:
mean height = 65 inches, standard deviation = 4 inches. Assuming the data
is approximately normally distributed, what proportion of students in the class
have heights greater than 70 inches?
Solution
Step 1: Find the z-score for a height of 70 inches using the formula:
z=x−µ
σ
where xis the given height, µis the mean height, and σis the standard deviation.
Plugging in the values, we get:
z=70 −65
4=5
4= 1.25
Step 2: Use a standard normal distribution table or calculator to find the
proportion of students with heights greater than 70 inches. From the z-score
table, we look up the area to the right of z= 1.25, which is approximately
0.1056.
Step 3: Therefore, approximately 10.56
Question 13
Question
For a given dataset, the mean is 25, variance is 16, and standard deviation is 4.
Calculate the coefficient of variation.
Solution
Step 1: The coefficient of variation (CV) is defined as the ratio of the standard
deviation to the mean, expressed as a percentage. Step 2: To calculate the CV,
we use the formula:
CV = Standard Deviation
Mean ×100
Step 3: Substituting the given values into the formula, we get:
CV = 4
25×100
10
Step 4: Simplifying the expression further, we have:
CV = 0.16 ×100
CV = 16%
Step 5: Therefore, the coefficient of variation for the given dataset is 16
Question 14
Question
Suppose a data set has a mean of 50, a median of 47, and a mode of 45. If the
data set has a total of 100 values, what is the range of the data set?
Solution
Step 1: Since the mode is 45, we know that there are more occurrences of 45
than any other value in the data set.
Step 2: Since the median is 47, we know that half of the data set values are
less than or equal to 47 and the other half are greater than or equal to 47.
Step 3: Since the mean is 50, we know that the sum of all values divided by
the total number of values is 50.
Step 4: Let’s first calculate the sum of all values in the data set. Since the
mean is 50 and there are 100 values in total, the sum of all values is 50 ×100 =
5000.
Step 5: Since the mode is 45 and there are more occurrences of 45 than
any other value, let’s assume there are ’m’ occurrences of 45. Thus, the sum of
values excluding the occurrences of 45 will be: 5000 −45 ×m.
Step 6: Since the median is 47, half of the values should be less than or equal
to 47. Since there are ’m’ occurrences of 45 and the rest of the values must be
greater than 45, we can consider the sum of these values as 47×(100−m−m) =
4700 −2×47m.
Step 7: Now we can set up the equation involving the sum of all values:
5000 −45m= 4700 −94m. Solving this equation, we get 49m= 300 and thus,
m= 300/49.
Step 8: Now we can find the range of the data set. The minimum value is
45 and the maximum value is 45 + 2 ×47 = 139, since there are 2 values to the
right of the median to reach the maximum value.
Step 9: Therefore, the range of the data set is 139 −45 = 94.
Question 15
Question
The following data represents the test scores of 20 students:
11
39, 46, 51, 62, 54, 38, 26, 47, 58, 61, 73, 52, 59, 63, 65, 52, 49, 50, 55, 42.
Calculate the interquartile range of the data set.
Solution
Step 1: First, we need to arrange the data in ascending order:
26,38,39,42,46,47,49,50,51,52,52,54,55,58,59,61,62,63,65,73
Step 2: Next, we find the median of the data set. Since we have an even
number of data points, the median is the average of the two middle values:
Median = 52 + 54
2= 53
Step 3: To find the lower quartile Q1, we find the median of the lower half
of the data set. Since there are an odd number of values in the lower half, Q1
is the middle value:
Q1 = 47
Step 4: To find the upper quartile Q3, we find the median of the upper
half of the data set. Q3 is the middle value of the upper half, which is an odd
number:
Q3 = 61
Step 5: Finally, we calculate the interquartile range (IQR) by subtracting
Q1 from Q3:
IQR = Q3−Q1 = 61 −47 = 14
Therefore, the interquartile range of the data set is 14.
Question 16
Question
A researcher collected the following data on the number of hours of sleep per
night for a group of 30 university students: 6, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9,
10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 13, 14.
Calculate the mean, median, mode, range, variance, and standard deviation
of the data set.
Solution
Step 1: Find the mean: The mean is calculated by summing up all the data
points and then dividing by the total number of data points. Mean = 6+7+7+...+14
30 =
238
30 = 7.93
Step 2: Find the median: To find the median, we first rearrange the data
set in ascending order: 6, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10,
12
11, 11, 11, 11, 11, 12, 12, 12, 13, 14. Since there are 30 data points, the median
will be the average of the 15th and 16th terms, which are both 10. Median = 10
Step 3: Find the mode: We can see that the mode of the data set is 10, as
it appears the most frequently.
Step 4: Find the range: The range is the difference between the highest and
lowest data points in the data set. Range = 14 −6 = 8
Step 5: Find the variance: The variance is the average of the squared differ-
ences between each data point and the mean. Variance = (6−7.93)2+(7−7.93)2+...+(14−7.93)2
30 ≈
5.90
Step 6: Find the standard deviation: The standard deviation is the square
root of the variance. Standard Deviation = √5.90 ≈2.43
Question 17
Question
Let’s say we have a dataset of 100 students’ test scores in a Statistics class. The
mean of the test scores is 75 with a standard deviation of 10. If a new student
joins the class and scores 85 on the test, calculate the new mean and standard
deviation of the dataset.
Solution
Step 1: Find the new mean after adding the new student: The formula to
calculate the new mean after adding a new value is:
New Mean = n1×Old Mean + x
n1
where: - n1is the total number of values in the dataset after adding the new
value, - Old Mean = 75 is the previous mean, - x= 85 is the score of the new
student. Plugging in the values:
New Mean = 100 ×75 + 85
101 =7500 + 85
101 =7585
101 = 75.10
Step 2: Find the new standard deviation after adding the new student: The
formula to calculate the new standard deviation after adding a new value is:
New SD = s(n1−1) ×(Old SD2+ Old Mean2) + x2
n1−New Mean2
where: - n1= 101 is the total number of values in the dataset after adding the
new value, - Old SD = 10 is the previous standard deviation, - Old Mean = 75 is
the previous mean, - x= 85 is the score of the new student, - New Mean = 75.10
is the new mean. Plugging in the values:
New SD = r(101 −1) ×(102+ 752) + 852
101 −75.102
13
=r100 ×(100 + 5625) + 7225
101 −75.102
=r10000 + 562500 + 7225
101 −5640.01 = r579725
101 −5640.01
=√5747.77 −5640.01 = √107.76 ≈10.38
Therefore, the new mean is approximately 75.10 and the new standard deviation
is approximately 10.38.
Question 18
Question
Let X={4,8,10,12,14}be a set of data points. Calculate the standard devia-
tion of the data set X.
Solution
To calculate the standard deviation of a data set, we can follow these steps:
Step 1: Find the mean of the data set. Step 2: Calculate the variance of the
data set. Step 3: Take the square root of the variance to find the standard
deviation.
Step 1: Find the mean of the data set. The mean ¯xof a data set is calculated
by summing all the data points and dividing by the number of data points.
¯x=4 + 8 + 10 + 12 + 14
5=48
5= 9.6
Step 2: Calculate the variance of the data set. The variance of a data set is
calculated by finding the average of the squared differences between each data
point and the mean.
Variance = 1
n
n
X
i=1
(xi−¯x)2
Variance = 1
5[(4 −9.6)2+ (8 −9.6)2+ (10 −9.6)2+ (12 −9.6)2+ (14 −9.6)2]
Variance = 1
5[(5.6)2+ (−1.6)2+ (0.4)2+ (2.4)2+ (4.4)2]
Variance = 1
5[31.36 + 2.56 + 0.16 + 5.76 + 19.36] = 59.2
5= 11.84
Step 3: Take the square root of the variance to find the standard deviation.
The standard deviation σis the square root of the variance.
σ=√11.84 = 3.44
Therefore, the standard deviation of the data set Xis 3.44.
14
Question 19
Question
A researcher collected the following data on the number of hours per week that
a sample of students studied for their final exams:
8,12,10,15,6,14,9,11,8,13,16,5,9,12,10
Calculate the mean, median, mode, range, variance, and standard deviation
for this data set.
Solution
Step 1: Calculate the Mean
Mean = 8+12+10+15+6+14+9+11+8+13+16+5+9+12+10
15
=142
15
= 9.47
Step 2: Calculate the Median First, arrange the data in ascending order:
5,6,8,8,9,9,10,10,11,12,12,13,14,15,16
Since there are 15 data points, the median is the middle value, which is the
8th value: 10.
Step 3: Calculate the Mode The mode is the value that appears most
frequently. In this case, the mode is 8 hours.
Step 4: Calculate the Range The range is the difference between the
maximum and minimum values.
Range = 16 −5 = 11
Step 5: Calculate the Variance To calculate the variance, first calculate
the squared differences between each data point and the mean:
(8 −9.47)2= 1.972,(12 −9.47)2= 2.532, . . .
Add up these squared differences and divide by the number of data points
to get the variance.
Step 6: Calculate the Standard Deviation The standard deviation is
the square root of the variance.
After completing these calculations, we would have all the requested descrip-
tive statistics for this data set.
15
Question 20
Question
A researcher collects data on the reading speeds (in words per minute) of two
groups of students, Group A and Group B. The mean reading speed for Group
A is 250 words per minute with a standard deviation of 20, while the mean
reading speed for Group B is 270 words per minute with a standard deviation
of 30. Assuming the reading speeds are normally distributed, which group has
a wider range of reading speeds? Justify your answer.
Solution
Step 1: Calculate the coefficient of variation for each group. The coefficient of
variation is defined as the standard deviation divided by the mean, expressed
as a percentage. It allows us to compare the variability between datasets with
different units or scales.
For Group A: Coefficient of variation = standard deviation
mean ×100%
Coefficient of variation for Group A = 20
250 ×100% = 8%
For Group B: Coefficient of variation for Group B = 30
270 ×100% = 11.11%
Step 2: Compare the coefficient of variation. The coefficient of variation for
Group B (11.11%) is higher than that of Group A (8%). This indicates that
Group B has a wider range of reading speeds compared to Group A. The higher
the coefficient of variation, the greater the variability in the data. Therefore,
Group B has a wider range of reading speeds than Group A.
Question 21
Question
Let’s say we have a dataset with 50 values. After calculating the mean, variance,
standard deviation, skewness, and kurtosis, we obtained the following results:
mean = 65, variance = 100, standard deviation = 10, skewness = -0.5, kurtosis
= 2.
Given this information, calculate the coefficient of variation for this dataset.
Solution
To find the coefficient of variation, we first need to compute the standard devi-
ation as a percentage of the mean.
Step 1: Calculate the coefficient of variation using the formula:
Coefficient of Variation = Standard Deviation
Mean ×100
16
Step 2: Substitute the values into the formula:
Coefficient of Variation = 10
65×100
Step 3: Simplify the expression:
Coefficient of Variation = 10
65 ×100
Coefficient of Variation = 2
13 ×100
Coefficient of Variation = 15.38%
Therefore, the coefficient of variation for this dataset is 15.38
Question 22
Question
Let Xbe a random variable representing the number of hours students study
per week. The data collected from a sample of 30 students is as follows:
2,4,6,6,8,8,8,10,10,10,10,12,12,12,12,12,14,14,14,16,16,16,18,18,18,18,20,20,20,20
Calculate the mean, median, mode, variance, standard deviation, range, and
interquartile range of the data.
Solution
Step 1: Calculate the mean:
Mean = Px
n
Mean = 2+4+6+. . . + 20
30
Mean = 302
30 = 10.07
Step 2: Calculate the median: Since there are 30 data points, the median is
the average of the 15th and 16th values when the data is arranged in ascending
order.
Median = 12 + 12
2= 12
Step 3: Calculate the mode: The mode is the value(s) that appear most
frequently in the data. In this case, since 12, 14, 16, and 18 appear most
17
frequently (each with a frequency of 5), the data is bimodal. Therefore, the
modes are 12, 14, 16, and 18.
Step 4: Calculate the variance:
Variance = P(x−Mean)2
n
Variance = (2 −10.07)2+ (4 −10.07)2+. . . + (20 −10.07)2
30
Variance ≈30.95
Step 5: Calculate the standard deviation:
Standard Deviation = √Variance ≈√30.95 ≈5.56
Step 6: Calculate the range: Range is the difference between the maximum
and minimum values.
Range = 20 −2 = 18
Step 7: Calculate the interquartile range: First, find the first quartile (Q1)
and third quartile (Q3) by finding the medians of the lower half and upper half
of the data, respectively.
Q1=8, Q3 = 16
Interquartile Range = Q3−Q1 = 16 −8=8
Question 23
Question
A statistics class conducted a survey to collect data on the number of hours per
week students spend studying. The results are as follows: 25, 15, 18, 20, 22, 30,
27, 19, 17, 16, 24. Calculate the mean, median, mode, variance, and standard
deviation of the data set.
Solution
Step 1: Calculate the Mean The mean is calculated by adding up all the
values and then dividing by the total number of values.
Mean = 25 + 15 + 18 + 20 + 22 + 30 + 27 + 19 + 17 + 16 + 24
11
Mean = 233
11 = 21.18
Step 2: Calculate the Median To find the median, we first need to arrange
the data in ascending order: 15, 16, 17, 18, 19, 20, 22, 24, 25, 27, 30. Since
there are 11 values, the median will be the middle value, which is the 6th value
in this case. Therefore, the median is 20.
18
Step 3: Calculate the Mode The mode is the value that appears most
frequently in the data set. In this case, there is no value that appears more
than once, so the data set has no mode.
Step 4: Calculate the Variance The variance is calculated by finding the
average of the squared differences between each data point and the mean.
Variance = (25 −21.18)2+ (15 −21.18)2+. . . + (24 −21.18)2
11
Variance = 16.6464 + 38.6464 + . . . + 7.3364
11
Variance = 191.84
11 = 17.44
Step 5: Calculate the Standard Deviation The standard deviation is
the square root of the variance.
Standard Deviation = √17.44 ≈4.18
Therefore, the mean is 21.18, the median is 20, the mode does not exist, the
variance is 17.44, and the standard deviation is approximately 4.18.
Question 24
Question
The monthly salaries, in dollars, of a sample of university professors are as
follows:
$
6200,
$
5800,
$
7000,
$
6500,
$
7200,
$
6800,
$
7500. Calculate the mean,
median, mode, variance, and standard deviation of this data set.
Solution
Step 1: First, calculate the mean of the data set. Step 2: Then, find the median
of the data set. Step 3: Next, identify the mode of the data set. Step 4:
Calculate the variance of the data set. Step 5: Finally, determine the standard
deviation of the data set.
Step 1: To find the mean, we sum up all the monthly salaries and divide
by the total number of salaries. Mean = 6200+5800+7000+6500+7200+6800+7500
7=
46000
7= 6571.43.
Step 2: To find the median, we arrange the salaries in ascending order:
$
5800,
$
6200,
$
6500,
$
6800,
$
7000,
$
7200,
$
7500 Since there are 7 salaries, the
median is the middle value, which is
$
6800.
Step 3: Since there is no repeating value in the data set, there is no mode.
Step 4: To find the variance, we first calculate the squared difference of each
salary from the mean: (6200 −6571.43)2, (5800 −6571.43)2, (7000 −6571.43)2,
(6500−6571.43)2, (7200−6571.43)2, (6800−6571.43)2, (7500−6571.43)2Then,
19
we sum these squared differences and divide by the total number of salaries mi-
nus 1: Variance = (6200−6571.43)2+(5800−6571.43)2+(7000−6571.43)2+(6500−6571.43)2+(7200−6571.43)2+(6800−6571.43)2+(7500−6571.43)2
7−1=
265714.29
6= 44285.71.
Step 5: Finally, the standard deviation is the square root of the variance:
Standard deviation = √44285.71 = 210.59.
Question 25
Question
Let Xbe a random variable with the following probability distribution:
x−2 1 3 5
P(X=x) 0.3 0.2k0.1
Determine the value of kand calculate the mean, variance, and standard
deviation of X.
Solution
Step 1: Determine the value of k. Since the sum of all probabilities must equal
1, we have:
0.3+0.2 + k+ 0.1=1
0.6 + k= 1
k= 0.4
Step 2: Calculate the mean of X. The mean of a discrete random variable
can be calculated as:
E(X) = Xx·P(X=x)
E(X)=(−2)(0.3) + (1)(0.2) + (3)(0.4) + (5)(0.1)
E(X) = −0.6+0.2+1.2+0.5
E(X)=1.3
Step 3: Calculate the variance of X. The variance of Xis given by:
V ar(X) = E(X2)−[E(X)]2
To find E(X2), we calculate:
E(X2) = Xx2·P(X=x)
E(X2)=(−2)2(0.3) + (1)2(0.2) + (3)2(0.4) + (5)2(0.1)
E(X2)=1.2+0.2+4.8+0.5
E(X2)=6.7
20
Now, we can find the variance:
V ar(X)=6.7−(1.3)2
V ar(X) = 6.7−1.69
V ar(X)=4.94
Step 4: Calculate the standard deviation of X. The standard deviation is
the square root of the variance:
SD(X) = pV ar(X)
SD(X) = √4.94
SD(X)≈2.22
Therefore, the value of kis 0.4, the mean of Xis 1.3, the variance of Xis
4.94, and the standard deviation of Xis approximately 2.22.
Question 26
Question
Let’s say we have a dataset representing the heights (in centimeters) of 1000
university students. The mean height is 170 cm with a standard deviation of 10
cm. If we assume that the heights are normally distributed, what percentage of
students are shorter than 180 cm?
Solution
Step 1: Convert the problem into a standard normal distribution problem by
calculating the z-score for the height of 180 cm using the formula:
z=x−µ
σ
where: - x= 180 cm (height in question) - µ= 170 cm (mean height) -
σ= 10 cm (standard deviation)
Plugging in the values:
z=180 −170
10 =10
10 = 1
Step 2: Look up the z-score in the standard normal distribution table to
find the percentage of values below this point. A z-score of 1 corresponds to
approximately 84.13
Therefore, approximately 84.13
21
Question 27
Question
A researcher collects data on the heights (in inches) of 50 students in a university.
The following summary statistics were obtained: mean height = 65 inches,
standard deviation = 3.5 inches, maximum height = 72 inches, minimum height
= 58 inches. The heights are approximately normally distributed. Find the
z-score for a student who is 70 inches tall.
Solution
Step 1: Calculate the z-score using the formula
z=x−¯x
s
where: - x= 70 inches is the height of the student, - ¯x= 65 inches is the mean
height, - s= 3.5 inches is the standard deviation.
Step 2: Substitute the values into the formula to find the z-score:
z=70 −65
3.5=5
3.5≈1.43
Step 3: Therefore, the z-score for a student who is 70 inches tall is approxi-
mately 1.43.
Question 28
Question
Let X={x1, x2, x3, x4, x5}be a set of data points with x1= 5, x2= 7, x3= 4,
x4= 5, x5= 9. Calculate the sample mean, sample variance, and sample
standard deviation of the data set.
Solution
Step 1: Calculate the sample mean
Sample mean (¯x) = 1
n
n
X
i=1
xi
=1
5(5+7+4+5+9)
=30
5
= 6
22
Step 2: Calculate the sample variance
Sample variance (s2) = 1
n−1
n
X
i=1
(xi−¯x)2
=1
4[(5 −6)2+ (7 −6)2+ (4 −6)2+ (5 −6)2+ (9 −6)2]
=1
4[(−1)2+ (1)2+ (−2)2+ (−1)2+ (3)2]
=1
4[1+1+4+1+9]
=1
4·16
= 4
Step 3: Calculate the sample standard deviation
Sample standard deviation (s) = √s2
=√4
= 2
Therefore, the sample mean is 6, the sample variance is 4, and the sample
standard deviation is 2 for the given data set.
Question 29
Question
Suppose we have the following dataset representing the number of hours students
spend studying for a final exam:
{2,3,4,5,6,7,8,9,10}
Calculate the sample variance for this dataset.
Solution
To calculate the sample variance, we will follow these steps: Step 1: Calculate
the mean of the dataset. Step 2: Calculate the squared differences between each
data point and the mean. Step 3: Sum the squared differences. Step 4: Divide
the sum by the number of data points minus one to find the sample variance.
Step 1: Calculate the mean of the dataset.
Mean = 2+3+4+5+6+7+8+9+10
9=54
9= 6
23
Step 2: Calculate the squared differences between each data point and the
mean. (2 −6)2= 16,
(3 −6)2= 9,
(4 −6)2= 4,
(5 −6)2= 1,
(6 −6)2= 0,
(7 −6)2= 1,
(8 −6)2= 4,
(9 −6)2= 9,
(10 −6)2= 16.
Step 3: Sum the squared differences.
16+9+4+1+0+1+4+9+16=60
Step 4: Calculate the sample variance.
Sample Variance = 60
9−1=60
8= 7.5
Therefore, the sample variance for the given dataset is 7.5.
Question 30
Question
Let X={2,5,7,11,13}and Y={1,3,5,7,9}be two sets of data. Calculate
the coefficient of variation for both sets.
Solution
Step 1: Calculate the mean for sets Xand Y.
Step 2: Calculate the standard deviation for sets Xand Y.
Step 3: Calculate the coefficient of variation for sets Xand Y.
Step 1: To find the mean of set X, we sum up the values and divide by the
total number of values.
¯
X=PX
n=2+5+7+11+13
5=38
5= 7.6
To find the mean of set Y,
¯
Y=PY
n=1+3+5+7+9
5=25
5= 5
24
Step 2: To find the standard deviation of set X, we use the formula
σX=rP(Xi−¯
X)2
n
σX=r(2 −7.6)2+ (5 −7.6)2+ (7 −7.6)2+ (11 −7.6)2+ (13 −7.6)2
5
σX=r29.6+6.76 + 0.36 + 12.96 + 29.6
5
σX=r79.28
5=√15.856 ≈3.98
To find the standard deviation of set Y,
σY=rP(Yi−¯
Y)2
n
σY=r(1 −5)2+ (3 −5)2+ (5 −5)2+ (7 −5)2+ (9 −5)2
5
σY=r16+4+0+4+16
5
σY=r40
5=√8=2.83
Step 3: The coefficient of variation for set Xis calculated as:
CVX=σX
¯
X×100 = 3.98
7.6×100 ≈52.37%
Similarly, the coefficient of variation for set Yis calculated as:
CVY=σY
¯
Y×100 = 2.83
5×100 = 56.6%
Therefore, the coefficient of variation for set Xis approximately 52.37% and
for set Yis 56.6%.
25
Question 8
Question
A researcher collected the following data on the number of hours students stud-
ied for an exam: 3, 5, 7, 6, 4, 8, 10, 6, 5, 7. Compute the mean, median, mode,
range, variance, and standard deviation of the data.
Solution
Step 1: Calculate the mean.
Mean = 3+5+7+6+4+8+10+6+5+7
10
=61
10
= 6.1
Step 2: Calculate the median. First, we need to arrange the data in ascend-
ing order: 3, 4, 5, 5, 6, 6, 7, 7, 8, 10
Since we have an even number of data points, the median will be the average
of the two middle numbers: Median = 6+6
2= 6.
Step 3: Calculate the mode. The mode is the number that appears most
frequently in the dataset. In this case, both 5 and 7 appear twice, so the data
is bimodal.
Step 4: Calculate the range. Range = Max value - Min value = 10 - 3 = 7.
Step 5: Calculate the variance.
Variance = (3 −6.1)2+ (5 −6.1)2+ (7 −6.1)2+ (6 −6.1)2+ (4 −6.1)2+ (8 −6.1)2+ (10 −6.1)2+ (6 −6.1)2+ (5 −6.1)2+ (7 −6.1)2
10 −1
=9.61 + 1.21 + 0.81 + 0.01 + 4.41 + 4.41 + 13.69 + 0.01 + 1.21 + 0.81
9
=35.57
9
≈3.95
Step 6: Calculate the standard deviation. Standard deviation = √Variance ≈
√3.95 ≈1.98.
Question 9
Question
Suppose you have a dataset with the following values: 1, 2, 3, 4, 5, 6, 7, 8, 9, 100.
Calculate the mean, median, mode, range, variance, and standard deviation for
this dataset.
7
Solution
Step 1: Begin by arranging the dataset in ascending order: 1, 2, 3, 4, 5, 6, 7, 8,
9, 100.
Step 2: Calculate the mean: The mean is calculated by summing all values
and dividing by the number of values. Mean = 1+2+3+4+5+6+7+8+9+100
10 =145
10 =
14.5.
Step 3: Calculate the median: Since there are 10 values, the median is the
average of the 5th and 6th values. Median = 5+6
2=11
2= 5.5.
Step 4: Calculate the mode: In this dataset, there is no mode as each value
appears only once.
Step 5: Calculate the range: Range = Maximum value - Minimum value =
100 - 1 = 99.
Step 6: Calculate the variance: The variance is the average of the squared dif-
ferences from the mean. Variance = (1−14.5)2+(2−14.5)2+...+(100−14.5)2
10 =(198.25+161.29+...+2087.25)
10 =
3004.85
10 = 300.485.
Step 7: Calculate the standard deviation: The standard deviation is the
square root of the variance. Standard deviation = √300.485 ≈17.34.
Question 10
Question
The following data represents the test scores of students in two different classes:
Class A: 85, 90, 88, 92, 84
Class B: 78, 82, 88, 85, 90
Calculate the mean, median, mode, range, variance, and standard deviation
for each class. Compare the central tendency and dispersion measures between
the two classes.
Solution
Let’s calculate the mean, median, mode, range, variance, and standard deviation
for each class step by step.
Class A:
Mean: ¯x=85+90+88+92+84
5=439
5= 87.8
Median: Arrange the data in ascending order: 84, 85, 88, 90, 92. The
median is 88.
Mode: There is no mode since all values are unique.
Range: 92 −84 = 8
Variance: s2=1
nPn
i=1(xi−¯x)2
s2=(85−87.8)2+(90−87.8)2+(88−87.8)2+(92−87.8)2+(84−87.8)2
5=47.2
5= 9.44
8
Standard Deviation: s=√9.44 ≈3.08
Class B:
Mean: ¯x=78+82+88+85+90
5=423
5= 84.6
Median: Arrange the data in ascending order: 78, 82, 85, 88, 90. The
median is 85.
Mode: There is no mode since all values are unique.
Range: 90 −78 = 12
Variance: s2=1
nPn
i=1(xi−¯x)2
s2=(78−84.6)2+(82−84.6)2+(88−84.6)2+(85−84.6)2+(90−84.6)2
5=56.4
5= 11.28
Standard Deviation: s=√11.28 ≈3.36
Comparing the two classes: - Class A has a higher mean, median, and range
compared to Class B. - Class A has a lower variance and standard deviation
compared to Class B, indicating less dispersion in the data set.
Question 11
Question
Suppose a researcher collected data on the heights (in centimeters) of 100 adult
individuals. The mean height was found to be 170 cm with a standard devi-
ation of 10 cm. If the researcher wants to identify the interval within which
approximately 95
Solution
Step 1: Calculate the margin of error using the formula Z×σ
√n, where Zis
the Z-score corresponding to the desired level of confidence, σis the standard
deviation, and nis the sample size.
Given: - Z-score for 95- Standard deviation, σ= 10 cm. - Sample size,
n= 100.
Plugging in the values: Margin of error = 1.96 ×10
√100 = 1.96. Thus, the
margin of error is 1.96 cm.
Step 2: Find the lower and upper bounds of the interval by adding and
subtracting the margin of error from the mean.
Lower bound: 170 −1.96 = 168.04. Upper bound: 170 + 1.96 = 171.96.
Therefore, the interval within which approximately 95
9
Question 12
Question
A researcher collects data on the heights (in inches) of students in a mathe-
matics class. The following descriptive statistics were computed for the data:
mean height = 65 inches, standard deviation = 4 inches. Assuming the data
is approximately normally distributed, what proportion of students in the class
have heights greater than 70 inches?
Solution
Step 1: Find the z-score for a height of 70 inches using the formula:
z=x−µ
σ
where xis the given height, µis the mean height, and σis the standard deviation.
Plugging in the values, we get:
z=70 −65
4=5
4= 1.25
Step 2: Use a standard normal distribution table or calculator to find the
proportion of students with heights greater than 70 inches. From the z-score
table, we look up the area to the right of z= 1.25, which is approximately
0.1056.
Step 3: Therefore, approximately 10.56
Question 13
Question
For a given dataset, the mean is 25, variance is 16, and standard deviation is 4.
Calculate the coefficient of variation.
Solution
Step 1: The coefficient of variation (CV) is defined as the ratio of the standard
deviation to the mean, expressed as a percentage. Step 2: To calculate the CV,
we use the formula:
CV = Standard Deviation
Mean ×100
Step 3: Substituting the given values into the formula, we get:
CV = 4
25×100
10
Step 4: Simplifying the expression further, we have:
CV = 0.16 ×100
CV = 16%
Step 5: Therefore, the coefficient of variation for the given dataset is 16
Question 14
Question
Suppose a data set has a mean of 50, a median of 47, and a mode of 45. If the
data set has a total of 100 values, what is the range of the data set?
Solution
Step 1: Since the mode is 45, we know that there are more occurrences of 45
than any other value in the data set.
Step 2: Since the median is 47, we know that half of the data set values are
less than or equal to 47 and the other half are greater than or equal to 47.
Step 3: Since the mean is 50, we know that the sum of all values divided by
the total number of values is 50.
Step 4: Let’s first calculate the sum of all values in the data set. Since the
mean is 50 and there are 100 values in total, the sum of all values is 50 ×100 =
5000.
Step 5: Since the mode is 45 and there are more occurrences of 45 than
any other value, let’s assume there are ’m’ occurrences of 45. Thus, the sum of
values excluding the occurrences of 45 will be: 5000 −45 ×m.
Step 6: Since the median is 47, half of the values should be less than or equal
to 47. Since there are ’m’ occurrences of 45 and the rest of the values must be
greater than 45, we can consider the sum of these values as 47×(100−m−m) =
4700 −2×47m.
Step 7: Now we can set up the equation involving the sum of all values:
5000 −45m= 4700 −94m. Solving this equation, we get 49m= 300 and thus,
m= 300/49.
Step 8: Now we can find the range of the data set. The minimum value is
45 and the maximum value is 45 + 2 ×47 = 139, since there are 2 values to the
right of the median to reach the maximum value.
Step 9: Therefore, the range of the data set is 139 −45 = 94.
Question 15
Question
The following data represents the test scores of 20 students:
11
39, 46, 51, 62, 54, 38, 26, 47, 58, 61, 73, 52, 59, 63, 65, 52, 49, 50, 55, 42.
Calculate the interquartile range of the data set.
Solution
Step 1: First, we need to arrange the data in ascending order:
26,38,39,42,46,47,49,50,51,52,52,54,55,58,59,61,62,63,65,73
Step 2: Next, we find the median of the data set. Since we have an even
number of data points, the median is the average of the two middle values:
Median = 52 + 54
2= 53
Step 3: To find the lower quartile Q1, we find the median of the lower half
of the data set. Since there are an odd number of values in the lower half, Q1
is the middle value:
Q1 = 47
Step 4: To find the upper quartile Q3, we find the median of the upper
half of the data set. Q3 is the middle value of the upper half, which is an odd
number:
Q3 = 61
Step 5: Finally, we calculate the interquartile range (IQR) by subtracting
Q1 from Q3:
IQR = Q3−Q1 = 61 −47 = 14
Therefore, the interquartile range of the data set is 14.
Question 16
Question
A researcher collected the following data on the number of hours of sleep per
night for a group of 30 university students: 6, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9,
10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 13, 14.
Calculate the mean, median, mode, range, variance, and standard deviation
of the data set.
Solution
Step 1: Find the mean: The mean is calculated by summing up all the data
points and then dividing by the total number of data points. Mean = 6+7+7+...+14
30 =
238
30 = 7.93
Step 2: Find the median: To find the median, we first rearrange the data
set in ascending order: 6, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10,
12
11, 11, 11, 11, 11, 12, 12, 12, 13, 14. Since there are 30 data points, the median
will be the average of the 15th and 16th terms, which are both 10. Median = 10
Step 3: Find the mode: We can see that the mode of the data set is 10, as
it appears the most frequently.
Step 4: Find the range: The range is the difference between the highest and
lowest data points in the data set. Range = 14 −6 = 8
Step 5: Find the variance: The variance is the average of the squared differ-
ences between each data point and the mean. Variance = (6−7.93)2+(7−7.93)2+...+(14−7.93)2
30 ≈
5.90
Step 6: Find the standard deviation: The standard deviation is the square
root of the variance. Standard Deviation = √5.90 ≈2.43
Question 17
Question
Let’s say we have a dataset of 100 students’ test scores in a Statistics class. The
mean of the test scores is 75 with a standard deviation of 10. If a new student
joins the class and scores 85 on the test, calculate the new mean and standard
deviation of the dataset.
Solution
Step 1: Find the new mean after adding the new student: The formula to
calculate the new mean after adding a new value is:
New Mean = n1×Old Mean + x
n1
where: - n1is the total number of values in the dataset after adding the new
value, - Old Mean = 75 is the previous mean, - x= 85 is the score of the new
student. Plugging in the values:
New Mean = 100 ×75 + 85
101 =7500 + 85
101 =7585
101 = 75.10
Step 2: Find the new standard deviation after adding the new student: The
formula to calculate the new standard deviation after adding a new value is:
New SD = s(n1−1) ×(Old SD2+ Old Mean2) + x2
n1−New Mean2
where: - n1= 101 is the total number of values in the dataset after adding the
new value, - Old SD = 10 is the previous standard deviation, - Old Mean = 75 is
the previous mean, - x= 85 is the score of the new student, - New Mean = 75.10
is the new mean. Plugging in the values:
New SD = r(101 −1) ×(102+ 752) + 852
101 −75.102
13
=r100 ×(100 + 5625) + 7225
101 −75.102
=r10000 + 562500 + 7225
101 −5640.01 = r579725
101 −5640.01
=√5747.77 −5640.01 = √107.76 ≈10.38
Therefore, the new mean is approximately 75.10 and the new standard deviation
is approximately 10.38.
Question 18
Question
Let X={4,8,10,12,14}be a set of data points. Calculate the standard devia-
tion of the data set X.
Solution
To calculate the standard deviation of a data set, we can follow these steps:
Step 1: Find the mean of the data set. Step 2: Calculate the variance of the
data set. Step 3: Take the square root of the variance to find the standard
deviation.
Step 1: Find the mean of the data set. The mean ¯xof a data set is calculated
by summing all the data points and dividing by the number of data points.
¯x=4 + 8 + 10 + 12 + 14
5=48
5= 9.6
Step 2: Calculate the variance of the data set. The variance of a data set is
calculated by finding the average of the squared differences between each data
point and the mean.
Variance = 1
n
n
X
i=1
(xi−¯x)2
Variance = 1
5[(4 −9.6)2+ (8 −9.6)2+ (10 −9.6)2+ (12 −9.6)2+ (14 −9.6)2]
Variance = 1
5[(5.6)2+ (−1.6)2+ (0.4)2+ (2.4)2+ (4.4)2]
Variance = 1
5[31.36 + 2.56 + 0.16 + 5.76 + 19.36] = 59.2
5= 11.84
Step 3: Take the square root of the variance to find the standard deviation.
The standard deviation σis the square root of the variance.
σ=√11.84 = 3.44
Therefore, the standard deviation of the data set Xis 3.44.
14
Question 19
Question
A researcher collected the following data on the number of hours per week that
a sample of students studied for their final exams:
8,12,10,15,6,14,9,11,8,13,16,5,9,12,10
Calculate the mean, median, mode, range, variance, and standard deviation
for this data set.
Solution
Step 1: Calculate the Mean
Mean = 8+12+10+15+6+14+9+11+8+13+16+5+9+12+10
15
=142
15
= 9.47
Step 2: Calculate the Median First, arrange the data in ascending order:
5,6,8,8,9,9,10,10,11,12,12,13,14,15,16
Since there are 15 data points, the median is the middle value, which is the
8th value: 10.
Step 3: Calculate the Mode The mode is the value that appears most
frequently. In this case, the mode is 8 hours.
Step 4: Calculate the Range The range is the difference between the
maximum and minimum values.
Range = 16 −5 = 11
Step 5: Calculate the Variance To calculate the variance, first calculate
the squared differences between each data point and the mean:
(8 −9.47)2= 1.972,(12 −9.47)2= 2.532, . . .
Add up these squared differences and divide by the number of data points
to get the variance.
Step 6: Calculate the Standard Deviation The standard deviation is
the square root of the variance.
After completing these calculations, we would have all the requested descrip-
tive statistics for this data set.
15
Question 20
Question
A researcher collects data on the reading speeds (in words per minute) of two
groups of students, Group A and Group B. The mean reading speed for Group
A is 250 words per minute with a standard deviation of 20, while the mean
reading speed for Group B is 270 words per minute with a standard deviation
of 30. Assuming the reading speeds are normally distributed, which group has
a wider range of reading speeds? Justify your answer.
Solution
Step 1: Calculate the coefficient of variation for each group. The coefficient of
variation is defined as the standard deviation divided by the mean, expressed
as a percentage. It allows us to compare the variability between datasets with
different units or scales.
For Group A: Coefficient of variation = standard deviation
mean ×100%
Coefficient of variation for Group A = 20
250 ×100% = 8%
For Group B: Coefficient of variation for Group B = 30
270 ×100% = 11.11%
Step 2: Compare the coefficient of variation. The coefficient of variation for
Group B (11.11%) is higher than that of Group A (8%). This indicates that
Group B has a wider range of reading speeds compared to Group A. The higher
the coefficient of variation, the greater the variability in the data. Therefore,
Group B has a wider range of reading speeds than Group A.
Question 21
Question
Let’s say we have a dataset with 50 values. After calculating the mean, variance,
standard deviation, skewness, and kurtosis, we obtained the following results:
mean = 65, variance = 100, standard deviation = 10, skewness = -0.5, kurtosis
= 2.
Given this information, calculate the coefficient of variation for this dataset.
Solution
To find the coefficient of variation, we first need to compute the standard devi-
ation as a percentage of the mean.
Step 1: Calculate the coefficient of variation using the formula:
Coefficient of Variation = Standard Deviation
Mean ×100
16
Step 2: Substitute the values into the formula:
Coefficient of Variation = 10
65×100
Step 3: Simplify the expression:
Coefficient of Variation = 10
65 ×100
Coefficient of Variation = 2
13 ×100
Coefficient of Variation = 15.38%
Therefore, the coefficient of variation for this dataset is 15.38
Question 22
Question
Let Xbe a random variable representing the number of hours students study
per week. The data collected from a sample of 30 students is as follows:
2,4,6,6,8,8,8,10,10,10,10,12,12,12,12,12,14,14,14,16,16,16,18,18,18,18,20,20,20,20
Calculate the mean, median, mode, variance, standard deviation, range, and
interquartile range of the data.
Solution
Step 1: Calculate the mean:
Mean = Px
n
Mean = 2+4+6+. . . + 20
30
Mean = 302
30 = 10.07
Step 2: Calculate the median: Since there are 30 data points, the median is
the average of the 15th and 16th values when the data is arranged in ascending
order.
Median = 12 + 12
2= 12
Step 3: Calculate the mode: The mode is the value(s) that appear most
frequently in the data. In this case, since 12, 14, 16, and 18 appear most
17
frequently (each with a frequency of 5), the data is bimodal. Therefore, the
modes are 12, 14, 16, and 18.
Step 4: Calculate the variance:
Variance = P(x−Mean)2
n
Variance = (2 −10.07)2+ (4 −10.07)2+. . . + (20 −10.07)2
30
Variance ≈30.95
Step 5: Calculate the standard deviation:
Standard Deviation = √Variance ≈√30.95 ≈5.56
Step 6: Calculate the range: Range is the difference between the maximum
and minimum values.
Range = 20 −2 = 18
Step 7: Calculate the interquartile range: First, find the first quartile (Q1)
and third quartile (Q3) by finding the medians of the lower half and upper half
of the data, respectively.
Q1=8, Q3 = 16
Interquartile Range = Q3−Q1 = 16 −8=8
Question 23
Question
A statistics class conducted a survey to collect data on the number of hours per
week students spend studying. The results are as follows: 25, 15, 18, 20, 22, 30,
27, 19, 17, 16, 24. Calculate the mean, median, mode, variance, and standard
deviation of the data set.
Solution
Step 1: Calculate the Mean The mean is calculated by adding up all the
values and then dividing by the total number of values.
Mean = 25 + 15 + 18 + 20 + 22 + 30 + 27 + 19 + 17 + 16 + 24
11
Mean = 233
11 = 21.18
Step 2: Calculate the Median To find the median, we first need to arrange
the data in ascending order: 15, 16, 17, 18, 19, 20, 22, 24, 25, 27, 30. Since
there are 11 values, the median will be the middle value, which is the 6th value
in this case. Therefore, the median is 20.
18
Step 3: Calculate the Mode The mode is the value that appears most
frequently in the data set. In this case, there is no value that appears more
than once, so the data set has no mode.
Step 4: Calculate the Variance The variance is calculated by finding the
average of the squared differences between each data point and the mean.
Variance = (25 −21.18)2+ (15 −21.18)2+. . . + (24 −21.18)2
11
Variance = 16.6464 + 38.6464 + . . . + 7.3364
11
Variance = 191.84
11 = 17.44
Step 5: Calculate the Standard Deviation The standard deviation is
the square root of the variance.
Standard Deviation = √17.44 ≈4.18
Therefore, the mean is 21.18, the median is 20, the mode does not exist, the
variance is 17.44, and the standard deviation is approximately 4.18.
Question 24
Question
The monthly salaries, in dollars, of a sample of university professors are as
follows:
$
6200,
$
5800,
$
7000,
$
6500,
$
7200,
$
6800,
$
7500. Calculate the mean,
median, mode, variance, and standard deviation of this data set.
Solution
Step 1: First, calculate the mean of the data set. Step 2: Then, find the median
of the data set. Step 3: Next, identify the mode of the data set. Step 4:
Calculate the variance of the data set. Step 5: Finally, determine the standard
deviation of the data set.
Step 1: To find the mean, we sum up all the monthly salaries and divide
by the total number of salaries. Mean = 6200+5800+7000+6500+7200+6800+7500
7=
46000
7= 6571.43.
Step 2: To find the median, we arrange the salaries in ascending order:
$
5800,
$
6200,
$
6500,
$
6800,
$
7000,
$
7200,
$
7500 Since there are 7 salaries, the
median is the middle value, which is
$
6800.
Step 3: Since there is no repeating value in the data set, there is no mode.
Step 4: To find the variance, we first calculate the squared difference of each
salary from the mean: (6200 −6571.43)2, (5800 −6571.43)2, (7000 −6571.43)2,
(6500−6571.43)2, (7200−6571.43)2, (6800−6571.43)2, (7500−6571.43)2Then,
19
we sum these squared differences and divide by the total number of salaries mi-
nus 1: Variance = (6200−6571.43)2+(5800−6571.43)2+(7000−6571.43)2+(6500−6571.43)2+(7200−6571.43)2+(6800−6571.43)2+(7500−6571.43)2
7−1=
265714.29
6= 44285.71.
Step 5: Finally, the standard deviation is the square root of the variance:
Standard deviation = √44285.71 = 210.59.
Question 25
Question
Let Xbe a random variable with the following probability distribution:
x−2 1 3 5
P(X=x) 0.3 0.2k0.1
Determine the value of kand calculate the mean, variance, and standard
deviation of X.
Solution
Step 1: Determine the value of k. Since the sum of all probabilities must equal
1, we have:
0.3+0.2 + k+ 0.1=1
0.6 + k= 1
k= 0.4
Step 2: Calculate the mean of X. The mean of a discrete random variable
can be calculated as:
E(X) = Xx·P(X=x)
E(X)=(−2)(0.3) + (1)(0.2) + (3)(0.4) + (5)(0.1)
E(X) = −0.6+0.2+1.2+0.5
E(X)=1.3
Step 3: Calculate the variance of X. The variance of Xis given by:
V ar(X) = E(X2)−[E(X)]2
To find E(X2), we calculate:
E(X2) = Xx2·P(X=x)
E(X2)=(−2)2(0.3) + (1)2(0.2) + (3)2(0.4) + (5)2(0.1)
E(X2)=1.2+0.2+4.8+0.5
E(X2)=6.7
20
Now, we can find the variance:
V ar(X)=6.7−(1.3)2
V ar(X) = 6.7−1.69
V ar(X)=4.94
Step 4: Calculate the standard deviation of X. The standard deviation is
the square root of the variance:
SD(X) = pV ar(X)
SD(X) = √4.94
SD(X)≈2.22
Therefore, the value of kis 0.4, the mean of Xis 1.3, the variance of Xis
4.94, and the standard deviation of Xis approximately 2.22.
Question 26
Question
Let’s say we have a dataset representing the heights (in centimeters) of 1000
university students. The mean height is 170 cm with a standard deviation of 10
cm. If we assume that the heights are normally distributed, what percentage of
students are shorter than 180 cm?
Solution
Step 1: Convert the problem into a standard normal distribution problem by
calculating the z-score for the height of 180 cm using the formula:
z=x−µ
σ
where: - x= 180 cm (height in question) - µ= 170 cm (mean height) -
σ= 10 cm (standard deviation)
Plugging in the values:
z=180 −170
10 =10
10 = 1
Step 2: Look up the z-score in the standard normal distribution table to
find the percentage of values below this point. A z-score of 1 corresponds to
approximately 84.13
Therefore, approximately 84.13
21
Question 27
Question
A researcher collects data on the heights (in inches) of 50 students in a university.
The following summary statistics were obtained: mean height = 65 inches,
standard deviation = 3.5 inches, maximum height = 72 inches, minimum height
= 58 inches. The heights are approximately normally distributed. Find the
z-score for a student who is 70 inches tall.
Solution
Step 1: Calculate the z-score using the formula
z=x−¯x
s
where: - x= 70 inches is the height of the student, - ¯x= 65 inches is the mean
height, - s= 3.5 inches is the standard deviation.
Step 2: Substitute the values into the formula to find the z-score:
z=70 −65
3.5=5
3.5≈1.43
Step 3: Therefore, the z-score for a student who is 70 inches tall is approxi-
mately 1.43.
Question 28
Question
Let X={x1, x2, x3, x4, x5}be a set of data points with x1= 5, x2= 7, x3= 4,
x4= 5, x5= 9. Calculate the sample mean, sample variance, and sample
standard deviation of the data set.
Solution
Step 1: Calculate the sample mean
Sample mean (¯x) = 1
n
n
X
i=1
xi
=1
5(5+7+4+5+9)
=30
5
= 6
22
Step 2: Calculate the sample variance
Sample variance (s2) = 1
n−1
n
X
i=1
(xi−¯x)2
=1
4[(5 −6)2+ (7 −6)2+ (4 −6)2+ (5 −6)2+ (9 −6)2]
=1
4[(−1)2+ (1)2+ (−2)2+ (−1)2+ (3)2]
=1
4[1+1+4+1+9]
=1
4·16
= 4
Step 3: Calculate the sample standard deviation
Sample standard deviation (s) = √s2
=√4
= 2
Therefore, the sample mean is 6, the sample variance is 4, and the sample
standard deviation is 2 for the given data set.
Question 29
Question
Suppose we have the following dataset representing the number of hours students
spend studying for a final exam:
{2,3,4,5,6,7,8,9,10}
Calculate the sample variance for this dataset.
Solution
To calculate the sample variance, we will follow these steps: Step 1: Calculate
the mean of the dataset. Step 2: Calculate the squared differences between each
data point and the mean. Step 3: Sum the squared differences. Step 4: Divide
the sum by the number of data points minus one to find the sample variance.
Step 1: Calculate the mean of the dataset.
Mean = 2+3+4+5+6+7+8+9+10
9=54
9= 6
23
Step 2: Calculate the squared differences between each data point and the
mean. (2 −6)2= 16,
(3 −6)2= 9,
(4 −6)2= 4,
(5 −6)2= 1,
(6 −6)2= 0,
(7 −6)2= 1,
(8 −6)2= 4,
(9 −6)2= 9,
(10 −6)2= 16.
Step 3: Sum the squared differences.
16+9+4+1+0+1+4+9+16=60
Step 4: Calculate the sample variance.
Sample Variance = 60
9−1=60
8= 7.5
Therefore, the sample variance for the given dataset is 7.5.
Question 30
Question
Let X={2,5,7,11,13}and Y={1,3,5,7,9}be two sets of data. Calculate
the coefficient of variation for both sets.
Solution
Step 1: Calculate the mean for sets Xand Y.
Step 2: Calculate the standard deviation for sets Xand Y.
Step 3: Calculate the coefficient of variation for sets Xand Y.
Step 1: To find the mean of set X, we sum up the values and divide by the
total number of values.
¯
X=PX
n=2+5+7+11+13
5=38
5= 7.6
To find the mean of set Y,
¯
Y=PY
n=1+3+5+7+9
5=25
5= 5
24
Step 2: To find the standard deviation of set X, we use the formula
σX=rP(Xi−¯
X)2
n
σX=r(2 −7.6)2+ (5 −7.6)2+ (7 −7.6)2+ (11 −7.6)2+ (13 −7.6)2
5
σX=r29.6+6.76 + 0.36 + 12.96 + 29.6
5
σX=r79.28
5=√15.856 ≈3.98
To find the standard deviation of set Y,
σY=rP(Yi−¯
Y)2
n
σY=r(1 −5)2+ (3 −5)2+ (5 −5)2+ (7 −5)2+ (9 −5)2
5
σY=r16+4+0+4+16
5
σY=r40
5=√8=2.83
Step 3: The coefficient of variation for set Xis calculated as:
CVX=σX
¯
X×100 = 3.98
7.6×100 ≈52.37%
Similarly, the coefficient of variation for set Yis calculated as:
CVY=σY
¯
Y×100 = 2.83
5×100 = 56.6%
Therefore, the coefficient of variation for set Xis approximately 52.37% and
for set Yis 56.6%.
25
Question 8
Question
A researcher collected the following data on the number of hours students stud-
ied for an exam: 3, 5, 7, 6, 4, 8, 10, 6, 5, 7. Compute the mean, median, mode,
range, variance, and standard deviation of the data.
Solution
Step 1: Calculate the mean.
Mean = 3+5+7+6+4+8+10+6+5+7
10
=61
10
= 6.1
Step 2: Calculate the median. First, we need to arrange the data in ascend-
ing order: 3, 4, 5, 5, 6, 6, 7, 7, 8, 10
Since we have an even number of data points, the median will be the average
of the two middle numbers: Median = 6+6
2= 6.
Step 3: Calculate the mode. The mode is the number that appears most
frequently in the dataset. In this case, both 5 and 7 appear twice, so the data
is bimodal.
Step 4: Calculate the range. Range = Max value - Min value = 10 - 3 = 7.
Step 5: Calculate the variance.
Variance = (3 −6.1)2+ (5 −6.1)2+ (7 −6.1)2+ (6 −6.1)2+ (4 −6.1)2+ (8 −6.1)2+ (10 −6.1)2+ (6 −6.1)2+ (5 −6.1)2+ (7 −6.1)2
10 −1
=9.61 + 1.21 + 0.81 + 0.01 + 4.41 + 4.41 + 13.69 + 0.01 + 1.21 + 0.81
9
=35.57
9
≈3.95
Step 6: Calculate the standard deviation. Standard deviation = √Variance ≈
√3.95 ≈1.98.
Question 9
Question
Suppose you have a dataset with the following values: 1, 2, 3, 4, 5, 6, 7, 8, 9, 100.
Calculate the mean, median, mode, range, variance, and standard deviation for
this dataset.
7
Solution
Step 1: Begin by arranging the dataset in ascending order: 1, 2, 3, 4, 5, 6, 7, 8,
9, 100.
Step 2: Calculate the mean: The mean is calculated by summing all values
and dividing by the number of values. Mean = 1+2+3+4+5+6+7+8+9+100
10 =145
10 =
14.5.
Step 3: Calculate the median: Since there are 10 values, the median is the
average of the 5th and 6th values. Median = 5+6
2=11
2= 5.5.
Step 4: Calculate the mode: In this dataset, there is no mode as each value
appears only once.
Step 5: Calculate the range: Range = Maximum value - Minimum value =
100 - 1 = 99.
Step 6: Calculate the variance: The variance is the average of the squared dif-
ferences from the mean. Variance = (1−14.5)2+(2−14.5)2+...+(100−14.5)2
10 =(198.25+161.29+...+2087.25)
10 =
3004.85
10 = 300.485.
Step 7: Calculate the standard deviation: The standard deviation is the
square root of the variance. Standard deviation = √300.485 ≈17.34.
Question 10
Question
The following data represents the test scores of students in two different classes:
Class A: 85, 90, 88, 92, 84
Class B: 78, 82, 88, 85, 90
Calculate the mean, median, mode, range, variance, and standard deviation
for each class. Compare the central tendency and dispersion measures between
the two classes.
Solution
Let’s calculate the mean, median, mode, range, variance, and standard deviation
for each class step by step.
Class A:
Mean: ¯x=85+90+88+92+84
5=439
5= 87.8
Median: Arrange the data in ascending order: 84, 85, 88, 90, 92. The
median is 88.
Mode: There is no mode since all values are unique.
Range: 92 −84 = 8
Variance: s2=1
nPn
i=1(xi−¯x)2
s2=(85−87.8)2+(90−87.8)2+(88−87.8)2+(92−87.8)2+(84−87.8)2
5=47.2
5= 9.44
8
Standard Deviation: s=√9.44 ≈3.08
Class B:
Mean: ¯x=78+82+88+85+90
5=423
5= 84.6
Median: Arrange the data in ascending order: 78, 82, 85, 88, 90. The
median is 85.
Mode: There is no mode since all values are unique.
Range: 90 −78 = 12
Variance: s2=1
nPn
i=1(xi−¯x)2
s2=(78−84.6)2+(82−84.6)2+(88−84.6)2+(85−84.6)2+(90−84.6)2
5=56.4
5= 11.28
Standard Deviation: s=√11.28 ≈3.36
Comparing the two classes: - Class A has a higher mean, median, and range
compared to Class B. - Class A has a lower variance and standard deviation
compared to Class B, indicating less dispersion in the data set.
Question 11
Question
Suppose a researcher collected data on the heights (in centimeters) of 100 adult
individuals. The mean height was found to be 170 cm with a standard devi-
ation of 10 cm. If the researcher wants to identify the interval within which
approximately 95
Solution
Step 1: Calculate the margin of error using the formula Z×σ
√n, where Zis
the Z-score corresponding to the desired level of confidence, σis the standard
deviation, and nis the sample size.
Given: - Z-score for 95- Standard deviation, σ= 10 cm. - Sample size,
n= 100.
Plugging in the values: Margin of error = 1.96 ×10
√100 = 1.96. Thus, the
margin of error is 1.96 cm.
Step 2: Find the lower and upper bounds of the interval by adding and
subtracting the margin of error from the mean.
Lower bound: 170 −1.96 = 168.04. Upper bound: 170 + 1.96 = 171.96.
Therefore, the interval within which approximately 95
9
Question 12
Question
A researcher collects data on the heights (in inches) of students in a mathe-
matics class. The following descriptive statistics were computed for the data:
mean height = 65 inches, standard deviation = 4 inches. Assuming the data
is approximately normally distributed, what proportion of students in the class
have heights greater than 70 inches?
Solution
Step 1: Find the z-score for a height of 70 inches using the formula:
z=x−µ
σ
where xis the given height, µis the mean height, and σis the standard deviation.
Plugging in the values, we get:
z=70 −65
4=5
4= 1.25
Step 2: Use a standard normal distribution table or calculator to find the
proportion of students with heights greater than 70 inches. From the z-score
table, we look up the area to the right of z= 1.25, which is approximately
0.1056.
Step 3: Therefore, approximately 10.56
Question 13
Question
For a given dataset, the mean is 25, variance is 16, and standard deviation is 4.
Calculate the coefficient of variation.
Solution
Step 1: The coefficient of variation (CV) is defined as the ratio of the standard
deviation to the mean, expressed as a percentage. Step 2: To calculate the CV,
we use the formula:
CV = Standard Deviation
Mean ×100
Step 3: Substituting the given values into the formula, we get:
CV = 4
25×100
10
Step 4: Simplifying the expression further, we have:
CV = 0.16 ×100
CV = 16%
Step 5: Therefore, the coefficient of variation for the given dataset is 16
Question 14
Question
Suppose a data set has a mean of 50, a median of 47, and a mode of 45. If the
data set has a total of 100 values, what is the range of the data set?
Solution
Step 1: Since the mode is 45, we know that there are more occurrences of 45
than any other value in the data set.
Step 2: Since the median is 47, we know that half of the data set values are
less than or equal to 47 and the other half are greater than or equal to 47.
Step 3: Since the mean is 50, we know that the sum of all values divided by
the total number of values is 50.
Step 4: Let’s first calculate the sum of all values in the data set. Since the
mean is 50 and there are 100 values in total, the sum of all values is 50 ×100 =
5000.
Step 5: Since the mode is 45 and there are more occurrences of 45 than
any other value, let’s assume there are ’m’ occurrences of 45. Thus, the sum of
values excluding the occurrences of 45 will be: 5000 −45 ×m.
Step 6: Since the median is 47, half of the values should be less than or equal
to 47. Since there are ’m’ occurrences of 45 and the rest of the values must be
greater than 45, we can consider the sum of these values as 47×(100−m−m) =
4700 −2×47m.
Step 7: Now we can set up the equation involving the sum of all values:
5000 −45m= 4700 −94m. Solving this equation, we get 49m= 300 and thus,
m= 300/49.
Step 8: Now we can find the range of the data set. The minimum value is
45 and the maximum value is 45 + 2 ×47 = 139, since there are 2 values to the
right of the median to reach the maximum value.
Step 9: Therefore, the range of the data set is 139 −45 = 94.
Question 15
Question
The following data represents the test scores of 20 students:
11
39, 46, 51, 62, 54, 38, 26, 47, 58, 61, 73, 52, 59, 63, 65, 52, 49, 50, 55, 42.
Calculate the interquartile range of the data set.
Solution
Step 1: First, we need to arrange the data in ascending order:
26,38,39,42,46,47,49,50,51,52,52,54,55,58,59,61,62,63,65,73
Step 2: Next, we find the median of the data set. Since we have an even
number of data points, the median is the average of the two middle values:
Median = 52 + 54
2= 53
Step 3: To find the lower quartile Q1, we find the median of the lower half
of the data set. Since there are an odd number of values in the lower half, Q1
is the middle value:
Q1 = 47
Step 4: To find the upper quartile Q3, we find the median of the upper
half of the data set. Q3 is the middle value of the upper half, which is an odd
number:
Q3 = 61
Step 5: Finally, we calculate the interquartile range (IQR) by subtracting
Q1 from Q3:
IQR = Q3−Q1 = 61 −47 = 14
Therefore, the interquartile range of the data set is 14.
Question 16
Question
A researcher collected the following data on the number of hours of sleep per
night for a group of 30 university students: 6, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9,
10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 13, 14.
Calculate the mean, median, mode, range, variance, and standard deviation
of the data set.
Solution
Step 1: Find the mean: The mean is calculated by summing up all the data
points and then dividing by the total number of data points. Mean = 6+7+7+...+14
30 =
238
30 = 7.93
Step 2: Find the median: To find the median, we first rearrange the data
set in ascending order: 6, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10,
12
11, 11, 11, 11, 11, 12, 12, 12, 13, 14. Since there are 30 data points, the median
will be the average of the 15th and 16th terms, which are both 10. Median = 10
Step 3: Find the mode: We can see that the mode of the data set is 10, as
it appears the most frequently.
Step 4: Find the range: The range is the difference between the highest and
lowest data points in the data set. Range = 14 −6 = 8
Step 5: Find the variance: The variance is the average of the squared differ-
ences between each data point and the mean. Variance = (6−7.93)2+(7−7.93)2+...+(14−7.93)2
30 ≈
5.90
Step 6: Find the standard deviation: The standard deviation is the square
root of the variance. Standard Deviation = √5.90 ≈2.43
Question 17
Question
Let’s say we have a dataset of 100 students’ test scores in a Statistics class. The
mean of the test scores is 75 with a standard deviation of 10. If a new student
joins the class and scores 85 on the test, calculate the new mean and standard
deviation of the dataset.
Solution
Step 1: Find the new mean after adding the new student: The formula to
calculate the new mean after adding a new value is:
New Mean = n1×Old Mean + x
n1
where: - n1is the total number of values in the dataset after adding the new
value, - Old Mean = 75 is the previous mean, - x= 85 is the score of the new
student. Plugging in the values:
New Mean = 100 ×75 + 85
101 =7500 + 85
101 =7585
101 = 75.10
Step 2: Find the new standard deviation after adding the new student: The
formula to calculate the new standard deviation after adding a new value is:
New SD = s(n1−1) ×(Old SD2+ Old Mean2) + x2
n1−New Mean2
where: - n1= 101 is the total number of values in the dataset after adding the
new value, - Old SD = 10 is the previous standard deviation, - Old Mean = 75 is
the previous mean, - x= 85 is the score of the new student, - New Mean = 75.10
is the new mean. Plugging in the values:
New SD = r(101 −1) ×(102+ 752) + 852
101 −75.102
13
=r100 ×(100 + 5625) + 7225
101 −75.102
=r10000 + 562500 + 7225
101 −5640.01 = r579725
101 −5640.01
=√5747.77 −5640.01 = √107.76 ≈10.38
Therefore, the new mean is approximately 75.10 and the new standard deviation
is approximately 10.38.
Question 18
Question
Let X={4,8,10,12,14}be a set of data points. Calculate the standard devia-
tion of the data set X.
Solution
To calculate the standard deviation of a data set, we can follow these steps:
Step 1: Find the mean of the data set. Step 2: Calculate the variance of the
data set. Step 3: Take the square root of the variance to find the standard
deviation.
Step 1: Find the mean of the data set. The mean ¯xof a data set is calculated
by summing all the data points and dividing by the number of data points.
¯x=4 + 8 + 10 + 12 + 14
5=48
5= 9.6
Step 2: Calculate the variance of the data set. The variance of a data set is
calculated by finding the average of the squared differences between each data
point and the mean.
Variance = 1
n
n
X
i=1
(xi−¯x)2
Variance = 1
5[(4 −9.6)2+ (8 −9.6)2+ (10 −9.6)2+ (12 −9.6)2+ (14 −9.6)2]
Variance = 1
5[(5.6)2+ (−1.6)2+ (0.4)2+ (2.4)2+ (4.4)2]
Variance = 1
5[31.36 + 2.56 + 0.16 + 5.76 + 19.36] = 59.2
5= 11.84
Step 3: Take the square root of the variance to find the standard deviation.
The standard deviation σis the square root of the variance.
σ=√11.84 = 3.44
Therefore, the standard deviation of the data set Xis 3.44.
14
Question 19
Question
A researcher collected the following data on the number of hours per week that
a sample of students studied for their final exams:
8,12,10,15,6,14,9,11,8,13,16,5,9,12,10
Calculate the mean, median, mode, range, variance, and standard deviation
for this data set.
Solution
Step 1: Calculate the Mean
Mean = 8+12+10+15+6+14+9+11+8+13+16+5+9+12+10
15
=142
15
= 9.47
Step 2: Calculate the Median First, arrange the data in ascending order:
5,6,8,8,9,9,10,10,11,12,12,13,14,15,16
Since there are 15 data points, the median is the middle value, which is the
8th value: 10.
Step 3: Calculate the Mode The mode is the value that appears most
frequently. In this case, the mode is 8 hours.
Step 4: Calculate the Range The range is the difference between the
maximum and minimum values.
Range = 16 −5 = 11
Step 5: Calculate the Variance To calculate the variance, first calculate
the squared differences between each data point and the mean:
(8 −9.47)2= 1.972,(12 −9.47)2= 2.532, . . .
Add up these squared differences and divide by the number of data points
to get the variance.
Step 6: Calculate the Standard Deviation The standard deviation is
the square root of the variance.
After completing these calculations, we would have all the requested descrip-
tive statistics for this data set.
15
Question 20
Question
A researcher collects data on the reading speeds (in words per minute) of two
groups of students, Group A and Group B. The mean reading speed for Group
A is 250 words per minute with a standard deviation of 20, while the mean
reading speed for Group B is 270 words per minute with a standard deviation
of 30. Assuming the reading speeds are normally distributed, which group has
a wider range of reading speeds? Justify your answer.
Solution
Step 1: Calculate the coefficient of variation for each group. The coefficient of
variation is defined as the standard deviation divided by the mean, expressed
as a percentage. It allows us to compare the variability between datasets with
different units or scales.
For Group A: Coefficient of variation = standard deviation
mean ×100%
Coefficient of variation for Group A = 20
250 ×100% = 8%
For Group B: Coefficient of variation for Group B = 30
270 ×100% = 11.11%
Step 2: Compare the coefficient of variation. The coefficient of variation for
Group B (11.11%) is higher than that of Group A (8%). This indicates that
Group B has a wider range of reading speeds compared to Group A. The higher
the coefficient of variation, the greater the variability in the data. Therefore,
Group B has a wider range of reading speeds than Group A.
Question 21
Question
Let’s say we have a dataset with 50 values. After calculating the mean, variance,
standard deviation, skewness, and kurtosis, we obtained the following results:
mean = 65, variance = 100, standard deviation = 10, skewness = -0.5, kurtosis
= 2.
Given this information, calculate the coefficient of variation for this dataset.
Solution
To find the coefficient of variation, we first need to compute the standard devi-
ation as a percentage of the mean.
Step 1: Calculate the coefficient of variation using the formula:
Coefficient of Variation = Standard Deviation
Mean ×100
16
Step 2: Substitute the values into the formula:
Coefficient of Variation = 10
65×100
Step 3: Simplify the expression:
Coefficient of Variation = 10
65 ×100
Coefficient of Variation = 2
13 ×100
Coefficient of Variation = 15.38%
Therefore, the coefficient of variation for this dataset is 15.38
Question 22
Question
Let Xbe a random variable representing the number of hours students study
per week. The data collected from a sample of 30 students is as follows:
2,4,6,6,8,8,8,10,10,10,10,12,12,12,12,12,14,14,14,16,16,16,18,18,18,18,20,20,20,20
Calculate the mean, median, mode, variance, standard deviation, range, and
interquartile range of the data.
Solution
Step 1: Calculate the mean:
Mean = Px
n
Mean = 2+4+6+. . . + 20
30
Mean = 302
30 = 10.07
Step 2: Calculate the median: Since there are 30 data points, the median is
the average of the 15th and 16th values when the data is arranged in ascending
order.
Median = 12 + 12
2= 12
Step 3: Calculate the mode: The mode is the value(s) that appear most
frequently in the data. In this case, since 12, 14, 16, and 18 appear most
17
frequently (each with a frequency of 5), the data is bimodal. Therefore, the
modes are 12, 14, 16, and 18.
Step 4: Calculate the variance:
Variance = P(x−Mean)2
n
Variance = (2 −10.07)2+ (4 −10.07)2+. . . + (20 −10.07)2
30
Variance ≈30.95
Step 5: Calculate the standard deviation:
Standard Deviation = √Variance ≈√30.95 ≈5.56
Step 6: Calculate the range: Range is the difference between the maximum
and minimum values.
Range = 20 −2 = 18
Step 7: Calculate the interquartile range: First, find the first quartile (Q1)
and third quartile (Q3) by finding the medians of the lower half and upper half
of the data, respectively.
Q1=8, Q3 = 16
Interquartile Range = Q3−Q1 = 16 −8=8
Question 23
Question
A statistics class conducted a survey to collect data on the number of hours per
week students spend studying. The results are as follows: 25, 15, 18, 20, 22, 30,
27, 19, 17, 16, 24. Calculate the mean, median, mode, variance, and standard
deviation of the data set.
Solution
Step 1: Calculate the Mean The mean is calculated by adding up all the
values and then dividing by the total number of values.
Mean = 25 + 15 + 18 + 20 + 22 + 30 + 27 + 19 + 17 + 16 + 24
11
Mean = 233
11 = 21.18
Step 2: Calculate the Median To find the median, we first need to arrange
the data in ascending order: 15, 16, 17, 18, 19, 20, 22, 24, 25, 27, 30. Since
there are 11 values, the median will be the middle value, which is the 6th value
in this case. Therefore, the median is 20.
18
Step 3: Calculate the Mode The mode is the value that appears most
frequently in the data set. In this case, there is no value that appears more
than once, so the data set has no mode.
Step 4: Calculate the Variance The variance is calculated by finding the
average of the squared differences between each data point and the mean.
Variance = (25 −21.18)2+ (15 −21.18)2+. . . + (24 −21.18)2
11
Variance = 16.6464 + 38.6464 + . . . + 7.3364
11
Variance = 191.84
11 = 17.44
Step 5: Calculate the Standard Deviation The standard deviation is
the square root of the variance.
Standard Deviation = √17.44 ≈4.18
Therefore, the mean is 21.18, the median is 20, the mode does not exist, the
variance is 17.44, and the standard deviation is approximately 4.18.
Question 24
Question
The monthly salaries, in dollars, of a sample of university professors are as
follows:
$
6200,
$
5800,
$
7000,
$
6500,
$
7200,
$
6800,
$
7500. Calculate the mean,
median, mode, variance, and standard deviation of this data set.
Solution
Step 1: First, calculate the mean of the data set. Step 2: Then, find the median
of the data set. Step 3: Next, identify the mode of the data set. Step 4:
Calculate the variance of the data set. Step 5: Finally, determine the standard
deviation of the data set.
Step 1: To find the mean, we sum up all the monthly salaries and divide
by the total number of salaries. Mean = 6200+5800+7000+6500+7200+6800+7500
7=
46000
7= 6571.43.
Step 2: To find the median, we arrange the salaries in ascending order:
$
5800,
$
6200,
$
6500,
$
6800,
$
7000,
$
7200,
$
7500 Since there are 7 salaries, the
median is the middle value, which is
$
6800.
Step 3: Since there is no repeating value in the data set, there is no mode.
Step 4: To find the variance, we first calculate the squared difference of each
salary from the mean: (6200 −6571.43)2, (5800 −6571.43)2, (7000 −6571.43)2,
(6500−6571.43)2, (7200−6571.43)2, (6800−6571.43)2, (7500−6571.43)2Then,
19
we sum these squared differences and divide by the total number of salaries mi-
nus 1: Variance = (6200−6571.43)2+(5800−6571.43)2+(7000−6571.43)2+(6500−6571.43)2+(7200−6571.43)2+(6800−6571.43)2+(7500−6571.43)2
7−1=
265714.29
6= 44285.71.
Step 5: Finally, the standard deviation is the square root of the variance:
Standard deviation = √44285.71 = 210.59.
Question 25
Question
Let Xbe a random variable with the following probability distribution:
x−2 1 3 5
P(X=x) 0.3 0.2k0.1
Determine the value of kand calculate the mean, variance, and standard
deviation of X.
Solution
Step 1: Determine the value of k. Since the sum of all probabilities must equal
1, we have:
0.3+0.2 + k+ 0.1=1
0.6 + k= 1
k= 0.4
Step 2: Calculate the mean of X. The mean of a discrete random variable
can be calculated as:
E(X) = Xx·P(X=x)
E(X)=(−2)(0.3) + (1)(0.2) + (3)(0.4) + (5)(0.1)
E(X) = −0.6+0.2+1.2+0.5
E(X)=1.3
Step 3: Calculate the variance of X. The variance of Xis given by:
V ar(X) = E(X2)−[E(X)]2
To find E(X2), we calculate:
E(X2) = Xx2·P(X=x)
E(X2)=(−2)2(0.3) + (1)2(0.2) + (3)2(0.4) + (5)2(0.1)
E(X2)=1.2+0.2+4.8+0.5
E(X2)=6.7
20
Now, we can find the variance:
V ar(X)=6.7−(1.3)2
V ar(X) = 6.7−1.69
V ar(X)=4.94
Step 4: Calculate the standard deviation of X. The standard deviation is
the square root of the variance:
SD(X) = pV ar(X)
SD(X) = √4.94
SD(X)≈2.22
Therefore, the value of kis 0.4, the mean of Xis 1.3, the variance of Xis
4.94, and the standard deviation of Xis approximately 2.22.
Question 26
Question
Let’s say we have a dataset representing the heights (in centimeters) of 1000
university students. The mean height is 170 cm with a standard deviation of 10
cm. If we assume that the heights are normally distributed, what percentage of
students are shorter than 180 cm?
Solution
Step 1: Convert the problem into a standard normal distribution problem by
calculating the z-score for the height of 180 cm using the formula:
z=x−µ
σ
where: - x= 180 cm (height in question) - µ= 170 cm (mean height) -
σ= 10 cm (standard deviation)
Plugging in the values:
z=180 −170
10 =10
10 = 1
Step 2: Look up the z-score in the standard normal distribution table to
find the percentage of values below this point. A z-score of 1 corresponds to
approximately 84.13
Therefore, approximately 84.13
21
Question 27
Question
A researcher collects data on the heights (in inches) of 50 students in a university.
The following summary statistics were obtained: mean height = 65 inches,
standard deviation = 3.5 inches, maximum height = 72 inches, minimum height
= 58 inches. The heights are approximately normally distributed. Find the
z-score for a student who is 70 inches tall.
Solution
Step 1: Calculate the z-score using the formula
z=x−¯x
s
where: - x= 70 inches is the height of the student, - ¯x= 65 inches is the mean
height, - s= 3.5 inches is the standard deviation.
Step 2: Substitute the values into the formula to find the z-score:
z=70 −65
3.5=5
3.5≈1.43
Step 3: Therefore, the z-score for a student who is 70 inches tall is approxi-
mately 1.43.
Question 28
Question
Let X={x1, x2, x3, x4, x5}be a set of data points with x1= 5, x2= 7, x3= 4,
x4= 5, x5= 9. Calculate the sample mean, sample variance, and sample
standard deviation of the data set.
Solution
Step 1: Calculate the sample mean
Sample mean (¯x) = 1
n
n
X
i=1
xi
=1
5(5+7+4+5+9)
=30
5
= 6
22
Step 2: Calculate the sample variance
Sample variance (s2) = 1
n−1
n
X
i=1
(xi−¯x)2
=1
4[(5 −6)2+ (7 −6)2+ (4 −6)2+ (5 −6)2+ (9 −6)2]
=1
4[(−1)2+ (1)2+ (−2)2+ (−1)2+ (3)2]
=1
4[1+1+4+1+9]
=1
4·16
= 4
Step 3: Calculate the sample standard deviation
Sample standard deviation (s) = √s2
=√4
= 2
Therefore, the sample mean is 6, the sample variance is 4, and the sample
standard deviation is 2 for the given data set.
Question 29
Question
Suppose we have the following dataset representing the number of hours students
spend studying for a final exam:
{2,3,4,5,6,7,8,9,10}
Calculate the sample variance for this dataset.
Solution
To calculate the sample variance, we will follow these steps: Step 1: Calculate
the mean of the dataset. Step 2: Calculate the squared differences between each
data point and the mean. Step 3: Sum the squared differences. Step 4: Divide
the sum by the number of data points minus one to find the sample variance.
Step 1: Calculate the mean of the dataset.
Mean = 2+3+4+5+6+7+8+9+10
9=54
9= 6
23
Step 2: Calculate the squared differences between each data point and the
mean. (2 −6)2= 16,
(3 −6)2= 9,
(4 −6)2= 4,
(5 −6)2= 1,
(6 −6)2= 0,
(7 −6)2= 1,
(8 −6)2= 4,
(9 −6)2= 9,
(10 −6)2= 16.
Step 3: Sum the squared differences.
16+9+4+1+0+1+4+9+16=60
Step 4: Calculate the sample variance.
Sample Variance = 60
9−1=60
8= 7.5
Therefore, the sample variance for the given dataset is 7.5.
Question 30
Question
Let X={2,5,7,11,13}and Y={1,3,5,7,9}be two sets of data. Calculate
the coefficient of variation for both sets.
Solution
Step 1: Calculate the mean for sets Xand Y.
Step 2: Calculate the standard deviation for sets Xand Y.
Step 3: Calculate the coefficient of variation for sets Xand Y.
Step 1: To find the mean of set X, we sum up the values and divide by the
total number of values.
¯
X=PX
n=2+5+7+11+13
5=38
5= 7.6
To find the mean of set Y,
¯
Y=PY
n=1+3+5+7+9
5=25
5= 5
24
Step 2: To find the standard deviation of set X, we use the formula
σX=rP(Xi−¯
X)2
n
σX=r(2 −7.6)2+ (5 −7.6)2+ (7 −7.6)2+ (11 −7.6)2+ (13 −7.6)2
5
σX=r29.6+6.76 + 0.36 + 12.96 + 29.6
5
σX=r79.28
5=√15.856 ≈3.98
To find the standard deviation of set Y,
σY=rP(Yi−¯
Y)2
n
σY=r(1 −5)2+ (3 −5)2+ (5 −5)2+ (7 −5)2+ (9 −5)2
5
σY=r16+4+0+4+16
5
σY=r40
5=√8=2.83
Step 3: The coefficient of variation for set Xis calculated as:
CVX=σX
¯
X×100 = 3.98
7.6×100 ≈52.37%
Similarly, the coefficient of variation for set Yis calculated as:
CVY=σY
¯
Y×100 = 2.83
5×100 = 56.6%
Therefore, the coefficient of variation for set Xis approximately 52.37% and
for set Yis 56.6%.
25
