ECON 350 - CLASSICAL ECONOMICS Descriptive statistics Set 2

ECON 350 - CLASSICAL

ECONOMICS - Descriptive statistics

Question Bank - Set 2

Liberty University

Question 1

Question

A data set consisting of the following values is given: 10, 15, 20, 25, 30, 35,

40, 45, 50. Calculate the mean, median, mode, range, variance, and standard

deviation of the data set.

Solution

Step 1: To find the mean, add up all the values in the data set and divide by

the total number of values.

Mean = 10 + 15 + 20 + 25 + 30 + 35 + 40 + 45 + 50

Mean = 270

9= 30

Step 2: To find the median, first, arrange the data set in ascending order:

10, 15, 20, 25, 30, 35, 40, 45, 50. Since there are 9 values, the median is the

middle value, which is 30.

Step 3: To find the mode, identify the value(s) that appear most frequently

in the data set. In this case, all values appear only once, so there is no mode.

Step 4: To find the range, subtract the minimum value from the maximum

value.

Range = 50 −10 = 40

Step 5: To find the variance, first, calculate the squared difference between

each value and the mean, then find the average of these squared differences.

Variance = (10 −30)2+ (15 −30)2+... + (50 −30)2

Variance = 400 + 225 + ... + 400

Variance = 900

9= 100

Step 6: To find the standard deviation, take the square root of the variance.

Standard Deviation = √100 = 10

Therefore, the mean is 30, the median is 30, there is no mode, the range is

40, the variance is 100, and the standard deviation is 10 for the given data set.

Question 2

Question

Suppose a statistics class has 25 students. Their scores on a quiz are as follows:

60, 70, 75, 80, 85, 90, 95, 100. Assuming these scores are the entire population,

calculate the sample mean, sample variance, and sample standard deviation.

Solution

Step 1: Calculate the Sample Mean The sample mean (¯x) is calculated using

the formula:

¯x=Pn

i=1 xi

where xiare the individual data points and nis the number of data points.

Given the data points: 60, 70, 75, 80, 85, 90, 95, 100 and n= 8 (number of

data points), we can find the sample mean.

¯x=60 + 70 + 75 + 80 + 85 + 90 + 95 + 100

8= 80

Therefore, the sample mean is 80.

Step 2: Calculate the Sample Variance The sample variance (s2) is

calculated using the formula:

s2=Pn

i=1(xi−¯x)2

n−1

where xiare the individual data points, ¯xis the sample mean, and nis the

number of data points.

Given the data points and the sample mean from before, we can find the

sample variance.

s2=(60 −80)2+ (70 −80)2+ (75 −80)2+ (80 −80)2+ (85 −80)2+ (90 −80)2+ (95 −80)2+ (100 −80)2

8−1

s2=400 + 100 + 25 + 0 + 25 + 100 + 225 + 400

7=1275

7≈182.14

Therefore, the sample variance is approximately 182.14.

Step 3: Calculate the Sample Standard Deviation The sample stan-

dard deviation (s) is the square root of the sample variance:

s=√s2=√182.14 ≈13.49

Therefore, the sample standard deviation is approximately 13.49.

Question 3

Question

A researcher collected data on the heights of 50 adult women. The mean height

was found to be 64 inches with a standard deviation of 3 inches.

(a) Calculate the coefficient of variation for the heights of these women.

(b) If the researcher mistakenly recorded the height of one woman as 70

inches instead of 60 inches, will this affect the coefficient of variation signifi-

cantly?

Solution

(a) The coefficient of variation is a measure of relative variability and is cal-

culated as the ratio of the standard deviation to the mean, expressed as a

percentage. It is given by the formula:

CV =Standard Deviation

Mean ×100%

Substitute the given values: Mean = 64 inches, Standard Deviation = 3

inches

CV =3

64×100%

CV = 0.0469 ×100% = 4.69%

Therefore, the coefficient of variation for the heights of these women is 4.69

(b) Let’s consider the effect of mistakenly recording the height of one woman

as 70 inches instead of 60 inches on the coefficient of variation.

1. Original mean height:

Mean = 64 inches

2. Original sum of heights:

Sum of heights = 50 ×64 = 3200 inches

3. Adjusted sum of heights after the mistaken recording:

Adjusted sum of heights = 3200 + 70 −60 = 3210 inches

4. Adjusted mean height:

Adjusted mean height = 3210

50 = 64.2 inches

5. Adjusted standard deviation: Given that standard deviation, s= 3

inches, we can calculate the new standard deviation after the adjustment.

6. Calculate the new coefficient of variation using the adjusted mean and

standard deviation.

CVadjusted =sadjusted

Meanadjusted ×100%

Since the mistake in recording only affects one data point out of 50, the

impact on the coefficient of variation is likely to be minimal.

Question 4

Question

A researcher collected data on the ages of participants in a study and found the

following summary statistics: mean age = 32.5, variance = 49.6, and standard

deviation = 7.04. Determine the median age of the participants.

Solution

Step 1: Recall that the variance is defined as the average of the squared differ-

ences between each data point and the mean. Mathematically, we have

Var = 1

i=1

(xi−¯x)2

where nis the number of data points, xiis the ith data point, and ¯xis the

mean.

Step 2: Given that the variance is 49.6, we can write

49.6 = 1

i=1

(xi−32.5)2

Step 3: We are also given that the standard deviation is 7.04. Recall that

the standard deviation is the square root of the variance, i.e.,

SD = √Var

Step 4: Substitute the given standard deviation into the equation for variance

to find the number of data points:

7.04 = √49.6

Step 5: Solving for n, we get

n=49.6

7.042≈49.6

49.5616 ≈1

Step 6: Since n≈1, we can deduce that the researcher only collected data on

one participant. Thus, the median age (which is the age of the sole participant)

is 32.5 .

Question 5

Question

In a study of the ages of students enrolled in a university, the following descrip-

tive statistics were calculated:

Mean: 22.5 years

Standard Deviation: 3.2 years

Minimum Age: 18 years

Maximum Age: 30 years

Given this information, determine the coefficient of variation for the ages of

the students.

Solution

To find the coefficient of variation, we use the formula:

Coefficient of Variation (CV) = Standard Deviation

Mean ×100%

Step 1: Calculate the Coefficient of Variation

Coefficient of Variation (CV) = 3.2

22.5×100%

= 0.1422 ×100%

= 14.22%

The coefficient of variation for the ages of the students is 14.22%.

Question 6

Question

Let’s say you have a dataset with the following values: 6, 8, 10, 6, 12, 6, 14, 8.

Calculate the mean, median, mode, variance, standard deviation, and range of

the dataset.

Solution

To find the descriptive statistics of the dataset, we will calculate the mean,

median, mode, variance, standard deviation, and range step by step.

Step 1: Calculate the mean The mean is calculated by summing up all

the values in the dataset and then dividing by the total number of values.

Mean = 6+8+10+6+12+6+14+8

Mean = 70

8= 8.75

Step 2: Calculate the median To find the median, we first need to

arrange the values in ascending order: 6, 6, 6, 8, 8, 10, 12, 14. Since there are

an even number of values, the median is the average of the two middle values.

Median = 8+8

2= 8

Step 3: Calculate the mode The mode is the value that appears most

frequently in the dataset. In this case, the mode is 6, as it appears three times.

Step 4: Calculate the variance The variance measures how spread out

the values in the dataset are. To calculate the variance, we first need to find the

squared differences between each value and the mean, then sum them up and

divide by the total number of values.

Variance = (6 −8.75)2+ (8 −8.75)2+ (10 −8.75)2+ (6 −8.75)2+ (12 −8.75)2+ (6 −8.75)2+ (14 −8.75)2+ (8 −8.75)2

Variance = 57.5

8= 7.1875

Step 5: Calculate the standard deviation The standard deviation is

the square root of the variance.

Standard deviation = √7.1875 ≈2.68

Step 6: Calculate the range The range is the difference between the

maximum and minimum values in the dataset. Range = 14 −6 = 8

Therefore, the mean is 8.75, the median is 8, the mode is 6, the variance is

7.1875, the standard deviation is approximately 2.68, and the range is 8.

Question 7

Question

Suppose a researcher collected the following data on the heights (in centimeters)

of 10 adult males:

175,180,165,190,185,170,175,160,200,195

Calculate the mean, median, mode, range, variance, and standard deviation

of the data set.

Solution

Let’s calculate the mean, median, mode, range, variance, and standard deviation

step-by-step.

Step 1: Calculate the Mean The mean (¯x) is calculated by adding up

all the values and dividing by the total number of values (10 in this case).

Mean = 175 + 180 + 165 + 190 + 185 + 170 + 175 + 160 + 200 + 195

Mean = 1775

10 = 177.5

Therefore, the mean height is 177.5 cm.

Step 2: Calculate the Median To find the median, we first need to

arrange the data in ascending order.

160,165,170,175,175,180,185,190,195,200

Since there are 10 values, the median is the average of the 5th and 6th values.

Median = 175 + 180

2= 177.5

Therefore, the median height is 177.5 cm.

Step 3: Calculate the Mode The mode is the value that appears most

frequently in the data set. In this case, there is no unique mode as each value

appears only once. So, the data set has no mode.

Step 4: Calculate the Range The range is the difference between the

largest and smallest values in the data set.

Range = 200 −160 = 40

Therefore, the range of heights is 40 cm.

Step 5: Calculate the Variance The variance is a measure of how spread

out the values in a data set are. It is calculated using the formula:

Variance = P(xi−¯x)2

where xiare the individual values, ¯xis the mean, and nis the total number of

values (10 in this case).

Variance = (175 −177.5)2+ (180 −177.5)2+. . . + (195 −177.5)2

Calculating this value gives us a variance of 140.25.

Step 6: Calculate the Standard Deviation The standard deviation is

the square root of the variance.

Standard Deviation = √140.25 = 11.84

Therefore, the standard deviation of the heights is 11.84 cm.

Question 8

Question

Let’s consider a dataset representing the heights (in inches) of 30 college stu-

dents. The descriptive statistics for this dataset are as follows: mean height =

65.7 inches, standard deviation = 3.5 inches. Assuming the heights are normally

distributed, calculate the z-score for a student who is 70 inches tall.

Solution

Step 1: Recall that the z-score formula is given by z=x−µ

σ, where xis the

value we are interested in, µis the mean, and σis the standard deviation.

Step 2: Substituting the given values into the formula, we have:

z=70 −65.7

3.5

Step 3: Calculate the z-score:

z=4.3

3.5= 1.2286

Step 4: Therefore, the z-score for a student who is 70 inches tall is ap-

proximately 1.2286. This means that the student’s height is 1.2286 standard

deviations above the mean height of the dataset.

Question 9

Question

Let X={8,12,15,21,24}and Y={10,13,17,22,28}be two data sets. Calcu-

late the median, mode, and range for each data set.

Solution

Step 1: To find the median of a data set, we first need to arrange the data set

in ascending order. For set X, the data set is {8,12,15,21,24}. When arranged

in ascending order, we get {8,12,15,21,24}. The median is the middle value,

which is 15.

For set Y, the data set is {10,13,17,22,28}. When arranged in ascending

order, we get {10,13,17,22,28}. The median is the middle value, which is 17.

Step 2: To find the mode of a data set, we need to identify the value that

appears most frequently. If there are multiple modes, the data set is considered

multimodal. For set X, all values appear only once, so there is no mode.

For set Y, all values appear only once, so there is no mode.

Step 3: To find the range of a data set, we need to find the difference between

the maximum and minimum values. For set X, the minimum value is 8 and the

maximum value is 24. Therefore, the range is 24 −8 = 16.

For set Y, the minimum value is 10 and the maximum value is 28. Therefore,

the range is 28 −10 = 18.

In conclusion, - The median of set Xis 15 and the mode is undefined. - The

median of set Yis 17 and the mode is undefined. - The range of set Xis 16

and the range of set Yis 18.

Question 10

Question

In a study of the heights of students in a university, the following data was

collected: 160 cm, 165 cm, 170 cm, 172 cm, 175 cm, 180 cm, 185 cm, 190 cm,

195 cm. Calculate the mean, median, mode, range, variance, and standard

deviation of the heights.

Solution

Step 1: Calculate the mean. The mean is calculated using the formula:

Mean = Pn

i=1 xi

where xiare the individual heights and nis the number of observations.

Calculating the mean:

Mean = 160 + 165 + 170 + 172 + 175 + 180 + 185 + 190 + 195

9=1592

9= 176.9 cm

Step 2: Calculate the median. To find the median, first arrange the data in

ascending order: 160, 165, 170, 172, 175, 180, 185, 190, 195.

Since there are 9 observations, the median is the middle value, which is the

fifth value: Median = 175 cm.

Step 3: Calculate the mode. The mode is the value that appears most

frequently in the data set. In this case, all values appear only once, so there is

no mode.

Step 4: Calculate the range. The range is the difference between the highest

and lowest values in the data set. Range = 195 cm - 160 cm = 35 cm.

Step 5: Calculate the variance. The variance is calculated using the formula:

Variance = Pn

i=1(xi−Mean)2

Calculating the variance:

Variance = (160 −176.9)2+ (165 −176.9)2+. . . + (195 −176.9)2

9=829.6

9≈92.178 cm2

Step 6: Calculate the standard deviation. The standard deviation is the

square root of the variance:

Standard Deviation = √Variance = √92.178 ≈9.6 cm

Therefore, the mean height is 176.9 cm, the median height is 175 cm, there

is no mode, the range is 35 cm, the variance is approximately 92.178 cm2, and

the standard deviation is approximately 9.6 cm.

Question 11

Question

Suppose a researcher collected data on the test scores of students in two different

classes. The mean test score for Class A was 85 with a standard deviation of 10,

while the mean test score for Class B was 78 with a standard deviation of 8. If

the two classes had the same number of students, calculate the pooled standard

deviation for the combined data.

Solution

Step 1: Calculate the pooled standard deviation using the formula:

Pooled standard deviation = s(n1−1)s2

1+ (n2−1)s2

n1+n2−2

where n1and n2are the number of observations in Class A and Class B re-

spectively, and s1and s2are the standard deviations for Class A and Class

Given: n1=n2,s1= 10, s2= 8

Substitute the values into the formula:

Pooled standard deviation = s(n1−1)(10)2+ (n1−1)(8)2

2n1−2

Step 2: Simplify the equation further:

Pooled standard deviation = s100(n1−1) + 64(n1−1)

2n1−2

Pooled standard deviation = s164(n1−1)

2n1−2

Step 3: Expand and simplify:

Pooled standard deviation = r164n1−164

2n1−2

Pooled standard deviation = s4(41n1−41)

2(2n1−1)

Pooled standard deviation = s4(41(n1−1)

2(2(n1−1) + 1)

Pooled standard deviation = s4(41)

2(2) + 1

Pooled standard deviation = r164

Pooled standard deviation = √32.8

Pooled standard deviation ≈5.72

Therefore, the pooled standard deviation for the combined data is approxi-

mately 5.72.

Question 12

Question

A researcher collected data on the monthly salaries of employees at a large

company. The data is summarized in the following table:

Mean $5000

Median $4800

Mode $4500

Standard Deviation $600

Based on the given information, is the distribution of monthly salaries skewed

to the left, skewed to the right, or approximately symmetric?

Solution

Step 1: Since the mean, median, and mode are all different, this indicates that

the distribution of monthly salaries is not symmetric.

Step 2: To determine whether the distribution is skewed to the left, skewed to

the right, or approximately symmetric, we can compare the mean and median.

Step 3: If the mean is less than the median, the distribution is skewed to

the left. If the mean is greater than the median, the distribution is skewed to

the right. If the mean is approximately equal to the median, the distribution is

approximately symmetric.

Step 4: In this case, the mean is $5000 and the median is $4800. Since the

mean is greater than the median, the distribution of monthly salaries is skewed

to the right.

Question 13

Question

Let X={3,5,7,8,11,15}and Y={2,4,6,9,12,14}. Compute the sample

mean, sample variance, and sample standard deviation for each set.

Solution

Step 1: Calculate the mean. The sample mean ¯

Xfor set Xis given by

X=1

i=1

xi.

Similarly, the sample mean ¯

Yfor set Yis given by

Y=1

j=1

yj.

Step 2: Calculate the variance. The sample variance s2

Xfor set Xis given

X=1

n−1

i=1

(xi−¯

X)2.

Similarly, the sample variance s2

Yfor set Yis given by

Y=1

m−1

j=1

(yj−¯

Y)2.

Step 3: Calculate the standard deviation. The sample standard deviation

sXfor set Xis simply the square root of the variance: sX=ps2

X. Similarly,

the sample standard deviation sYfor set Yis sY=ps2

Question 14

Question

A researcher collected the following data on the number of hours spent studying

for an exam by a group of students: 2, 3, 4, 5, 6, 7, 8, 9, 10. Calculate the mean,

median, variance, and standard deviation of the data set.

Solution

Step 1: Calculate the mean. The mean is calculated by summing all the data

points and dividing by the total number of data points.

Mean = 2+3+4+5+6+7+8+9+10

9=54

9= 6

Step 2: Calculate the median. To find the median, first arrange the data

in ascending order: 2, 3, 4, 5, 6, 7, 8, 9, 10. Since there are 9 data points, the

median is the middle value, which is the 5th value.

Median = 6

Step 3: Calculate the variance. The variance is the average of the squared

differences between each data point and the mean.

Variance = (2 −6)2+ (3 −6)2+ (4 −6)2+ (5 −6)2+ (6 −6)2+ (7 −6)2+ (8 −6)2+ (9 −6)2+ (10 −6)2

=16+9+4+1+0+1+4+9+16

9=60

9≈6.67

Step 4: Calculate the standard deviation. The standard deviation is the

square root of the variance.

Standard deviation = √6.67 ≈2.58

Therefore, the mean is 6, the median is 6, the variance is 6.67, and the

standard deviation is approximately 2.58.

Question 15

Question

In a research study involving a sample of 50 individuals, the heights (in inches)

were recorded and the following descriptive statistics were calculated:



Mean: 65.7 inches



Standard Deviation: 2.5 inches

Assuming the heights are normally distributed, calculate the z-score for a height

of 70 inches.

Solution

Step 1: Calculate the z-score using the formula: z=x−µ

σwhere xis the value,

µis the mean, and σis the standard deviation.

z=70 −65.7

2.5

z=4.3

2.5

z= 1.72

Therefore, the z-score for a height of 70 inches is 1.72.

Question 16

Question

The data below represents the number of hours spent studying per week by a

sample of 20 university students:

3,4,6,9,10,11,13,15,16,17,18,19,20,21,22,23,25,26,27,30

Calculate the mean, median, mode, range, variance, and standard deviation

for the given data.

Solution

Step 1: Calculate the mean. The mean is calculated by summing up all the

values and then dividing by the total number of values.

Mean = 3+4+6+9+10+11+13+15+16+17+18+19+20+21+22+23+25+26+27+30

Mean = 331

20 = 16.55

Step 2: Calculate the median. To find the median, we need to arrange the

data in ascending order and then find the middle value. Arranging the data:

3,4,6,9,10,11,13,15,16,17,18,19,20,21,22,23,25,26,27,30

Since there are 20 values, the median is the average of the 10th and 11th values.

Median = 16+17

2= 16.5

Step 3: Calculate the mode. The mode is the value(s) that occur most

frequently. In this dataset, there is no mode as all values appear only once.

Step 4: Calculate the range. The range is found by subtracting the minimum

value from the maximum value. Range = 30 - 3 = 27

Step 5: Calculate the variance. The variance is the average of the squared

differences between each value and the mean.

Variance = P(xi−Mean)2

n−1

Variance = (3 −16.55)2+ (4 −16.55)2+... + (30 −16.55)2

Calculating this gives the variance as 64.6474.

Step 6: Calculate the standard deviation. The standard deviation is the

square root of the variance.

Standard Deviation = √64.6474 ≈8.038

Question 17

Question

Let’s say we have a dataset with 50 observations. After calculating the mean,

median, range, and standard deviation, we noticed that the mean and median

are close but the range is very large, and the standard deviation is relatively

small. Discuss possible reasons for this discrepancy in descriptive statistics.

Solution

To understand the discrepancy in descriptive statistics, we need to consider the

characteristics of the dataset and how each measure is computed.

Step 1: Mean and Median - The mean is calculated as the sum of all

observations divided by the total number of observations. - The median is the

middle value in a dataset when the observations are ordered from smallest to

largest. - When the mean and median are close, it suggests that the dataset is

approximately symmetric.

Step 2: Range - The range is the difference between the maximum and

minimum values in the dataset. - A large range can occur when there are outliers

in the dataset, especially if these outliers are at the extremes.

Step 3: Standard Deviation - The standard deviation is a measure of

the dispersion of data points around the mean. - A relatively small standard

deviation suggests that the data points are clustered closely around the mean,

with little variability.

Step 4: Possible Reasons for Discrepancy - The dataset may have

outliers at the extremes, causing a large range but not affecting the mean sig-

nificantly due to the mean’s sensitivity to extreme values. - The dataset may

be approximately symmetric, explaining the closeness between the mean and

median. - The small standard deviation indicates low variability among the

data points, which is why the data points are clustered closely around the mean

despite the outliers.

In summary, the discrepancy in the descriptive statistics can be attributed

to outliers influencing the range but not affecting the mean significantly, the

dataset being approximately symmetric, and the low variability among data

points reflected in the small standard deviation.

Question 18

Question

Let Xbe a random variable with probability mass function given by:

P(X= 1) = 0.2, P (X= 2) = 0.3, P (X= 3) = 0.1, P (X= 4) = 0.4

Find the mean, variance, and standard deviation of the random variable X.

Solution

Step 1: Calculate the mean The mean of a discrete random variable is given

by:

µ=E[X] = X

x·P(X=x)

Substitute the given probabilities into the formula:

µ= 1 ·0.2+2·0.3+3·0.1+4·0.4

µ= 0.2+0.6+0.3+1.6=2.7

Therefore, the mean of the random variable Xis 2.7.

Step 2: Calculate the variance The variance of a discrete random variable

is given by:

σ2=E[(X−µ)2] = X

(x−µ)2·P(X=x)

Substitute the given probabilities and the calculated mean into the formula:

σ2= (1 −2.7)2·0.2 + (2 −2.7)2·0.3 + (3 −2.7)2·0.1 + (4 −2.7)2·0.4

σ2= 1.69 ·0.2+0.49 ·0.3+0.09 ·0.1+1.69 ·0.4

σ2= 0.338 + 0.147 + 0.009 + 0.676 = 1.17

Therefore, the variance of the random variable Xis 1.17.

Step 3: Calculate the standard deviation The standard deviation is the

square root of the variance:

σ=√σ2=√1.17 ≈1.08

Therefore, the standard deviation of the random variable Xis approximately

1.08.

Question 19

Question

Let’s say we have two datasets, dataset A and dataset B. Dataset A consists

of 20 data points with a mean of 45 and a standard deviation of 5. Dataset B

consists of 15 data points with a mean of 50 and a standard deviation of 8.

Given this information, can you determine which dataset has more variability

in the data? Justify your answer using the standard deviations of the datasets.

Solution

To determine which dataset has more variability in the data, we can compare

the standard deviations of the datasets. The standard deviation measures the

dispersion of data points in a dataset. A larger standard deviation indicates

more variability in the data.

Step 1: Calculate the Coefficients of Variation (CV) for each

dataset

The Coefficient of Variation (CV) is a relative measure of variability that

allows us to compare the standard deviations of datasets with different mean

values. It is calculated by dividing the standard deviation by the mean.

For Dataset A:

CVA=5

45 = 0.1111

For Dataset B:

CVB=8

50 = 0.16

Step 2: Compare the Coefficients of Variation

Since Dataset B has a larger coefficient of variation (0.16 ¿ 0.1111), this

indicates that Dataset B has more variability in the data compared to Dataset

Therefore, Dataset B has more variability in the data than Dataset A based

on the standard deviations and coefficients of variation.

Question 20

Question

A researcher is conducting a study on the relationship between hours spent

studying per week and final exam scores for a group of university students.

After collecting the data, the descriptive statistics for hours studied per week

are as follows: mean = 15, median = 14, mode = 16, range = 10, and standard

deviation = 3.2. If a student studied 18 hours per week, how many standard

deviations is this value from the mean?

Solution

Step 1: Calculate the z-score using the formula:

z=x−¯x

Where: - x= 18 (hours studied per week by the student) - ¯x= 15 (mean

hours studied per week) - s= 3.2 (standard deviation)

Step 2: Substitute the values into the formula:

z=18 −15

3.2

Step 3: Calculate the z-score:

z=3

3.2= 0.9375

Step 4: Therefore, the student who studied 18 hours per week is 0.9375

standard deviations above the mean.

Question 21

Question

Let’s consider a dataset of 50 students’ scores on a midterm exam in a statistics

course. The data are as follows:

70,65,85,90,72,60,78,82,88,75,79,92,85,69,71,83,77,81,73,80,64,70,75,78,82,79,68,76,84,87,90,74,71,75,79,81,74,73,80,72,76,68,85,77,70,82,83,67,74

Calculate the following descriptive statistics for the dataset: 1. Mean 2.

Median 3. Variance 4. Standard deviation

Solution

1. Calculate the mean: Step 1: Add up all the values in the dataset.

70+65+85+90+72+60+78+82+88+75+79+92+85+69+71+83+77+81+73+80+64+70+75+78+82+79+68+76+84+87+90+74+71+75+79+81+74+73+80+72+76+68+85+77+70+82+83+67+74 = 3925

Step 2: Divide the sum by the total number of values in the dataset (50) to

find the mean.

Mean = 3925

50 = 78.5

2. Calculate the median: Step 1: Arrange the values in ascending order:

60,64,65,67,68,68,69,70,70,70,71,71,72,72,73,73,74,74,74,75,75,75,75,76,76,77,77,78,78,79,79,79,79,80,80,81,81,82,82,82,83,83,84,85,85,85,87,88,90,90,92

Step 2: Find the middle value. Since we have 50 values, the median will be

the average of the two middle values.

Median = 75 + 76

2= 75.5

3. Calculate the variance: Step 1: Find the squared difference between each

value and the mean, then sum these squared differences.

i=1

(xi−¯x)2

i=1

(xi−78.5)2= 1127.1

Step 2: Divide the sum by the total number of values in the dataset (50) to

find the variance.

Variance = 1127.1

50 = 22.542

4. Calculate the standard deviation:

Standard deviation = √22.542 ≈4.75

Question 22

Question

A researcher collected data on the heights (in inches) of 50 students in a uni-

versity. The sample mean height was found to be 65 inches with a standard

deviation of 3 inches. Furthermore, it was observed that the heights were nor-

mally distributed. Using this information, calculate the z-score for a student

who is 70 inches tall.

Solution

Step 1: Calculate the z-score using the formula for z-score:

z=x−¯x

where: x= student’s height = 70 inches, ¯x= sample mean height = 65 inches,

σ= standard deviation = 3 inches.

Step 2: Substitute the values into the formula:

z=70 −65

z=5

z≈1.67

Step 3: Therefore, the z-score for a student who is 70 inches tall is approxi-

mately 1.67.

Question 23

Question

A researcher collected data on the starting salaries of 100 graduates from a

university. The mean starting salary was

50,000 with a standard deviation of

5,000. Assuming the data is normally distributed, calculate the z-score for a

starting salary of

45,000.

Solution

Let’s first recall the formula for calculating the z-score:

z=x−µ

where: - xis the individual data point, - µis the mean of the data, - σis the

standard deviation of the data.

Step 1: Substitute the given values into the formula:

z=45000 −50000

5000

Step 2: Calculate the z-score:

z=−5000

5000

z=−1

Step 3: Answer: The z-score for a starting salary of

45,000 is -1.

Question 24

Question

Given the following dataset:

12,15,18,21,24,27,30,33,36,39

Calculate the mean, median, mode, variance, standard deviation, range, and

interquartile range.

Solution

Step 1: Calculate the mean:

Mean = 12 + 15 + 18 + 21 + 24 + 27 + 30 + 33 + 36 + 39

10 =255

10 = 25.5

Step 2: Calculate the median: Since we have an even number of data points,

the median is the average of the two middle values.

Median = 24 + 27

2=51

2= 25.5

Step 3: Calculate the mode: Since all values are unique, there is no mode.

Step 4: Calculate the variance:

Variance = P(Xi−¯

X)2

n=(12 −25.5)2+ (15 −25.5)2+. . . + (39 −25.5)2

Variance = 1582.5

10 = 158.25

Step 5: Calculate the standard deviation:

Standard deviation = √158.25 = 12.58

Step 6: Calculate the range:

Range = largest value −smallest value = 39 −12 = 27

Step 7: Calculate the interquartile range: First, find the first quartile (Q1)

and the third quartile (Q3). Since we have 10 data points, Q1 will be the average

of the 3rd and 4th values, and Q3 will be the average of the 8th and 9th values.

Q1 = 18 + 21

2= 19.5

Q3 = 33 + 36

2= 34.5

Then, calculate the interquartile range:

Interquartile range = Q3−Q1 = 34.5−19.5 = 15

Therefore, the mean is 25.5, median is 25.5, mode does not exist, variance

is 158.25, standard deviation is 12.58, range is 27, and the interquartile range

is 15.

Question 25

Question

A researcher conducted a study on the relationship between the number of hours

of study per week and final exam scores for a group of university students. The

data collected is as follows:

Hours of Study (x) Exam Score (y)

9 75

14 82

11 78

10 80

Calculate the mean, median, range, variance, and standard deviation for

both hours of study and exam scores.

Solution

Step 1: Calculate the mean for both hours of study and exam scores.

For hours of study,

Mean = 9 + 14 + 11 + 10

4=44

4= 11

For exam scores,

Mean = 75 + 82 + 78 + 80

4=315

4= 78.75

Step 2: Calculate the median for both hours of study and exam scores.

For hours of study, since the data is already arranged from smallest to largest,

the median is the average of the two middle values:

Median = 10 + 11

2=21

2= 10.5

For exam scores, since the data is already arranged from smallest to largest,

the median is the average of the two middle values:

Median = 78 + 80

2=158

2= 79

Step 3: Calculate the range for both hours of study and exam scores.

For hours of study,

Range = 14 −9 = 5

For exam scores,

Range = 82 −75 = 7

Step 4: Calculate the variance for both hours of study and exam scores.

For hours of study,

Variance = (9 −11)2+ (14 −11)2+ (11 −11)2+ (10 −11)2

4−1=4+9+0+1

3=14

3≈4.67

For exam scores,

Variance = (75 −78.75)2+ (82 −78.75)2+ (78 −78.75)2+ (80 −78.75)2

4−1=16.5625 + 6.0625 + 0.5625 + 1.5625

3=24.75

3≈8.25

Step 5: Calculate the standard deviation for both hours of study and exam

scores.

For hours of study,

Standard deviation = √4.67 ≈2.16

For exam scores,

Standard deviation = √8.25 ≈2.87

Question 26

Question

Suppose we have the following dataset representing the scores of 20 students in

a statistics exam:

60,75,82,67,89,93,78,71,76,85,69,73,79,81,88,70,64,77,84,72

Determine the mean, median, mode, range, variance, and standard deviation

of the dataset.

Solution

Step 1: Calculate the mean

Mean = 1

i=1

Mean = 1

20×(60+75+82+67+89+93+78+71+76+85+69+73+79+81+88+70+64+77+84+72)

Mean = 1460

Mean = 73

Step 2: Calculate the median Since the dataset is already sorted from

least to greatest, the median will be the average of the 10th and 11th values.

Median = (73 + 76)

Median = 74.5

Step 3: Calculate the mode The mode is the score that appears most

frequently in the dataset. In this case, there is no score that appears more than

once, so there is no mode.

Step 4: Calculate the range

Range = Maximum value −Minimum value

Range = 93 −60

Range = 33

Step 5: Calculate the variance

Variance = Pn

i=1(xi−Mean)2

Variance = (60 −73)2+ (75 −73)2+. . . + (72 −73)2

Variance = 340.6

Variance = 17.03

Step 6: Calculate the standard deviation

Standard Deviation = √Variance

Standard Deviation = √17.03

Standard Deviation ≈4.13

Therefore, the mean is 73, the median is 74.5, the mode does not exist, the

range is 33, the variance is 17.03, and the standard deviation is approximately

4.13.

Question 27

Question

Consider the following dataset representing the scores of 20 students on a physics

exam:

73,78,82,65,88,91,72,95,79,84,77,73,86,90,68,81,76,89,74,80

Calculate the mean, median, mode, variance, standard deviation, and range

of the scores.

Solution

Step 1: Calculate the mean:

Mean = 1

i=1

Mean = 1

20(73+78+82+65+88+91+72+95+79+84+77+73+86+90+68+81+76+89+74+80)

Mean = 1569

20 = 78.45

Step 2: Calculate the median: To find the median, we first need to arrange

the data in ascending order:

65,68,72,73,73,74,76,77,78,79,80,81,82,84,86,88,89,90,91,95

Since there are 20 values, the median is the average of the 10th and 11th values:

Median = 80 + 81

2= 80.5

Step 3: Calculate the mode: From the dataset, we can see that the mode is

73 since it occurs twice, which is more frequent than any other value.

Step 4: Calculate the variance:

Variance = 1

i=1

(xi−Mean)2

Variance = 1

20[(73 −78.45)2+ (78 −78.45)2+... + (80 −78.45)2]

Variance ≈36.9475

Step 5: Calculate the standard deviation:

Standard deviation = √Variance ≈√36.9475 ≈6.08

Step 6: Calculate the range:

Range = Max −Min = 95 −65 = 30

Therefore, the mean is 78.45, median is 80.5, mode is 73, variance is ap-

proximately 36.9475, standard deviation is approximately 6.08, and the range

is 30.

Question 28

Question

Let’s consider a dataset that represents the heights (in inches) of 50 university

students. The following descriptive statistics were calculated for this dataset:



Mean height: 65 inches



Standard deviation: 3.5 inches

Given this information, what can you conclude about the distribution of heights

among the university students?

Solution

To draw conclusions about the distribution of heights among the university

students based on the provided descriptive statistics, we need to consider the

properties of the mean and standard deviation.

Step 1: Mean The mean height of 65 inches indicates that, on average, the

university students are 65 inches tall.

Step 2: Standard Deviation The standard deviation of 3.5 inches measures

the variability or spread of heights around the mean. A smaller standard devia-

tion implies that the data points are closer to the mean, while a larger standard

deviation indicates that the data points are more spread out.

Step 3: Conclusion Since the standard deviation is relatively small at 3.5

inches, we can conclude that the heights of the university students are clustered

around the mean height of 65 inches. This suggests that the distribution of

heights is relatively narrow and the majority of students have heights close to

the mean height.

Question 29

Question

Consider the following data set:

12,15,18,22,25,29,33,35,40,42,45

Find the mean, median, mode, variance, and standard deviation of the data

set.

Solution

Step 1: Find the Mean Calculate the mean by summing up all the values and

then dividing by the total number of values.

Mean = 12 + 15 + 18 + 22 + 25 + 29 + 33 + 35 + 40 + 42 + 45

Mean = 316

11 = 28.73

Step 2: Find the Median To find the median, first arrange the data in

ascending order:

12,15,18,22,25,29,33,35,40,42,45

Since there are 11 values, the median is the middle value, which in this case is

the 6th value:

Median = 29

Step 3: Find the Mode The mode is the value that appears most frequently

in the data set. In this case, each value appears only once, so the data set has

no mode.

Step 4: Find the Variance Calculate the variance by finding the average of

the squared differences between each value and the mean.

Variance = (12 −28.73)2+ (15 −28.73)2+. . . + (45 −28.73)2

Variance ≈195.21

Step 5: Find the Standard Deviation The standard deviation is the square

root of the variance.

Standard Deviation = √Variance ≈√195.21 ≈13.97

Question 30

Question

Suppose we have a dataset with the following values:

12,15,18,20,22,24,25,27,29,30

Calculate the following descriptive statistics for this dataset:



Mean



Median



Variance



Standard Deviation

Solution

Let’s start by arranging the dataset in ascending order:

12,15,18,20,22,24,25,27,29,30

Step 1: Calculate the Mean The mean is calculated by summing all

values and dividing by the total number of values:

Mean = 12 + 15 + 18 + 20 + 22 + 24 + 25 + 27 + 29 + 30

10 = 22.2

Step 2: Calculate the Median Since we have an even number of values,

the median is the average of the two middle values:

Median = 22 + 24

2= 23

Step 3: Calculate the Variance The variance is calculated by finding the

average of the squared differences between each value and the mean:

Variance = (12 −22.2)2+ (15 −22.2)2+. . . + (30 −22.2)2

Variance ≈37.56

Step 4: Calculate the Standard Deviation The standard deviation is

the square root of the variance:

Standard Deviation = √37.56 ≈6.13

Therefore, for the given dataset, the mean is 22.2, median is 23, variance is

37.56, and standard deviation is approximately 6.13.

Question 6

Question

Let’s say you have a dataset with the following values: 6, 8, 10, 6, 12, 6, 14, 8.

Calculate the mean, median, mode, variance, standard deviation, and range of

the dataset.

Solution

To find the descriptive statistics of the dataset, we will calculate the mean,

median, mode, variance, standard deviation, and range step by step.

Step 1: Calculate the mean The mean is calculated by summing up all

the values in the dataset and then dividing by the total number of values.

Mean = 6+8+10+6+12+6+14+8

Mean = 70

8= 8.75

Step 2: Calculate the median To find the median, we first need to

arrange the values in ascending order: 6, 6, 6, 8, 8, 10, 12, 14. Since there are

an even number of values, the median is the average of the two middle values.

Median = 8+8

2= 8

Step 3: Calculate the mode The mode is the value that appears most

frequently in the dataset. In this case, the mode is 6, as it appears three times.

Step 4: Calculate the variance The variance measures how spread out

the values in the dataset are. To calculate the variance, we first need to find the

squared differences between each value and the mean, then sum them up and

divide by the total number of values.

Variance = (6 −8.75)2+ (8 −8.75)2+ (10 −8.75)2+ (6 −8.75)2+ (12 −8.75)2+ (6 −8.75)2+ (14 −8.75)2+ (8 −8.75)2

Variance = 57.5

8= 7.1875

Step 5: Calculate the standard deviation The standard deviation is

the square root of the variance.

Standard deviation = √7.1875 ≈2.68

Step 6: Calculate the range The range is the difference between the

maximum and minimum values in the dataset. Range = 14 −6 = 8

Therefore, the mean is 8.75, the median is 8, the mode is 6, the variance is

7.1875, the standard deviation is approximately 2.68, and the range is 8.

Question 7

Question

Suppose a researcher collected the following data on the heights (in centimeters)

of 10 adult males:

175,180,165,190,185,170,175,160,200,195

Calculate the mean, median, mode, range, variance, and standard deviation

of the data set.

Solution

Let’s calculate the mean, median, mode, range, variance, and standard deviation

step-by-step.

Step 1: Calculate the Mean The mean (¯x) is calculated by adding up

all the values and dividing by the total number of values (10 in this case).

Mean = 175 + 180 + 165 + 190 + 185 + 170 + 175 + 160 + 200 + 195

Mean = 1775

10 = 177.5

Therefore, the mean height is 177.5 cm.

Step 2: Calculate the Median To find the median, we first need to

arrange the data in ascending order.

160,165,170,175,175,180,185,190,195,200

Since there are 10 values, the median is the average of the 5th and 6th values.

Median = 175 + 180

2= 177.5

Therefore, the median height is 177.5 cm.

Step 3: Calculate the Mode The mode is the value that appears most

frequently in the data set. In this case, there is no unique mode as each value

appears only once. So, the data set has no mode.

Step 4: Calculate the Range The range is the difference between the

largest and smallest values in the data set.

Range = 200 −160 = 40

Therefore, the range of heights is 40 cm.

Step 5: Calculate the Variance The variance is a measure of how spread

out the values in a data set are. It is calculated using the formula:

Variance = P(xi−¯x)2

where xiare the individual values, ¯xis the mean, and nis the total number of

values (10 in this case).

Variance = (175 −177.5)2+ (180 −177.5)2+. . . + (195 −177.5)2

Calculating this value gives us a variance of 140.25.

Step 6: Calculate the Standard Deviation The standard deviation is

the square root of the variance.

Standard Deviation = √140.25 = 11.84

Therefore, the standard deviation of the heights is 11.84 cm.

Question 8

Question

Let’s consider a dataset representing the heights (in inches) of 30 college stu-

dents. The descriptive statistics for this dataset are as follows: mean height =

65.7 inches, standard deviation = 3.5 inches. Assuming the heights are normally

distributed, calculate the z-score for a student who is 70 inches tall.

Solution

Step 1: Recall that the z-score formula is given by z=x−µ

σ, where xis the

value we are interested in, µis the mean, and σis the standard deviation.

Step 2: Substituting the given values into the formula, we have:

z=70 −65.7

3.5

Step 3: Calculate the z-score:

z=4.3

3.5= 1.2286

Step 4: Therefore, the z-score for a student who is 70 inches tall is ap-

proximately 1.2286. This means that the student’s height is 1.2286 standard

deviations above the mean height of the dataset.

Question 9

Question

Let X={8,12,15,21,24}and Y={10,13,17,22,28}be two data sets. Calcu-

late the median, mode, and range for each data set.