Student Student Name: Jenna Brown
Week 2 – Statistics Exercise
Complete the following exercises and submit for grading by the end of this week. No statistical
software is required; you should be able to execute all of the mathematical operations with a
standard calculator. Type your answers directly into this document and Save.
Suppose you have magically changed places with the professor teaching this course and that you
have just administered an examination that consists of 100 multiple-choice items (where 1 point is
awarded for each correct answer). The distribution of scores for the 25 students enrolled in your
class could theoretically range from 0 (none correct) to 100 (all correct). Below are your student’s
scores. You will use this raw data to complete all of the calculations in this assignment:
One task at hand is to communicate the test results to your class. You want to do that in a way that
will help students understand how their performance on the test compared with the performances
of other students. Probably, the first step is to arrange the data by converting it from a casual
listing of raw scores into something that immediately provides a little more information.
Display (in descending order) the test scores and complete the table below. (6 points)
Scores from
Your Test (X)
Score f
(frequen
cy)
f(X)
96
1
96
94
1
94
92
1
92
87
1
87
85
1
85
84
1
84
83
1
83
82
1
82
79
1
79
78
1
78
76
1
76
73
1
73
72
1
72
69
2
138
67
2
134
66
1
66
63
1
63
62
1
62
61
2
122
51
1
51
44
1
44
42
1
42
(2 points each):
1. Identify the median of the frequency distribution. Median = 72
2. Identify the mode in the frequency distribution. Mode(s) = 69, 67, 61
3. What is the range of this frequency distribution? Range = 54
On average, how much does each score in this distribution vary from the mean score?
The steps for calculating the average deviation (AD) of a frequency distribution is as
follows: i. Determine the deviation scores for each score in the frequency
distribution (in other words, how much does each individual
score vary from the mean score?).
ii. Find the sum of the deviation scores.
iii. Divide the sum of the deviation scores by the total number of
scores to obtain the average deviation.
Complete the table below (10 points).
Scores from
Score f
Absolute
Your Test
(freque
Value
(X)
ncy)
of
(X−x)
96
1
24
94
1
22
92
1
20
87
1
15
85
1
13
84
1
12
83
1
11
82
1
10
79
1
7
78
1
6
76
1
5
73
1
1
72
1
0
69
2
6
67
2
10
66
1
6
63
1
9
62
1
10
61
2
11
51
1
21
44
1
28
42
1
30
MEAN=72.12
(I DID THE AVERAGE DEVIATION WITH THE RAW SCORES)
4. The sum of the absolute value of deviation scores = 277
5. The total number of scores in the frequency distribution = 25
6. Therefore, average deviation (AD) = 11.08 or rounded to 11
What is the standard deviation of this distribution?
The standard deviation is equal to the square root of the average squared deviations
about the mean. More succintly, it is equal to the square root of the variance. So one
way to calculate the standard deviation of a frequency distribution is to calculate the
variance. Complete the table below as the first step in calculating the variance:
(10 points)
X
f
X−x
(X−x)
2
96
1
24
576
94
1
22
484
92
1
20
400
87
1
15
225
85
1
13
169
84
1
12
144
83
1
11
121
82
1
10
100
79
1
7
49
78
1
6
36
76
1
5
25
73
1
1
1
72
1
0
0
69
2
6
36
67
2
10
100
66
1
6
36
63
1
9
81
62
1
10
100
61
2
11
121
51
1
21
441
44
1
28
784
42
1
30
900
(2 points each)
7. The sum of the squared values of deviation scores = 4,929
8. Variance = Sum of the squared values of deviation scores ÷ total number of scores
9. Therefore, variance = 197.16
10.
Standard deviation = √Variance = 14.04
Think about how you will communicate this data to the class.
(2 points each)
11. What type of frequency distribution would you use? I would use grouped frequency
distribution, that way I could group the scores into class intervals.
12. Which type of graph would you use to represent the data? I would use a bar graph
and label the x-axis with my group intervals that I just mentioned, and then I would
label the y-axis with “number of cases”
13. Which measure of central tendency would you use to represent the data?I would use
the arithmetic mean to represent the data because it takes into account the actual
numerical value of each score. I believe the best way to represent this data would be to
tell my class the average score.
14. Which measure of variability would you use to represent the data? I would use the
average deviation to show how each score deviated from the mean of the all of the
scores.
It may be meaningful to your students to reference a normal curve when communicating the
results. This may be accomplished by calculating z scores and T scores.
Z scores
The formula for calculating z scores is as follows:
In the equation, x is the mean of the frequency distribution and S is the standard deviation
of the frequency distribution. Complete the table by calculating the z score.
(25 points)
X
f
X
−
x
z
=
(X−x)
÷
S
96
1
24
1.71
94
1
22
1.57
92
1
20
1.42
87
1
15
1.07
85
1
13
.93
84
1
12
.85
83
1
11
.78
82
1
10
.71
79
1
7
.5
78
1
6
.64
76
1
5
.36
73
1
1
.07
72
1
0
0
69
2
6
.43
67
2
10
.71
66
1
6
.43
63
1
9
.64
62
1
10
.71
61
2
11
.78
51
1
21
1.5
44
1
28
1.99
42
1
30
2.14
T scores = 10z + 50
Complete the table by calculating the Tscore.
(25 points)
Score
f(frequen
cy)
T=10z+50
1.71
1
67.1
1.57
1
65.7
1.42
1
64.2
1.07
1
60.7
.93
1
59.3
.85
1
58.5
.78
1
57.8
.71
1
57.1
.5
1
55
.64
1
56.4
.36
1
53.6
.07
1
50.7
0
1
50
.43
2
54.3
.71
2
57.1
.43
1
54.3
.64
1
56.4
.71
1
57.1
.71
2
57.1
.78
1
57.8
1.5
1
65
1.99
1
69.9
2.14
1
71.4
