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Chapter 3: A Statistics Refresher
Student Name: Landyn Beal
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:
Cohen: Psychological Testing and Assessment, 9e
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Chapter 3: A Statistics Refresher
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 (frequency) 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
Cohen: Psychological Testing and Assessment, 9e
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Chapter 3: A Statistics Refresher
72 1 72
69 2 138
67 2 134
66 1 66
63 1 63
62 1 62
61 2 122
51 1 51
11 1 44
42 1 42
(2 points each):
1. Identify the of the frequency distribution. Median = 72median
2. Identify the in the frequency distribution. Mode(s) = 69, 67, 61mode
3. What is the of this frequency distribution? Range = 54range
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 Your Test
(X)
Score f
(frequency)
Absolute Value
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
Cohen: Psychological Testing and Assessment, 9e
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Chapter 3: A Statistics Refresher
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
4. The sum of the absolute value of deviation scores = 277 ( )2 points
5. The total number of scores in the frequency distribution = 25
6. Therefore, average deviation (AD) = 11.1 (2 points)
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
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Chapter 3: A Statistics Refresher
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 = 4929
8. Variance = Sum of the squared values of deviation scores ÷ total number of scores
9. Therefore, variance = 197.2
10. Standard deviation = √Variance = 14.0
Think about how you will communicate this data to the class.
(2 points each)
11. What type of frequency distribution would you use?
- Grouped Frequency
12. Which type of would you use to represent the data?graph
- Bar Graph
13. Which measure of would you use to represent the data?central tendency
- Arithmetic
14. Which measure of would you use to represent the data?variability
- Average Deviatoin
It may be meaningful to your students to reference a normal curve when communicating the results. This
may be accomplished by calculating and z scores T scores.
z scores
The formula for calculating scores is as follows:z
Cohen: Psychological Testing and Assessment, 9e
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Chapter 3: A Statistics Refresher
In the equation, is the mean of the frequency distribution and is the standard deviation of the x S
frequency distribution. Complete the table by calculating the score. z
(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 score.T
(25 points)
Score f (frequency) T
1.71 1 67.1
1.57 1 65.7
1.42 1 64.2
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Chapter 3: A Statistics Refresher
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
Cohen: Psychological Testing and Assessment, 9e
