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Statistics Refresher Week 2
Statistics in Psychology (Liberty University)
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PSYC 421
STATISTICS REFRESHER TEMPLATE
Student Name: Christopher Hill
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:
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PSYC 421
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) The first calculation
is completed for you, in red, as an example. To calculate f(X) multiply the score times it’s frequency.
Example: 96(1) = 96.
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
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PSYC 421
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
Calculating the Average Deviation
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). The first calculation is completed for you, in red, as an example.
Scores from Your Test
(X)
Score f
(frequency)
Absolute Value
of (X−x)
96
1
23.88
94
92
87
85
1
1
1
1
21.88
19.88
14.88
12.88
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PSYC 421
84
83
82
79
78
76
73
72
69
69
67
67
66
63
62
61
61
51
44
42
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
11.88
10.88
9.88
6.88
5.88
3.88
0.88
0.12
-3.12
-3.12
-5.12
-5.12
-6.12
-9.12
-10.12
-11.12
-11.12
-21.12
-28.12
-30.12
4.
The sum of the absolute value of deviation scores =
287.12
5. The total number of scores in the frequency distribution = 25
6. Therefore, average deviation (AD) =
11.48
(2 points)
(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
succinctly, 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: (The first calculation is completed for you, in red, as an example.)
(10 points)
X
f
X−x
(X−x)2
96
1
23.88
570.25
94
1
21.88
478.73
92
1
19.88
395.21
87
1
14.88
221.41
85
1
12.88
165.89
84
1
11.88
141.13
83
1
10.88
118.37
82
1
9.88
97.61
79
1
6.88
47.33
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PSYC 421
78
1
5.88
34.57
76
1
3.88
15.05
73
1
0.88
0.77
72
1
0.12
0.01
69
1
-3.12
9.73
69
1
-3.12
9.73
67
1
-5.12
26.21
67
1
-5.12
26.21
66
1
-6.12
37.45
63
1
-9.12
83.17
62
1
-10.12
102.41
61
1
-11.12
123.65
61
1
-11.12
123.65
51
1
-21.12
446.05
44
1
-28.12
790.73
42
1
-30.12
907.21
(2 points each)
7. The sum of the squared values of deviation scores = 4972.53
8. Variance = Sum of the squared values of deviation scores ÷ total number of scores
9. Therefore, variance = 198.90
10. Standard deviation = √Variance = 14.10
Think about how you will communicate this data to the class.
(2 points each)
11. What type of frequency distribution would you use?
Simple frequency distribution
12. Which type of graph would you use to represent the data?
Histogram
13. Which measure of central tendency would you use to represent the data?
The mean
14. Which measure of variability would you use to represent the data?
Standard deviation
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.
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PSYC 421
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. The first calculation is completed
for you, in red, as an example.
(25 points)
X
f
X−x
z = (X−x) ÷ S
96
1
23.88
1.69
94
1
21.88
1.55
92
1
19.88
1.41
87
1
14.88
1.06
85
1
12.88
0.91
84
1
11.88
0.84
83
1
10.88
0.77
82
1
9.88
0.70
79
1
6.88
0.49
78
1
5.88
0.42
76
1
3.88
0.28
73
1
0.88
0.06
72
1
0.12
0.01
69
1
3.12
0.22
69
1
3.12
0.22
67
1
5.12
0.36
67
1
5.12
0.36
66
1
6.12
0.43
63
1
9.12
0.65
62
1
10.12
0.72
61
1
11.12
0.79
61
1
11.12
0.79
51
1
21.12
1.49
44
1
28.12
1.99
42
1
30.12
2.14
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PSYC 421
T scores = 10z + 50
Complete the table by calculating the T score. The first calculation is completed for you, in red, as an
example.
(25 points)
Score
f (frequency)
96
1
94
92
87
85
84
83
82
79
78
76
73
72
69
69
67
67
66
63
62
61
61
51
44
42
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
T
67 (66.90)
65.5
64.1
60.6
59.1
58.4
57.7
57
55 (54.90)
54.2
52.8
50.6
50.1
52.2
52.2
53.6
53.6
54.3
56.5
57.2
58 (57.90)
58 (57.90)
65 (64.90)
70 (69.90)
71.4
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