Okun
PSY 230
STUDY GUIDE # 4
Locating the Relative Standing of an Individual Score in a Distribution
I. Percentile Ranks and Percentiles
1. What is a percentile rank? How can a percentile rank be useful? How can the percentile rank
of a score be computed?
Percentile ranks. A percentile rank of a score is a single number that indicates the
percent of participants in a specific reference group scoring at or below a given score. A
percentile rank is one strategy for locating the relative standing of an individual’s score in the
distribution of scores. By converting a raw score to a percentile rank, we can address the
question, what percentage of scores are at or below a given score?
Frequency Distribution for Recall of Names from High School Year Book
X
f
cf
_________________
98
1
40
87
2
39
85
2
37
84
1
35
83
1
34
76
1
33
75
1
32
74
1
31
69
1
30
67
1
29
65
4
28
64
1
24
63
1
23
57
2
22
56
1
20
54
2
19
53
1
17
49
2
16
48
1
14
47
2
13
46
1
11
45
1
10
44
1
9
38
1
8
37
2
7
34
2
5
1
33
3
3
2
With a percentile rank problem, we start with a raw score and end with a percentile rank between
1 and 100.
STEPS INVOLVED IN CONVERTING A SCORE TO A PERCENTILE RANK
1.
Construct a cumulative frequency distribution from the simple frequency distribution.
2.
Find the cumulative frequency of the score.
3.
Divide the cumulative frequency by N.
4.
Multiply this quotient by 100.
2. What is a percentile? How can a percentile be useful? How can the score associated with a
percentile be computed?
A percentile is the score at or below which a given percent of the cases lie. A percentile is
useful when we want to identify a cut score in a distribution. For example, if a University
has room to admit only the top 25% of the applicants, it would want to know what score is
associated with the 75th percentile. By converting a percentile to a raw score, we can address
the question, what score is associated with a given percentile?
With a percentile problem, we start with a percentile and end up with a raw score.
STEPS INVOLVED in CONVERTING A PERCENTILE TO A SCORE
1.
Construct a cumulative frequency distribution from the simple frequency distribution.
2.
Multiply the percentile by N to determine how far you need to “walk” into the
distribution.
3.
Use the cumulative frequency column to count in the number of scores as determined in
STEP 2.
4.
Determine the score across from the cumulative frequency where you have stopped.
3.
Which pieces of information are needed to construct a box-and-whisker plot? How are
box-and-whisker plots constructed?
To compute a box-and-whisker plot, we need 5 pieces of information--the lowest score, P25, P50,
P75, and the highest score.
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4.
What information can be gleaned from a box-and-whisker plot?
Section 1: 60, 65, 70, 86, 88, 90, 92, 94, 95, 96, 97, and 98
Section 2: 60, 62, 64, 66, 68, 70, 72, 74, 76, 77, 79, and 98
II. The Normal Curve
5. How can empirical and theoretical normal distributions be distinguished from each other?
There are two ways to generate a normal distribution--empirically and theoretically. When we
get a normal distribution by plotting data that have been collected, we have used the empirical
route. When we get a normal distribution by using a mathematical equation, we have used the
theoretical route. The theoretical normal curve is defined by an equation that includes both the
population mean and the population standard deviation.
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6. Why did statisticians decide to build a normal curve table? What is the name given to this
normal curve table?
7. What must be done in order to use the standard normal curve table?
8. What process is used to transform raw scores so that predetermined values are obtained for
the mean and the standard deviation of the new distribution? What are the scores in the new
distribution called?
9. What are standard scores with a  of 0 and a  of 1 called? What are the properties of a
distribution of z scores?
Properties of a distribution of z scores:
(1)  = 0;
(2)  = 1; and
(3) The shape of a set of z scores is identical to the shape of the raw scores.
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10. How can a distribution of raw scores be transformed to z scores?
For a population, zi = [Xi - x] / x
Steps in Computing z scores:
1) Determine x
2) Determine x
3) Subtract x from each Xi (Column # 2) and
4) Divide each deviation score, [Xi - x], by x (Column #3).
TRANSFORMING RAW SCORES TO z SCORES:
BOWLING AVERAGES OF 20 PLAYERS
(x = 180, x = 5.34)
Xi
189
188
187
186
185
184
183
182
181
180
180
179
178
177
176
175
174
173
172
171
Xi - x
9
8
7
6
5
4
3
2
1
0
0
-1
-2
-3
-4
-5
-6
-7
-8
-9
zi = [Xi - x] / x
1.69
1.50
1.31
1.12
0.94
0.75
0.56
0.37
0.19
0.00
0.00
-0.19
-0.37
-0.56
-0.75
-0.94
-1.12
-1.31
-1.50
-1.69
11. What information is conveyed by (a) the sign of a z score; and (b) the value of a z score.
A z score tells us how far above or below the mean an individual’s score is in standard
deviation units. The sign of the z score tells us whether the raw score is below or above the
mean. The value of the z score tells us how far from the mean the raw score falls in standard
deviation units.
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12. What information is provided for positive z scores in columns B and C of the Standard
Normal Curve Table?
13. What information is provided for negative z scores in columns B’ and C’ of the Standard
Normal Curve Table?
Summary
Column B indicates the proportion of cases between the mean and a positive z score.
Column B’ indicates the proportion of cases between the mean and a negative z score.
Column C indicates the proportion of cases above a positive z score.
Column C’ indicates the proportion of cases below a negative z score.
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14. How can we solve problems involving 1 score using the Standard Normal Curve Table?
Types of Problems Involving a Single Score and Strategies for Solution
Proportion Requested
Above
The Score
At or Below
The Score
+
Type I
Column C
Type II
1-Column C
-
Type III
1- Column C’
Type IV
Column C’
z Score Sign
The following problems involve Verbal SAT scores (x = 500, x = 100).
A.
Type I: What proportion of students score above 640?
B.
Type II: What proportion of students score at or below 540?
C.
Type III: What proportion of students score above 410?
D.
Type IV: What proportion of students score at or below 470?
15. How can we solve problems involving 2 scores (1 below the mean and 1 above the mean)
using the Standard Normal Curve Table?
Types of Problems Involving 1 Score Below and 1 Score Above the Mean
Type
Proportion Requested
Strategy For Solution
V
Between the Scores
Column B’ + Column B
VI
Beyond the Scores
Column C’ + Column C
E.
Type V: What proportion of students score between 490 and 600?
F.
Type VI: What proportion of students score either below 304 or above 680?
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16. Given the means and standard deviations for two or more distributions of raw scores, how
can we compare raw scores from these distributions? When would an individual prefer to have
(a) a small standard deviation; and (b) a large standard deviation?
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