Psychology 290
Research Methods & Statistics
Lab 9
Z-score distribution Work Sheet
January 23 - 25, 2007
1.
Compute either z-scores or raw scores based on the given information. a.
Mean = 75, Standard Deviation = 10
Determine the z-score associated with a raw score of 87. z = (87 – 75) = 12 = 1.2
10 10
Determine the z-score associated with a raw score of 46. z = (46 – 75) = -29 = -2.9
10 10
Determine the raw score associated with a z-score of 0.79. x = 0.79(10) + 75 = 7.9 +75 = 82.9
Determine the raw score associated with a z-score of -0.20. x = -0.20(10) + 75 = -2 +75 = 73 b.
Mean = 30, Variance = 25 (Standard Deviation = Square root of variance)
Determine raw score associated with z-score of -0.42. x = -0.42(5) + 30 = -2.1 +30 = 27.9
Determine z-score associated with raw score of 42. z = (42 – 30) = 12 = 2.4
5 5
What percentage of scores are less than a raw score of 30? z = (30 – 30) = 0 = 0
5 5
If you look up 0 in the z-table, larger and smaller portion is equal to 0.50
(50%). Also, Mean to z is equal to 0.00 (0%). This makes sense since 30 is the mean so half the scores will be above and the other half will be below.
In knowing this ahead of time, you can save time as the calculations were essentially unnecessary (i.e. if raw score = mean then z score = 0).

2.
In a particular very large sample of IQ scores, the mean was found to be 100 and the standard deviation was 16 (Each of these problems can be solved in more than one way. Question a. provides multiple ways you could solve the problem but for the remaining questions, the easiest/quickest method will be used). a.
What percentage of these people have an IQ below 132? z = (132 – 100) = 32 = 2.0
16 16
“Larger Portion” for a Z-score of 2.0 = 0.977
(A second way to answer this question is to determine the area from Mean to z, then add 0.50

Mean to z = 0.477 then add 0.50 equals 0.977)
(A third way to answer the question is to determine the “Smaller Portion” and subtract that value from 1.00

Smaller Portion = 0.023; subtracted from 1.00 equals 0.977).
% of people below a score of 132 = 0.977
or 97.7% b.
What percentage of these people have scores below 91? z = (91 – 100) = -9 = -0.56
16 16
Those who did worst than 91 are only those within the “Smaller Portion” region.
Area of Smaller Portion for a z-score 0f -0.56 is 0.288.

% of people below a score of 91
 0.288
or 28.8% c.
What percentage have scores above 80? z = (80 – 100) = -20 = -1.25
16 16
Those who scored above 80 are only those within the “Larger Portion” region.
Area of Smaller Portion for a z-score 0f -1.25 is 0.894.
% of scores above a score of 80
 0.894
or 89.4% d.
What percentage have IQs between 90 and 120? z = (90 – 100) = -10 = -0.625
16 16 z = (120 – 100) = 20 = 1.25
16 16
To get the area between, for this problem, look up the area for “Mean to z” for both values and add them together.
Mean to z for a z-score of -0.625 is 0.236.
Mean to z for a z-score of 1.25 is 0.394

% of scores between 90 and 120

0.236 + 0.394 = 0.63
or 63% e.
One IQ was at the 75 th percentile. What was the person’s raw score on the IQ test?
Remember that a percentile rank refers to the percent of scores below that point on the distribution (e.g. 75 th
percentile means that 75% of the scores were below the associated raw or z-score)
.
75 th
percentile = 75% = 0.75
In the z-table, find 0.75
under the “Larger Portion” column and determine the associated z-score (use the value closest to 0.750).
The associated z-score is approximately 0.67
. x = z(S) + mean

x = 0.67(16) + 100 = 10.7 + 100

Raw score = 110.7
Someone who scored at the 75 th
percentile achieved a score of 110.7
f.
Another IQ was at the 45 th percentile. What was that person’s raw score?
45 th
percentile = 45% = 0.45
(Since the percentile is less than 50, that means the value is less than the mean. Because the raw score is below the mean, the associated z-score is a negative value).

In the z-table, find 0.45
under the “Smaller Portion” column and determine the associated z-score (use the value closest to 0.450).
The associated z-score is approximately 0.13
. x = z(S) + mean

x = -0.13(16) + 100 = -2.08 + 100

Raw score = 97.92
Someone who scored at the 45 th
percentile achieved a score of 97.92
g.
What is the percentile rank of a score of 88? z = (88 – 100) = -12 = -0.75
16 16
Look up “Smaller Portion” for a z-score of -0.75 =
0.227
A raw score of 88 is associated with the 22.7 or 23 rd percentile . h.
What percentage have IQs between 105 and 135? z = (105 – 100) = 5 = 0.31
16 16

z = (135 – 100) = 35 = 2.18
16 16
Find the area between the Mean and z for both values.
Mean to z for a z-score of 0.31 is 0.122.
Mean to z for a z-score of 1.25 is 0.485
Since you are interested in the area between 105 and 135, you subtract the area between the Mean and z for the value of 105 from the area between the Mean and z for 135.
Therefore:
0.485 – 0.122 = 0.363

36.3%
% of scores between 105 and 135 = 0.363
or 36.3%