PSY 2010 Lecture Notes
Standard Scores Corty Ch 4
Standard Score
Standard Score: A way of representing performance on a test or some other measure so that persons
familiar with the standard score know immediately how well the person did relative to others taking the
same test.
Need for Standard Scores:
Suppose I’ve decided to increase my income by creating a psychological test and marketing it.
I decide to create an Educational Achievement Test.
I call it the Biderman Educational Achievement Test, the BEAT.
I even license the rights to the Sonny and Cher song, The Beat Goes On, and use as back ground music for
my advertisements.
(Play excerpt from Sonny & Cher’s song here.)
Suppose you took the BEAT and made a score of 37.
How did you do?
If this were a bowling score, most people would know how well you did – terribly.
If it were an IQ, most people would know how good a score that is – not very good.
But it’s not a bowling score and it’s not an IQ score, it’s a BEAT score.
How can we know how well we did with an unfamiliar score?
With unfamiliar scores, we need some standardized way of reporting results, so that persons familiar with
the standardized ways of reporting can determine immediately how well they have performed.
There are 100s of psychological tests. All of them have different numbers of items and therefore, different
possible raw scores.
Standard scores are ways of reporting performance from such tests..
Biderman’s Handouts
Standard Scores - 1
2/6/2016
Types of Standard Scores
1. Percentile based
The Percentile Rank of a score: Percentage of Scores less than or equal to a score value.
PR X = 100*
i
100: Best possible value
50: Middle value
0: Worst possible value
No. of scores <= X i
N
We’ve already studied this as the Cumulative Relative Frequency of a score. Percentile Rank is just
another name for it.
From Steinberg, p. 102, Bob’s scores on three tests. Ten persons took each test.
9
Test A: 100 * --------------10
=
90.00
3
Test B: 100 * --------------10
=
30.00
9
Test C: 100 * --------------10
=
90.00
Based on the Percentile Rank, Bob scored equally well on Test A and on Test C.
Percentile ranks are great for getting a quick and rough idea of where you are in a distribution.
But they don’t take distance into account. Compare Bob’s performance on Test A and Test C. Bob was
close to the top of the distribution of Test A but far from the top of Test C.
Biderman’s Handouts
Standard Scores - 2
2/6/2016
2. Standard scores based on the distance of X to the mean. (Corty, p 102)
A. The Z-score
Generic Form: Z = (X – Mean) / SD or
X - Mean
Z = -----------------Standard Deviation
There is a population version
Z = (X - µ) / σ
X - µ
Z = ------------------
or
σ
And there is a sample version
Z = (X – X-bar) / SN-1
or
X – X-bar
Z = ------------------
SN-1
Interpretation of the Z score
Z tells us how many standard deviations X is above or below the mean.
Selected Z’s
+3
+2
+1
0
-1
-2
-3
Interpretation
X is 3 SDs above the mean
X is 2 SDs above the mean
X is 1 SD above the mean
X is equal to the mean (not the median)
X is 1 SD below the mean
X is 2 SDs below the mean
X is 3 SDs below the mean
Characterizing Zs – Suppose all Xs in a sample were converted to Zs.
Regardless of the original scores
If all the scores in a collection are converted to Z-scores . . .
1. The mean of the Zs will always be 0.
2. The standard deviation of the Zs will always be 1.
3. The shape of the distribution of Zs will be the same as the shape of the distribution of Xs.
Three useful facts about Zs
If all scores in a large (N = 30 or larger) unimodel symmetric distribution are converted to Zs
1. About 2/3 of the Zs will be between -1 and +1.
2. About 95% of the Zs will be between -2 and +2.
3. More than 99% of the Zs will be between -3 and +3.
Biderman’s Handouts
Standard Scores - 3
2/6/2016
Usual and Unusual Zs
Suppose you’re walking down the street and a strange-looking man approaches you. He has a coat on. He
grabs the coat so that he can open it, to show you what’s sewn on the inside. He says, “Want to see a Zscore?” in a menacing voice.
You say, “Sure.” He opens the coat, and there is the number, 36.4, sewn on the inside pocket.
Could that be a Z? Is it probably a Z?
The answer is: it could be, but it’s probably not a Z, because in any collection of scores, 99% of the Zs
will be between -3 and +3. So 36.4 would be a VERY unusual Z.
Of course, there are people who are so unusually talented that if that talent were converted to a Z, the Z
would be very far from 0.
Michael Jordan’s ability to play basketball when he was in his prime: Z = 5, perhaps
Mozart’s ability to compose: Z = +5, perhaps.
Einstein’s mathematical ability: Z = +5, perhaps.
Uses of Z scores
1. Comparing individual scores from different distributions, just as we did with percentile ranks above.
Am I taller than I am heavy?
Answer: Compute my Z for height, about +.75. Compute my Z for weight: about +2.5. So I’m much
heavier than I am tall.
2. As a standardized test statistic.
We conduct research and obtain an outcome.
We compute a “Z statistic” which is essentially: (Obtained outcome – Expected Outcome) / SD.
If that “Z” is close to 0, we’ll conclude that our obtained outcome was essentially what it was expected to
be.
But if the “Z” is far from 0 (like 36.4), then we conclude that the obtained outcome was “significantly
different” from what it was expected to be. More on this in the section on hypothesis testing.
Problems with Zs
1. They’re usually numbers with digits on the right of the decimal point – non-engineering/math majors’
worst nightmare.
2. About half the Zs in a US collection will be negative – non-engineering/math majors’ second worst
nightmare.
Biderman’s Handouts
Standard Scores - 4
2/6/2016
Alternatives to Z Scores . . ..
T scores
Definition of T: T = 10*Z + 50, rounded to nearest whole number.
This definition shows the relationship of T to Z, but is not actually used by people whose tests are reported
as T scores. Instead, the persons who devise the tests figure out ways to go directly from the raw score to T,
without going through Z.
Central Tendency and Variability of Ts
1. Mean of Ts = 50.
2. Standard deviation of Ts = 10.
SAT scores
Definition: SAT = 100*Z + 500 rounded to nearest whole number.
Central Tendency and Variability of SATs
1. Mean of SATs = 500.
2. Standard Deviation of SATs = 100.
IQ scores
Definition: IQ = 15*Z + 100 rounded to nearest whole number
Central Tendency and Variability of IQs
1. Mean of IQs = 100.
2. Standard Deviation of IQs = 15.
Biderman’s Handouts
Standard Scores - 5
2/6/2016
Moving between Zs, Ts, SATs and IQs.
Suppose you took a verbal achievement test and scored Z = + 0.63.
Suppose you took a math achievement test and scored T = 63.
And suppose you took an abstract reasoning achievement test and scored SAT = 563.
And finally, suppose you took an IQ test and scored 130.
In which domain do you have the highest achievement?
Put the each score on the appropriate scale below.
3 SD's below mean
3 SD's above mean
2 SD's below mean
2 SD's above mean
1 SD above mean
1 SD below mean
The mean
Z
-3
-2
-1
0
1
2
3
T
20
30
40
50
60
70
80
SAT
200
300
400
500
600
700
800
IQ
55
70
85
100
115
Biderman’s Handouts
130
Standard Scores - 6
145
2/6/2016
Score Transformations and their effects
1. Effect of adding a constant to each score in a collection
New X = Old X + Constant
Example
New X = Old X + 10
Old Xs
46
44
44
43
40
40
Constant
10
10
10
10
10
10
New Xs
56
54
54
53
50
50
Questions we might ask . . .
If I add 10 to each X, what will be the effect on the Mean? the Median? the Mode?
When a constant is added to each X, all measures of Central Tendency change by the same amount.
What will be the effect on the Standard Deviation? the Range? the Interquartile Range?
When a constant is added to each X, all measures of Variability are not changed
.
Biderman’s Handouts
Standard Scores - 7
2/6/2016
2. Effect of multiplying each score by a constant
New X = Constant * Old X
Example
New X = Constant * 5
Old Xs
46
44
44
43
40
40
Constant
5
5
5
5
5
5
New Xs
230
220
220
215
200
200
Questions we might ask . . .
If I add 10 to each X, what will be the effect on the Mean? the Median? the Mode?
When a constant is added to each X, all measures of Central Tendency change by the same amount.
What will be the effect on the Standard Deviation? the Range? the Interquartile Range?
When each X is multiplied by a constant X,
a) Standard deviation changes by the same amount
b) Range changes by the same amount
c) Interquartile Range changes by the same amount
but
d) Variance changes by the square of the amount.
Biderman’s Handouts
Standard Scores - 8
2/6/2016