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..
Types of Standard Scores
1. Percentile based
The Percentile Rank of a score: Percentage of Scores less than or equal to a score value.
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 * ------= 90.00
10
3
Test B: 100 * ------= 30.00
10
9
Test C: 100 * ------= 90.00
10
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.
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
There is a population version
Z = (X - µ)/ σ or
And there is a sample version
Z = (X – X-bar) / SN-1or
Interpretation of the Z score
Z tells us how many standard deviations X is above or below the mean.
Selected Z’sInterpretation
+3X is 3 SDs above the mean
+2X is 2 SDs above the mean
+1X is 1 SD above the mean
0X is equal to the mean (not the median)
-1X is 1 SD below the mean
-2X is 2 SDs below the mean
-3X 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.
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 Z-score?” 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.
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.
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.
IQ557085100115130145
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 XsConstantNew Xs
461056
441054
441054
431053
401050
401050
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
.
2. Effect of multiplying each score by a constant
New X = Constant * Old X
Example
New X = Constant * 5
Old XsConstantNew Xs
465230
445220
445220
435215
405200
405200
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 - 110/3/2018