Topic 6: Standard Scores



PSY 2010 Lecture Notes

Standard Scores

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: Start here on 9/20/12.

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.

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.

If it were an IQ, most people would know how good a score that is.

But with unfamiliar tests, 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.

Standard scores are such ways of reporting performance.

Types of Standard Scores

1. Percentile based

The Percentile Rank of a score: Percentage of Scores less than or equal to a score value.

[pic]

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

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. Those based on a linear transformation of the relationship of X to the mean.

Linear Transformation: New score = Multiplicative constant • Old score + Additive constant

a. The Z Score. A score that represents how many SDs X is above or below the mean.

b. The T-score

T = 10 * Z+ 50 rounded to the nearest whole number.

c. The SAT score

SAT = 100 * Z+ 500 rounded to the nearest whole number.

Comparison of Scales of the three types of mean/SD based measures:

[pic]

The Z-score up close and personal

Generic Form: Z = (X – Mean) / SD

There is a population version

Z = (X - µ)/ σ

And there is a sample version

Z = (X – X-bar) / SN-1

Interpretation of the Z score

Z tells us how many standard deviations X is above or below the mean.

Selected Z’s Interpretation

+3 X is 3 SDs above the mean

+2 X is 2 SDs above the mean

+1 X is 1 SD above the mean

0 X is equal to the mean (not the median)

-1 X is 1 SD below the mean

-2 X is 2 SDs below the mean

-3 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) 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, 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

Mozart’s ability to compose: Z = +6.

Einstein’s mathematical ability: Z = +8.

Uses of Z scores

1. 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.

2. Comparing individual scores from different distributions.

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.

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.

So alternative standard scores have been proposed.

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?

Z-scores and the Normal Distribution.

When we want to compute percentages of scores in the Normal Distribution, we use Z-scores.

Appendix A –

If you want to know the percentage of scores less than some specific score in a Normal distribution, you must first convert that score to a Z. Then you can obtain that percentage from Appendix A.

For example,

What percentage of persons have IQs less than 135?

Convert 135 to a Z: Z = (135 – 100) / 15 = 2.33

Look up 2.33 in Appendix A, p. 518:

So, 99.01% of IQ scores are less than 135.

What percentage of persons have IQs less than 90?

Z = (90-100)/15 = -0.67.

Look up Z=-0.67 in Appendix A.

The answer is 25.14% of IQs are less than 90.

-----------------------

100: Best possible value

50: Middle value

0: Worst possible value

Where’d the 100 and 15 come from?

See the text, p. 92, Figure 7.4.

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