Chapter 4 Part 1

Translating to and from Z scores, the standard error of the mean, and confidence intervals around muT

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Where we have been: Z scores

The proportion above or below the score.The percentile rank equivalent.The proportion of scores between two Z

scores. The expected frequency of scores between

two Z scores

If you know the proportion from the mean to the score, then you can easily calculate:

Concepts behind Z scores Z scores represent standard deviations above and

below the mean.

Positive Z scores are scores higher than the mean. Negative Z scores are scores lower than the mean.

If you know the mean and standard deviation of a population, then you can always convert a raw score to a Z score.

If you know a Z score, the Z table will show you the proportion of the population between the mean and that Z score.

Synonyms

Scores above the mean can be called positive Z scores or high scores. The terms are synonymous.

Similarly, scores below the mean can be called negative Z scores or low scores. The terms are synonymous.

Raw scores to Z scores

Z =score - mean

standard deviation

If we know the mean and standard deviation of a population (mu and sigma), any score in that population can be translated into a Z score:

=X -

Z scores to other scores

Conversely, as long as you know the mean and standard deviation of a population, any Z score can be translated into equivalent score in that population.

Score = + ( Z * )

Calculating z scores

Z =score - mean

standard deviation

What is the Z score for someone 6’ tall, if the mean is 5’8” and the standard deviation is 3 inches?

Z =6’ - 5’8”

3”

=72 - 68

3 =

4

3 = 1.33

2100

2080 22802030 2330

Production

Frequency

units

2180

What is the Z score for a daily production of 2100, given a mean of 2180 units and a standard deviation of 50 units?

Z score = ( 2100 - 2180) / 50

3 2 1 0 1 2 3Standard

deviations

= -80 / 50

= -1.60

22302130

If you know a Z score, you can determine theoretical relative frequencies and expected frequencies using the Z table.You often start with raw or scale scores

and have to convert them to Z scores.Scale scores are public relations versions

of Z scores. They are Z scores with different means and standard deviations.

Concepts behind Scale Scores

Scale scores are Z scores expressed in a different way.

In a sense, the most basic scale score is the Z score itself, with mu = 0.00 and sigma = 1.00.

Raw scores can be converted to Z scores, which in turn can be converted to other scale scores.

And other scale scores can be converted to Z scores, that in turn can be converted to raw scores.

You need to memorize these scale scores

Z scores have been standardized so that they always have a mean of 0.00 and a standard deviation of 1.00.

Other scales that are Z scores in disguise use other means and standard deviations for historic or public relations reasons. Three frequently used scales are:

IQ - =100; = 15

SAT/GRE - =500; = 100

Normal scores - =50; = 10

200 800

470

300 700

For example: To solve the problem below, convert an SAT Score of 470 to a Z score, then use the Z table as usual.

Frequency

score

500

What percentage of test takers obtain a verbal score of470 or less, given a mean of 500 and a standard deviation of 100?

Z score = ( 470 - 500) / 100

3 2 1 0 1 2 3Standard

deviations

= -30 / 100

= -0.30

600400

Proportion mu to Z for Z score of -.30 = .1179

Proportion at/below score

= .5000 - .1179

= . 3821 = 38.21%

SAT to percentile – first transform to a Z scores

If a person scores 592 on the SATs, what percentile is she at?

Proportion mu to Z = .3212

592 500 92 100 0.92

SAT (X-) (X-)/

Percentile = (.5000 + .3212) * 100 = 82.12 = 82nd

Convert to IQ scores to Z scores to find the proportion of scores between two IQ scores.

IQ scores have mu = 100 and sigma = 15. What proportion of the scores falls between 85 and 115?

Z score = (85 - 100) / 15 = -15 / 15 = -1.00

Z score = (115 - 100) / 15 = 15 / 15 = 1.00

Proportion = .3413 + .3413 = .6826

What proportion of the scores falls between 95 and 110?Z score = (95 - 100) / 15 = -5 / 15 = -0.33

Z score = (110 - 100) / 15 = 10 / 15 = 0.67

Proportion = .2486 + .1293 = .3779

NOTICE: Equal sized intervals, close to and further from the mean: More scores close to the mean!

Given mu = 100 and sigma = 15, what proportion of the population falls between 95 and 105?

Z score = (95 - 100) / 15 = -5 / 15 = -.33

Z score = (115 - 100) / 15 = 5 / 15 = .33

Proportion = .1293 + .1293 = .2586

What proportion of the population falls between 105 and 115?Z score = (105 - 100) / 15 = 5 / 15 = 0.33

Z score = (115 - 100) / 15 = 105/ 15 = 1.00

Proportion = ..3413 - .1293 = .2120

Percentile equivalents of IQ scores: first translate to Z scores

Convert IQ scores of 120 & 80 to percentiles.

120 100 20.0 15 1.33 80 100 -20.0 15 -1.33mu-Z = .4082, .5000 + .4082 = .9082 = 91st percentile, Similarly 80 = .5000 - .4082 = 9th percentile

X (X-) (X-)/

Convert an IQ score of 100 to a percentile.An IQ of 100 is right at the mean and that’s the 50th percentile.

Going the other way – Z scores to scale scores

Remember:Score = + ( Z * )

Convert Z scores to IQ scores: Individual scale scores get rounded to nearest integer.

Z (Z*) IQ= + (Z * )

+2.67

-.060 15 -9.00 100 91

+2.67 15 40.05 +2.67 15 +2.67 15 40.05 100 +2.67 15 40.05 100 140

You try one. Let’s use normal scores (mu=50, sigma = 10).

Translate a Z score of –2.67 to a normal score.

Convert Z scores to normal scores: Individual scale scores get rounded to nearest integer.

Z (Z*) Score= + (Z * )

-2.67 10 –26.7 50 23

Tougher problems – like online quiz or midterm

If someone scores at the 58th percentile on the verbal part of the SAT, what is your best estimate of her SAT score?

Percentile to scale scoreIf someone scores at the 58th percentile on the SAT-verbal, what SAT-verbal score did he receive?

Look at Column 2 of the Z table on page 54. Closest Z score for area of .0800 is 0.20

58th Percentile is above the mean. This will be a positive Z score. The mean is the 50th percentile. So the 58th percentile is 8% or a proportion of .0800 above mu. So we have to find the Z score that gives us a proportion of .0800 of the scores between mu and Z.

0.20 100 20 500 520

Z (Z*) SAT= + (Z * )

Note that the only hard part is finding the Z score.

First, ask how far and which direction you are from mean of the scale score.

Second, use that percentage, expressed as a proportion to enter the Z table, looking in Column 2, 4, or 6 for a similar proportion.

Third, choose the Z score closest to that value

Finally, convert that to a scale score, (remembering on which side of the mean you are).

Slightly tougher –below the mean

Percentile to scale scoreIf someone scores at the 38th percentile on the SAT-verbal, what SAT-verbal score did he receive?

Look at Column 2 of the Z table on page 54. Closest Z score for area of .1200 is 0.31. Z is negative

38th percentile is below the mean. This will be a negative Z score. The mean is the 50th percentile. So the 38th percentile is 12% or a proportion of .1200 below mu. So we have to find the Z score that gives us a proportion of .1200 of the scores between mu and Z.

-0.31 100 -31 500 469

Z (Z*) SAT= + (Z * )

Double translations

35 25.00 10.00 6.00 1.67

On the verbal portion of the Wechsler IQ test, John scores 35 correct responses. The mean on this part of the IQ test is 25.00 and the standard deviation is 6.00. What is John’s verbal IQ score?

Raw (X- ) Scale Scale Scale

score (raw) (raw) Z score

6.00 1.67 100 15 125

IQ score = 100 + (1.67 * 15) = 125

Z score = 10.00 / 6.00 = 1.67