Chapter 4

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

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Where have we been

We have learned that you can describe how a distribution of numbers falls around its mean.

That description can be summary numbers, such as the population mean (mu), the variance or standard deviation (sigma2 or sigma) and how many scores there are in the population (N).

X

Equivalence of figural and tabular displaysWhen we want more detail about how

scores fall around their mean, we can use tabular displays (such as frequency distributions) or figural displays (such as histograms) as a description.

Both figures and tables can portray the tally, the way scores fall around their mean, in several ways. These included displaying absolute and relative frequencies in simple or cumulative forms.

Both theoretical and actual distributions can be displayed as figures or tables

The most important theoretical relative frequency distribution is the normal curve, also called the Z curve.

The normal curve (also called the Gaussian distribution or the normal error curve or the Z curve) is the most important theoretical frequency distribution.

Theoretical relative frequency distributions tell us how scores can be expected to fall around their mean.

As figures, theoretical relative frequency distributions show the proportion of scores that fall in a specific range around the mean.

Tabular forms of relative frequency distributions show culumative relative frequencies

The normal curve is symmetrical around the mean.

The Z table displays one side of the normal curve. It shows the cumulative proportion of scores between the mean and a specific score as that score moves further and further from the mean.

Z scores express number of standard deviations above or below the mean.

So the Z table shows the cumulative relative frequency of scores between the mean and a specific number of standard deviations above or below the mean.

Here is what it looks like, with the odd numbered columns showing distance from the mean expressed as a Z score and even numbered columns showing the cumulative relative frequency of scores between mu and Z.

The z tableThe Z table contains pairs of columns: columns of Z scores coordinated with columns of proportions from mu to Z.

The columns of proportions show the proportion of the scores that can be expected to lie between the mean and any other point on the curve. The Z table shows the cumulative relative frequencies for half the curve.

ZScore0.000.010.020.030.04

.1.9602.576

.3.904.004.505.00

Proportionmu to Z

.0000

.0040

.0080

.0120

.0160.

.4750

.4950.

.49995

.49997 .499997

.4999997

The Z table and normal curve are just different ways of presenting the same theoretical frequency distribution.

The normal curve is like a histogram whose intervals have been made very, very tiny. In fact, it is a frequency polygon, in which lines connect the midpoints of each interval.

You can make the intervals infinitely small, If you do that and connect the midpoints of those intervals, the result is a smooth line, the normal curve. If you have studied integral calculus you know precisely how this works.

If not, to keep it parallel to the Z table, you can think of it as a smoothed out histogram with one hundred divisions for each standard deviation.

Normal Curve – Basic Geography

Frequency

Measure

The mean

One standard deviation

|--------------49.87-----------------|------------------49.87------------|

|--------47.72----------|----------47.72--------|

-3.00 -2.00 -1.00 0.00 1.00 2.00 3.00Z scores

|---34.13--|--34.13---|Percentages

3 2 1 0 1 2 3Standard

deviations

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:

Z scores are the 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.

Symbols to memorize

Mu

Sigma

Standard error of the mean

Mean of a sample

X

X

Z scores are the scores!Raw scores to Z scores

Z =score - mean

standard deviation

If we know mu and sigma, any score can be translated into a Z score:

=X -

Z scores to other scores

Conversely, as long as you know mu and sigma, a Z score can be translated into any other type of score:

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

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, with preset means and standard deviations.

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

Concepts behind Scale Scores

Scale scores are raw scores expressed in a standardized way.

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 Scale scores can be converted to Z scores, those Z scores, 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 use other means and standard deviations.

Examples:

IQ - =100; = 15

SAT/GRE - =500; = 100

Normal scores - =50; = 10

To find percentiles or expected frequencies with scale scores, translate to Z scores and then use the Z table

200 800

470

300 700

For example: To solve the problem below, convert an SAT Score 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 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

PROPORTIONS BETWEEN TWO SCALE SCORES

Answer the following question without the table: Which of the following 10 point ranges will have the higher proportion of scores?1. IQ scores of 115 to 1252. IQ scores of 130 to 140

1. Without the table, the interval closer to the mean will have more of the scores.

Answer the following question with the table: What proportion of the scores fall into each of the 10 point ranges?

1. IQ scores of 115 to 1252. IQ scores of 130 to 140

Proportions

IQ scores of 115 and 125 = Z scores of 115-100/15 = 1.00 125-100/15=1.67Proportion Z1-Z2 = .4525 -.3413=.1112

IQ scores of 130 and 140 = Z scores of 130-100/15 = 2.00 140-100/15=2.67Proportion Z1-Z2 = .4962 -.4772=.0190

Now let’s compute percentile rank 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 -1.33Now do the translation from Z score to percentile rank:mu-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.

What is the percentile equivalent of a GRE score of 375?

Note: GRE is the same as SAT. mu = 500, sigma = 100

GRE score to percentile rank

Z=375-500/100 = -1.25Proportion between mean and Z = .3944We are below the mean, so to find

percentile rank subtract .3944 from .5000,

multiply by 100 and round if between the 1st & 99th percentiles.

.5000-.3944 = .1056 .1056*100 = 10.56 round to 11th 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

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 Z score 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 * )

Slightly tougher –below the mean

Percentile to Z score 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 * )

You solve one, if someone scores at the 20th percentile, what is their IQ score

Always use Z scores to translate scores

To go from a raw score to a scale score, you do the translation by turning the raw score into a Z score. Then you translate the Z score into a scale score,.

This is one part of a general rule, you can transform any kind of score to any other kind of score when you know mu and sigma of both scores and therefore can use Z scores as your translator.

Do this double translation.

On the GRE-Advanced Psychology exam, there are 225 questions.

The mean is 125.00 correct with a standard deviation of 12.00.John gets 116 correct. What is his GRE score on this test?

116 125.00 -9.00 12.00 -0.75

On the GRE-Advanced Psychology exam, there are 225 questions.The mean is 125.00 correct with a standard deviation of 12.00.John gets 116. What is his GRE score on this test?

Raw (X- ) Scale Scale Scale

score (raw) (raw) (raw) Z score

500 100 425