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Standard Scores

Richard S. Balkin, Ph.D., LPC-S, NCC

R. S. Balkin, 2008 2

Normal Distributions

While Best and Kahn (2003) indicatedthat the normal curve does notactually exist, measures ofpopulations tend to demonstrate thisdistribution

It is based on probability—the chanceof certain events occurring


Normal Curve The curve is

symmetrical 50% of scores are

above the mean,50% below

The mean, median,and mode have thesame value

Scores clusteraround the center


Normal distribution

"68-95-99" rule One standard deviation away from the mean in either direction (red)

includes about 68% of the data values. Another standard deviation out onboth sides includes about 27% more of the data (green). The third standarddeviation out adds another 4% of the data (blue).


The Normal Curve


Interpretations of the normalcurve Percentage of total space included between the mean

and a given standard deviation (z distance from themean)

Percentage of cases or n that fall between a givenmean and standard deviation

Probability that an event will occur between the meanand a given standard deviation

Calculate the percentile rank of scores in a normaldistribution

Normalize a frequency distribution Test the significance of observed measures in an

experiment


Interpretations of the normal curve

The normal curve has 2 important pieces ofinformation We can view information related to where scores

fall using the number line at the bottom of thecurve. I refer to this as the score world It can be expressed in raw scores or standard

scores—scores expressed in standard deviationunits

We can view information related to probability,percentages, and placement under the normalcurve. I refer to this as the area world


Interpretations of the normal curve

Area world

Score world


Does the normal curve reallyexist?

“…the normal distribution does notactually exist. It is not a fact ofnature. Rather, it is a mathematicalmodel—an idealization—that can beused to represent data collected inbehavioral research (Shavelson,1996, p. 120).

R. S. Balkin, 2008 10

Does the normal curve reallyexist? Glass & Hopkins (1996):

“God loves the normal curve” (p. 80). No set of empirical observations is ever perfectly

described by the normal distribution… but an independent measure taken repeatedly

will eventually resemble a normal distribution Many variables are definitely not normally

distributed (i.e. SES)“The normal curve has a smooth, altogether

handsome countenance—a thing of beauty”(p.83).

R. S. Balkin, 2008 11

Nonnormal distributions

Positively skewed—majority of thescores are near the lower numbers

Negatively skewed—the majority ofthe scores are near the highernumbers

Bimodal distributions have two modes

R. S. Balkin, 2008 12

Positively skewed

If a test was very difficult and almost everyone in theclass did very poorly on it, the resulting distributionwould most likely be positively skewed.

In the case of a positively skewed distribution, the modeis smaller than the median, which is smaller than themean. The mode is the point on the x-axiscorresponding to the highest point, that is the score withgreatest value, or frequency. The median is the point onthe x-axis that cuts the distribution in half, such that50% of the area falls on each side. The mean is pulledby the extreme scores on the right.

R. S. Balkin, 2008 13

Negatively skewed

A negatively skewed distribution is asymmetrical andpoints in the negative direction, such as would resultwith a very easy test. On an easy test, almost allstudents would perform well and only a few would dopoorly. The order of the measures of central tendencywould be the opposite of the positively skeweddistribution, with the mean being smaller than themedian, which is smaller than the mode.

R. S. Balkin, 2008 14

Normal Curve Summary

Unimodal Symmetry Points of inflection Tails that approach but never quite

touch the horizontal axis as theydeviate from the mean

R. S. Balkin, 2008 15

Standard scores

Standard scores assume a normaldistribution They provide a method of expressing any

score in a distribution in terms of itsdistance from the mean in standarddeviation units

Z score T score

R. S. Balkin, 2008 16

Z-score

A raw score by itself is rathermeaningless.

What gives the score meaning is itsdeviation from the mean

A Z score expresses this value instandard deviation units

R. S. Balkin, 2008 17

Z-score

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Raw score

Mean

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R. S. Balkin, 2008 18

Z score formula

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R. S. Balkin, 2008 19

Z score computation

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R. S. Balkin, 2008 20

T score

Another version of a standard score Converts from Z score Avoids the use of negative numbers

and decimals

R. S. Balkin, 2008 21

T score formula

T=50+10z Always rounded to the nearest whole

number A z score of 1.27=

R. S. Balkin, 2008 22

T score formula


number A z score of 1.27= T=50+10(1.27)=

R. S. Balkin, 2008 23

T score formula


number A z score of 1.27= T=50+10(1.27)=50+12.70=62.70=63

R. S. Balkin, 2008 24

More on standard scores

Any standard score can be convertedto standard deviation units

A test with a mean of 500 andstandard deviation of 100 would be…

R. S. Balkin, 2008 25

More on standard scores

Any standard scorecan be convertedto standarddeviation units

A test with a meanof 500 andstandard deviationof 100 would be…

zx

100500100500 +=+!

R. S. Balkin, 2008 26

Calculating the distribution that fallbefore, between, or beyond the meanand standard deviation

From this point on represents above z, 16%

From this point and below represents zb, 84th percentile

za

R. S. Balkin, 2008 27

Confidence Intervals Confidence intervals provide a range of

values given error in a score For example, if the mean = 20 then we can

be 68% confidence that the score will be plus or

minus 1 sd from the mean 95% confidence that the score will be plus or

minus 2 sd from the mean 99% confidence that the score will be plus or

minus 3 sd from the mean

R. S. Balkin, 2008 28

Confidence intervals

A mean of 20 with a sd of 5 68% confidence that the score will be

between 15 to 25 95% confidence that the score will be

between 10 to 30 99% confidence that the score will be

between 5 to 35

R. S. Balkin, 2008 29

Confidence Intervals

A mean of 48 and a sd of 2.75 68% confidence that the score will be

between ____ to ____ 95% confidence that the score will be


between ____ to ____

R. S. Balkin, 2008 30



between 45.25 to 50.75 95% confidence that the score will be



R. S. Balkin, 2008 31






R. S. Balkin, 2008 32





between 39.75 to 56.25

R. S. Balkin, 2008 33

Correlation Relationship between two or more paired

variables or two or more data sets Correlation = r or p Correlations range from -1.00 (perfect

negative correlation) to +1.00 (perfectpositive correlation)

A perfect correlation indicates that forevery unit of increase (or decrease) in onevariable there is an increase (or decrease)in another variable

R. S. Balkin, 2008 34

Positive correlation

R. S. Balkin, 2008 35

Negative correlation

R. S. Balkin, 2008 36

Low correlation

R. S. Balkin, 2008 37

Types of correlations

Pearson’s Product-Moment Coefficientof correlation Known as a Pearson’s r Most commonly used

Spearman Rank order coefficient ofcorrelation Known as Spearman rho (p) Only utilized with ordinal values

R. S. Balkin, 2008 38

Interpreting a correlationcoefficient

Be aware of outliers—scores thatdiffer markedly from the rest of thesample

Look at direction (positive ornegative) and magnitude (actualnumber)

Does not imply cause and effect See the table on p. 388 for

interpreting a correlation coefficient

R. S. Balkin, 2008 39


Using the table on p. 388 interpretthe following correlation coefficients:

+.52

-.78

+.12

R. S. Balkin, 2008 40


Using the table on p. 388 interpretthe following correlation coefficients:

+.52 Moderate

-.78 Substantial

+.12 Negligible

R. S. Balkin, 2008 41

Computing a Pearson r

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R. S. Balkin, 2008 42

Computing Pearson r

X Y x y xy

9 8

5 6

3 1

8 6

5 4

2

x2

y

R. S. Balkin, 2008 43

Computing Pearson r

X Y x y xy

9 8 3 9 3 9 9

5 6 -1 1 1 1 -1

3 1 -3 9 -4 16 12

8 6 2 4 1 1 2

5 4 -1 1 -1 1 1

mean = 6 5

sum=24 sum = 28 23

2

x 2

y

R. S. Balkin, 2008 44

Computing Pearson r

23

)28)(24(=r

R. S. Balkin, 2008 45

Computing Pearson r

8872.

23

)28)(24(

=

=

r

r

R. S. Balkin, 2008 46

Correlational Designs

We use correlations to explorerelationships between two variables

We can also use a correlation topredict an outcome—

In statistics this is known as a regressionanalysis

R. S. Balkin, 2008 47

Correlational Designs For example, a correlation of =.60 means

that for every increase in X there is a .60standard deviation unit increase in Y

Correlational designs are different fromexperimental designs

In a correlational design, we explorerelationships between two or morevariables that are interval or ratio

In a correlational design we do not comparegroups; we do not have randomassignment (though we do have randomsampling)

R. S. Balkin, 2008 48

Correlational Designs For example, maybe we want to know the

relationship between self-esteem anddepression. We could use two instruments, onethat measures self-esteem and one thatmeasures depression. Then we can conduct aregression analysis and see if the scores onone instrument predict scores on the other

I would hypothesize that high scores indepression correlation with low scores in self-esteem.

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