standard scores · Title: standard scores.ppt Author: Rick Balkin Created Date: 2/20/2008 9:10:39...
Transcript of standard scores · Title: standard scores.ppt Author: Rick Balkin Created Date: 2/20/2008 9:10:39...
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1
Standard Scores
Richard S. Balkin, Ph.D., LPC-S, NCC
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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
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R. S. Balkin, 2008 3
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
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R. S. Balkin, 2008 4
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).
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R. S. Balkin, 2008 5
The Normal Curve
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R. S. Balkin, 2008 6
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
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R. S. Balkin, 2008 7
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
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R. S. Balkin, 2008 8
Interpretations of the normal curve
Area world
Score world
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R. S. Balkin, 2008 9
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).
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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).
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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
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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.
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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.
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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
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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
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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
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R. S. Balkin, 2008 17
Z-score
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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
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R. S. Balkin, 2008 21
T score formula
T=50+10z Always rounded to the nearest whole
number A z score of 1.27=
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R. S. Balkin, 2008 22
T score formula
T=50+10z Always rounded to the nearest whole
number A z score of 1.27= T=50+10(1.27)=
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R. S. Balkin, 2008 23
T score formula
T=50+10z Always rounded to the nearest whole
number A z score of 1.27= T=50+10(1.27)=50+12.70=62.70=63
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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…
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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 +=+!
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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
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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
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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
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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 ____ 99% confidence that the score will be
between ____ to ____
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R. S. Balkin, 2008 30
Confidence Intervals
A mean of 48 and a sd of 2.75 68% confidence that the score will be
between 45.25 to 50.75 95% confidence that the score will be
between ____ to ____ 99% confidence that the score will be
between ____ to ____
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R. S. Balkin, 2008 31
Confidence Intervals
A mean of 48 and a sd of 2.75 68% confidence that the score will be
between 45.25 to 50.75 95% confidence that the score will be
between 42.5 to 53.5 99% confidence that the score will be
between ____ to ____
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R. S. Balkin, 2008 32
Confidence Intervals
A mean of 48 and a sd of 2.75 68% confidence that the score will be
between 45.25 to 50.75 95% confidence that the score will be
between 42.5 to 53.5 99% confidence that the score will be
between 39.75 to 56.25
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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
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R. S. Balkin, 2008 34
Positive correlation
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R. S. Balkin, 2008 35
Negative correlation
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R. S. Balkin, 2008 36
Low correlation
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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
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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
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R. S. Balkin, 2008 39
Interpreting a correlationcoefficient
Using the table on p. 388 interpretthe following correlation coefficients:
+.52
-.78
+.12
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R. S. Balkin, 2008 40
Interpreting a correlationcoefficient
Using the table on p. 388 interpretthe following correlation coefficients:
+.52 Moderate
-.78 Substantial
+.12 Negligible
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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
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5 6
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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
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R. S. Balkin, 2008 44
Computing Pearson r
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R. S. Balkin, 2008 45
Computing Pearson r
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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
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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)
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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.