Understanding the standard deviation: Both s and s represent a standard deviation: s (called in Pagano, the sample standard deviation), is the estimated standard deviation of a population based on a sample; s is the population standard deviation). The standard deviation is a measure of a set of scores. It is in the units of the measure of the individuals. It is a relatively stable measure that represents how variable the scores are, that is, how different they are from one another.
We have three basic issues associated with standard deviation
at this time.
1. What is it?1. What is the standard deviation? It is the positive square root of the variance. The variance is the average of the sum of the squared deviations from the mean. It is a measure in squared units of how far on the average each squared score deviates from the mean, a measure of the "location" of the set of scores.
2. How to compute it.
3. How to understand or interpret it.
The numerator of the variance is known as the Sum of Squares
and is often designated by SS. One would expect the denominator of the variance
to be N as the variance is an average. However due to certain mathematical
reasons, the denominator of the variance of a sample is n-1. The standard
deviation is the positive square root of the variance
2. Computation of SD was demonstrated in class. If you need more click here .
3. A z score can be computed for any individual by
taking the raw score, subtracting the mean and dividing by the standard deviation. 
, 
. If one computes z for every score in the set and plots the z scores, they
will give a distribution that is shaped like the original distribution but
has a mean of zero and a standard deviation of 1. This is essence transforms
the scores into a set of scores measured in z-score units. If the original
distribution was normal the z distribution is known as the unit normal curve.
All normal curves transformed into z-distributions, or unit normal cures
have exactly the same distribution properties. This fact allows for many
statistical computations and interpretations.
4. Although there are general principles interpreting
standard deviations and z scores, the most useful for understanding are those
that relate them to the normal curve. The standard deviation has an exact
location in relation to the mean and the distribution of scores in a normal
curve. Approximately 34% of the scores falls between the mean and the
score that is one standard deviation above the mean.
This URL gets you to
a normal distribution graph
, which you can be used to get proportions of scores between two points on
the normal curve, or probabilities of certain scores in a normal distribution.
It is by Gary McClelland of the University of Colorado. His home page for
this material is
http://psych.colorado.edu/~mcclella/java/zcalc.html
. Playing with this Java Applet should help understand the standard deviation
in relation to the normal curve. It gives the z scores and proportions between
two score values for any specified normal curve.
This Applet
, which is a little more interactive shows the same information. As does
this one
. This applet allows you to get numbers in any direction. This
last applet
duplicates the z-table in the back of the text book.
Important note: Although these applets can do many computations for you. You will be required to do some of them on the exams without the applets around. You must know how to compute all of the relations between Raw score, z-score, probabilities, percentiles, and number of individuals scoring above, below and between values. If you know the mean, standard deviation and number of individual scores involved, then given any specific value for an individual you should be able to derive the others.
Scoring systems: There are several standard
scoring systems based on z scores. T scoring has a mean of 50 and a standard
deviation of 10. Each score is T=50+10z. SAT and GRE scoring is originally
based on a mean of 500 and a standard deviation of 100. SAT =500+100z. Wechsler
IQ has a mean of 100 and a standard deviation of 15. Stanford Binet IQ has
a mean of 100 and a standard deviation of 16.
The reason
that this kind of scoring is used rather than only percentiles is that they
are based on a normal distribution which corresponds to most natural traits.
Percentiles give a rectangular distribution and misrepresents true differences
between adjacent scores. Percentile scoring does not meet therequirements
of interval scaling.