Objectives for Unit Four
Variability, Standard Scores, and Additional Descriptive Techniques
1. Know the meaning of variability.
Variability is the characteristic of a distribution indicating how spread out or homogeneous it is.
2. Know the meaning of sum of
In statistics, "sum of squares" usually refers to the sum of the squared deviations of scores from the mean.
3. Know the characteristics of
and be able to compute the range, interquartile range, and semi-interquartile
The range is the difference between the highest score and the lowest score in the distribution. If the highest score is 10 and the lowest score is 6, the range would be 4.
The interquartile range is the distance between the 1st and 3rd quartiles (25th and 75th percentiles).
The semi-interquartile range is half the interquartile range.
4. Know the characteristics of
standard deviation and variance.
The variance is the mean of the squared deviations from the mean (sum of squares).
The standard deviation is the square root of the variance (the variance is the square of the standard deviation). It is the more common term since it is measured in the same units as the raw scores from which it is computed while the variance is reported in squared units which is more difficult to use and interpret.
5. Know how to interpret the
standard deviation in terms of the normal distribution.
Although the variance is the "average" or mean squared distance of each score the mean, the standard deviation is not the mean "deviation." There is no simple way to interpret a standard deviation that will be the same for every distribution. A rough way to interpret the standard deviation is to remember that about 2/3 (actually 68%) of the cases in a normal distribution are within one standard deviation of the mean. This number (2/3) can be used as a rough estimate of where most scores in a distribution lie unless the distribution is extremely U-shaped or rectangular. Additional interpretations related to the normal distribution are found in a later objective in this unit.
6. Know situations when each
measure of variability is appropriate.
The standard deviation is the most common measure of variability. It is the best measure of the total group as it is sensitive to each score and is measured in the units of the raw scores. It is used in the situations where the mean is appropriate.
The variance is almost never used as a descriptor of the group. It has some statistical uses that are not practical.
The range is a crude measure of variability which is only effected by changes to two scores but if one of those changes drastically, the range is also drastically changed even though the rest of the distribution is unchanged. It is valuable when it is desired to know the extreme scores to be encountered when dealing with a group of scores. It has the same type of imprecision as the mode.
The interquartile range and semi-interquartile range are compromises between the crudeness of the range and the over-sensitivity of the standard deviation. When the effect of extreme scores is to be minimized, they are useful in the same way as the median. However, they are seldom used in research reports.
7. Know the effect changes in
individual scores, adding scores, or changing all scores by a constant
amount will change measures of variability.
If any single score is changed, the standard deviation changes. If the score is moved away from the mean the standard deviation increases. Moving the score toward the mean decreases the standard deviation.
If a score is added that is far from the mean the standard deviation increases. If the added score is close to the mean, the standard deviation decreases.
If a constant value is added or subtracted to each score, all measures of variability remain the same (even though all measures of central tendency will change by the constant amount). If a constant value is multiplied times or divided into each score, all measures of variability (except the variance) are changed by the same amount.
8. Know the lower limit for measures
All measures of variability cannot be less than 0. A value of zero for any of the measures indicates that there is no variability (all scores in the distribution are the same).
9. Know the relationship between
indices of central tendency and variability.
There is no direct relationship between indices of central tendency and variability. Adding or changing one or more scores may raise the mean but it would be equally likely to increase or decrease the standard deviation.
10. Know symbols for population
and sample standard deviation and variance.
The sample standard deviation is "s" (sample variance is "s²"). The population standard deviation is "" (population variance is "²").
11. Know population/sample and
definitional/computational formula differences for standard deviation and
variance and reasons for the differences.
The formulas for the population variance and standard deviation ( and ²) compute the mean sum of squares by dividing by "N". The formulas used to estimate the population variance and standard deviation (the sample values--s and s²) compute the mean sum of squares by dividing by "N-1".
N-1 is used in the sample formulas because the formulas using N are biased (if computed from an infinite number of samples from a given population, their mean value would not equal the population value). If N-1 was used the mean of the sample standard deviation values would equal the population standard deviation.
Definition formulas for population and sample standard deviation and variance use deviations to compute the sum of squares, whereas computational formulas use raw scores. Deviation formulas are easier to understand (it is easy to see that distance from the mean is the basic unit of analysis) but they are not as good for computational purposes due to rounding errors.
12. Know how to estimate the
standard deviation of a distribution and determine which of two distributions
has the greater variability.
The standard deviation of a sample can be estimated fairly accurately (if there are no very extreme scores) by dividing the range by a number between 3-6 depending on the size of the sample according to the following table.
Sample Size SD Size Compared to Range
13. Know the central tendency
and variability statistics needed to accurately describe a distribution.
If the distribution can be assumed to be approximately normally distributed, all that is needed to describe all relevant aspects of the distribution are the mean and standard deviation.
If the distribution is much different from a normal distribution five numbers (sometimes called a 5-number display) will describe the distribution fairly accurately. The numbers are the highest score, the lowest score, and the 25th, 50th, and 75th percentiles.
14. Know how to interpret a boxplot.
A standard boxplot (box and whiskers plot) has a box with the 25th and 75th percentiles at each end and the 50th percentile as a line somewhere inside the box, with "whiskers" extending from the 25th and 75th percentiles to the lowest and highest scores.
15. Know the meaning of a standard
A standard score is a score that conveys information concerning how far a score is in terms of standard deviation units.
16. Know the advantages and disadvantages
of using standard scores compared to raw scores, percentages, and percentile
A raw score cannot be interpreted very well without knowing the possible points or the largest possible score. For example, 10 points on a test (a raw score) might be good on a 10 point quiz or poor on a 100 point test. A raw score, however, has direct meaning (8 years old for a child has meaning without the need to provide any additional context).
A percentage score is frequently difficult to interpret without knowing other scores in the distribution. For example, 80% might be considered to be good if it were the highest score in the class on a hard test but poor if it were the lowest score in the class on an easy test. A percentage score can be useful for criterion-referenced purposes when scores of other persons or other scores for the same person are not relevant or needed.
Percentile ranks allow comparison with other persons but are not suitable for research since they cannot be used in calculations.
Standard scores can be used in calculations and provide comparative information with other persons or with other scores for the same person. For research purposes they are equal to and sometimes preferred over raw scores and better than percentile ranks.
Percentile ranks are preferred to standard scores for interpreting scores for individuals since you need to have some statistical expertise to interpret standard scores. Percentile ranks can usually simply explained in a few minutes without any statistical terms.
17. Know the meaning of and uses for z scores, T scores, and other common standard scores (IQ, GRE, SAT, etc.)
A z score is a number that tells directly how far a score is from the mean of a distribution in terms of standard deviations. A z score of 2.00 (or +2.00) would be two standard deviations above the mean. A z score of -2.00 would be two standard deviations below the mean. A z score of 0.00 would be equal to the mean. The mean z score is 0.00 with a standard deviation of 1.00.
A T score is a standard score with a mean of 50 and a standard deviation of 10. A T score of 70 is two standard deviations above the mean, a T score of 30 is two standard deviations below and the mean and a T score of 50 is equal to the mean.
Many standard scores such as GRE and SAT use 500 as the mean and 100 as the standard deviation. A GRE of 600 is one standard deviation above average. A complication with GRE scores and many other similar scores is that the "norm group" upon which the standard scores are based is different from the "norm group" upon which the percentile ranks are based. For this reason a GRE of 500 is seldom comparable to a percentile rank of 50 which is what you expect since GRE scores are approximately normally distributed.
Intelligence tests use IQ scores with a mean of 100 and a standard deviation usually of either 15 or 16. For example, an IQ of 115 would be one standard deviation above average.
z scores are used by statisticians for simple communication. Other standard scores are used primarily to avoid decimals and negative values.
18. Know how to convert raw scores
to z scores and z scores to other standard scores, given the mean and standard
deviation of the distribution with numbers selected so that a calculator
would not be necessary.
For example if a student received a raw score of 13 points on a test with a mean of 16 and a standard deviation of 3, the standard scores would be: z=-1.00, T=40, IQ=84, GRE=400.
19. Know how to convert standard
scores to raw scores and z scores, given the mean and standard deviation
of the distribution with numbers selected so that a calculator would not
If a student received a T score of 70 on a test in which the mean was 40 and the standard deviation was 6, the student's z score would be +2.00 and raw score would be 52.