Bit More About Dispersion

All measures of dispersion get larger when the distribution of scores is more widely varied. A narrow distribution (a lot of similar scores) has a small amount of dispersion. A wide distribution (lots of different scores) has a wide distribution. The more dispersion, the more heterogeneous (dissimilar) the scores will be.

 

Range

Range is easy to calculate. It is the highest score minus the lowest score. If the highest score is 11 and the lowest score is 3, the range equals 8.

 

Mean Absolute Deviation (MAD)

As the name suggests, mean variance (or mean absolute deviation) is a measure of variation from the mean. It is the average of the absolute values of the deviations from the mean. That is, the mean is subtracted from each raw score and the resulting deviations (called “little d’s”) are averaged (ignoring whether they are positive or negative).

 

Sum of Squares

Conceptually, Sum of Squares (abbreviated SS) is an extension of mean variance. Instead of taking the absolute values of the deviations, we square the critters (deviations), and add them up.

 

Variance

Variance of a population is always SS divided by N. This is true whether it is a large population or a small one. Variance of a large sample (N is larger than 30) is also calculated by Sum of Squares divided by N. If there are 40 or 400 in the sample, variance is SS divided by N.

However, if a sample is less than 30, it is easy to underestimate the variance of the population. Consequently, it is common practice to adjust the formula for a small sample variance. If N<30, variance is SS divided by N-1. Using N-1 instead of N results is a slightly larger estimate of variance and mitigates against the problem of using a small sample.

 

Standard deviation

This measure of dispersion is calculated by taking the square-root of variance. Regardless of whether you used N or N-1 to calculate variance, standard deviation is the square-root of variance. If variance is 7.22, the standard deviation is 2.69. If variance is 8.67, the standard deviation equals 2.94.

Technically, the square-root of a population variance is called sigma and the square-root of a sample variance is called the standard deviation. As a general rule, population measures use Greek symbols and sample parameters use English letters.

 

NOW YOU CHOOSE:
    Day 3: Dispersion
    A Bit More About Dispersion
    Even More About Dispersion
        Range
        MAD
        Sum of Squares
        Variance
        Standard Deviation
    How To Calculate
        Range
        MAD
        Sum of Squares
        Variance
        Standard Deviation
    Formulas For Dispersion
    Practice Problems
    More Practice Problems
    Basic Facts About Dispersion
    Vocabulary
    Quiz 3
    Summary