L.1.1 Basic statistics (Cont.)

Standard deviation. The sample standard deviation (SD or s) is a measure

of the variation of the data around the mean. The sample variance, s2, is given

by:

Σ x 2 - ( Σx )2/n

(L-2)

n-1

Standard error of the mean. The standard error of the mean (SE, or s//n)

estimates variation among sample means rather than among individual values.

The SE is an estimate of the SD among means that would be obtained from

several samples of n observations each. Most of the statistical tests in this

manual compare means with other means (e.g., soil treatment mean with refer-

ence mean) or with a fixed standard (e.g., FDA action level). Therefore, the

"natural" or "random" variation of sample means (estimated by SE), rather than

the variation among individual observations (estimated by s), is required for the

tests.

In addition to the summary statistics above, two other statistics derived from

the normal (bell-shaped) frequency distribution are central to statistical testing

and to the tests described in this Appendix. These two statistics are normal

deviates (z-scores) and Student's t.

Normal deviates (z). Z-scores or normal deviates measure distance from the

mean in standard deviation units in a normal distribution. For example, an

observation one standard deviation greater than the mean has a z-score of 1; the

mean has a z-score of 0. Z-scores are usually associated with a cumulative

probability or proportion. For example, suppose an investigator wants to know

the proportion of values in a normal distribution less than or equal to the mean

plus one standard deviation. In this situation z = 0.84, i.e., in a normal distribu-

tion, 84 percent of values will be less than or equal to the mean plus 1 standard

deviation. Alternatively, an investigator may want to determine the z-score

associated with a specific proportion or probability. For example, he or she may

want to know the range in which 95 percent of the values in a normal distribu-

tion should fall. That range is the mean " 1.96 standard deviation (z-scores from

-1.96 to +1.96).

Tables of z-scores can be found in most statistical texts, and bear titles such

as "Standard Normal Cumulative Probabilities," "Ordinates of the Normal

Curve," or "Normal Curve Areas." Typically the z-scores are listed in the

column (top) and row (left) margins, with the column marginal value being

added to the row marginal value to obtain the z-score. The body of the table

contains the probability associated with each z-score. However, depending on

the table, that probability may refer to the proportion of all values less than the z-

score, the proportion of values falling between zero and the z-score, or the

proportion of values greater than the z-score. For example, if the z-score is 1.96,

97.5 percent of the values in a normal distribution fall below the z-score

(Kleinbaum and Kupper 1978, Table A-1), 47.5 percent fall between zero and

the z-score (Rohlf and Sokal 1981, Table 11), and 2.5 percent fall above the

Appendix L

Statistical Methods