L.1.1 Basic statistics (Cont.)

z-score (Steel and Torrie 1980, Table A.4). It is important to distinguish which

probability is of interest.

Z-scores can also be obtained from functions in statistical software packages.

For example, in SAS the PROBIT function will return a z-score for a specified

probability, and the PROBNORM function will compute the proportion of

values less than a given z-score.

Student's t. Normal deviates can only be used to make inferences when the

standard deviation is known, rather than estimated. The true population mean

(F) and standard deviation (s ) are only known if the entire population is

sampled, which is rare. In most cases samples are taken randomly from the

population, and the s calculated from those samples is only an estimate of s .

Student's t-values account for this uncertainty, but are otherwise similar to

normal deviates. For example, an investigator may want to determine the range

in which 95 percent of the values in a population should fall, based on a sample

of 20 observations from that population. If the sample consisted of the entire

population, F and s would be known with certainty, and normal deviates would

be used to estimate the desired range (as in the above paragraph). However, if

the sample represented only a small proportion of the population, t-values would

be used to estimate the desired range. The degrees of freedom for the test, which

is defined as the sample size minus one (n - 1), must be used to obtain the correct

t-value. Student t-values decrease with increasing sample size, because larger

samples provide a more precise estimate of F and s . For a probability of 95

percent, the appropriate range of t-values is -2.09 to +2.09 when n = 20 (19

degrees of freedom). In other words, 95 percent of the values in the population

should lie within the range: sample mean " 2.09 s. Note that this is wider than

the corresponding range calculated using normal deviates. As sample size

increases, t-values converge on the z-scores for the same probability.

Tables of t-values typically give the degrees of freedom (df or v) in the row

(left) margin and probabilities or percentiles in the column (top) margin.

percentiles refer to the cumulative proportion of values less than t, whereas

probabilities (also known as a in this case) refer to the proportion of values less

than -t and/or greater than +t. A two-tailed probability refers to both "tails" of

the t-distribution curve, i.e., the probability of a value either >+t or <-t. A

one-tailed probability refers to only one of the tails of the curve, e.g., the proba-

bility of a value >+t.

When using a t table, it is crucial to determine whether the table is based on

one-tailed probabilities (such as Table V in McClave and Dietrich (1979), and

Table A-2 in Kleinbaum and Kupper (1978)), or two-tailed probabilities (such as

Table A.3 of Steel and Torrie (1980)). Some tables give both (such as Table B.3

of Zar (1984)). For most applications involving t-values in this Appendix, one-

tailed probabilities are desired. The body of the table contains the t-value for

each df and percentile (or a ). The t-value for a one-tailed probability may be

found in a two-tailed table by looking up t under the column for twice the

desired one-tailed probability. For example, the one-tailed t-value for a = 0.05

and df = 20 is 1.725, and is found in a two-tailed table using the column for

a = 0.10.

Appendix L

Statistical Methods