Reliability of s as a Measure of Precision

We describe several statistical tests that are used to test hypotheses, to produce confidence intervals for results, and to reject outlying data points. Most of these tests are based on sample standard deviations. The probability that these statistical tests provide correct results increases as the reliability of s becomes greater.

As N in Equation 1.1 increases, s becomes a better estimator of the population standard deviation, σ.

               (1.1)

Table 1.1  

When N is greater than about 20, s is usually a good estimator of σ, and these quantities can be assumed to be identical for most purposes. For example, if the 50 measurements in Table 1.1 are divided into 10 subgroups of 5 each, the value of s varies widely from one subgroup to another (0.0023 to 0.0079 mL) even though the average of the computed values of s is that of the entire set (0.0056 mL).

In contrast, the computed values of s for two subsets of 25 each are nearly identical (0.0054 and 0.0058 mL). The rapid improvement in the reliability of s with increases in N makes it feasible to obtain a good approximation of s when the method of measurement is not excessively time consuming and when an adequate supply of sample is available.

For example, if the pH of numerous solutions is to be measured in the course of an investigation, it is useful to evaluate s in a series of preliminary experiments. This measurement is simple, requiring only that a pair of rinsed and dried electrodes be immersed in the test solution and the pH read from a scale or a display.

To determine s, 20 to 30 portions of a buffer solution of fixed pH can be measured with all steps of the procedure being followed exactly. Normally, it is safe to assume that the random error in this test is the same as that in subsequent measurements. The value of s calculated from Equation 1.1 is a good estimator of the population value, σ.