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III-3. Distribution of standard deviations in samples from the same population. Thus one is likely to underestimate the standard devia- tion with small numbers of samples. This bias is particularly- important with fewer than 10 measurements. 5% low on the average. As the number of measure- ments available increases above ten, the form of the distribu- tion rapidly approaches normality with standard error: S. org/access_use#pd-google The fiducial limits for a standard deviation may then III. RELIABILITY OF ESTIMATES 27 be calculated in a similar manner to those of the average, with the reservation that the upper limit should be extended somewhat for standard deviations from small numbers of samples.

7 mg. of silver in a single analysis. 7 mg. of silver in a single analysis. This estimate of the odds relating to the hypo- thesis we have proposed is a test of significance. org/access_use#pd-google ity of this estimate of the odds is, however, subject to exami- II. STATISTICAL CONCEPTS 21 nation. 2 mg. 339 mg. as the standard deviation of the analy- sis. Intuitively it is obvious that these estimates based on only five measurements are not very reliable. org/access_use#pd-google consider the reliability of statistics.

34 STATISTICAL METHODS Student's t-Test for Paired Differences The results of an experiment on plasticizers for nylon is given in Table III-2. 3 probability levels not sufficiently improbable to refute the null hypothesis that there is no difference between the plasticizers in their rate of TABLE III-2 Plasticizer Analysis for Two Types of Plasticizers (initial plasticizer content = 20%) After 48 hr. at 60°C After 7 days at 50°C After 48 hr. at 70°C After 7 days at 70°C After 3 mo. Florida exposure....

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