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Recall that with a normal distribution, 95% of the distribution is within 1.96 standard deviations of the mean. Using the t distribution, if you have a sample size of only 5, 95% of the area is within 2.78 standard deviations of the mean.Therefore, the standard error of the mean would be multiplied by 2.78 rather than 1.96. The values of t to be used in a confidence interval can be looked up in a table of the t distribution.he value of 1.96 is based on the fact that 95% of the area of a normal distribution is within 1.96 standard deviations of the mean; 12 is the standard error of the mean. Figure 1. The sampling distribution of the mean for N=9. The middle 95% of the distribution is shaded.
Figure 1. Normal distribution with a mean of 50 and standard deviation of 10. 68% of the area is within one standard deviation (10) of the mean (50). Figure 2 shows a normal distribution with a mean of 100 and a standard deviation of 20.As in Figure 1, 68% of the distribution is within one standard deviation of the mean.Figure 2.igure 3 shows a normal distribution with a mean of 75 and a standard deviation of 10. The shaded area contains 95% of the area and extends from 55.4 to 94.6. For all normal distributions, 95% of the area is within 1.96 standard deviations of the mean.