proximations : two thirds of data lie within one standard deviation of the mean, and 95% lie within two standard deviations.
Although calculation of both the mean and standard deviation involves division by N, both are relatively independent of N. In other words, increasing the number of data points does not systematically increase or decrease either the mean or the standard deviation. Increasing the number of data points, however, does increase the usefulness of our calculated mean and standard deviation, because it increases the reliability of inferences drawn from them.
Based on visual examination of a histogram, it may be difficult to tell whether or not the data originate from a normally distributed parent population. For small N such as N=20 in Figure 1, random variations can cause substantial departures from the bell curve of Figure 3, and only the coarsest binning interval (Figure 1c) looks somewhat like a simple normal distribution. Even with N=50, the two samples in Figure 2 visually appear to be quite different, although both were drawn from the same table of random normal numbers. With N=100, the distribution begins to approximate closely the theoretical normal distribution (Figure 3) from which it was drawn. Fortunately, the mean and standard deviation are more robust; they are very similar for the samples of Figures 1 and 2.
The mean provides a much better estimate of the true value of X (the ‘true mean’ M) than do any of the individual measurements of X, because the mean averages out most of the random errors that cause differences between the individual measurements. How much better the mean is than the individual measurements depends on the dispersion (as represented by the standard deviation) and the number of measurements (N); more measurements and smaller standard deviation lead to greater accuracy of the calculated mean in estimating the true mean.
Our sample of N measurements is a subset of the parent population of potential measurements of X. We seek the value M of the parent population (the ‘true mean’). Finding the average X of our set of measurements (the ‘calculated mean’) is merely a means to that end. We are least interested in the value xi of any individual measurement, because it is affected strongly by unknown and extraneous sources of noise or scatter. If the data are normally distributed and unbiased, then the calculated mean is the most probable value of the true average of the parent population. Thus the mean is an extremely important quantity to calculate. Of course, if the data are biased such as would occur with a distorted yardstick, then our estimate of the true average is also biased. We will return later to the effects of a non-normal distribution.
Just as one can determine the mean and standard deviation of a set of N measurements, one can
imagine undertaking several groups of N measurements and then calculating a grand mean and
standard deviation of these groups. This grand mean would be closer to the true mean than most of
the individual means would be, and the scatter of the several group means would be smaller than the
scatter of the individual measurements. The standard deviation of the mean (σ X), also called the standard error of the mean, is: σ X = σ/N0.5. Note that unlike a sample standard deviation, the
standard deviation of the mean does decrease with increasing N. This standard deviation of the
mean has three far-reaching but underutilized applications: weighted averages, confidence limits for
the true mean, and determining how many measurements one should make.
