Properties of the Normal distribution

A remarkable property of the normal distribution is that, almost regardless of the distribution of x, sample averages x will approach the gaussian distribution as n gets large. Even for relatively small values of n, of about 10, the approximation in most cases is quite close. For example, sample averages of size 10 from the uniform distribution will have essentially a gaussian distribution. 

If we plot a Normal distribution for an arbitrary mean and standard deviation, as shown in Figure 4, it ean be shown that at lcr about the mean value, the area under the frequeney eurve is approximately 68.27% of the total, and at 2cr, the area is 95.45% of the total under the eurve, and so on. This property of the Normal distribution then beeomes useful in estimating the proportion of individuals within preseribed limits. 

In this case the summation is the sum of the squares of all the differences between the individual values and the mean. The standard deviation is the square root of this sum divided by n — 1 (although some definitions of standard deviation divide by n, n — 1 is preferred for small sample numbers as it gives a less biased estimate). The standard deviation is a property of the normal distribution, and is an expression of the dispersion (spread) of this distribution. Mathematically, (roughly) 65% of the area beneath the normal distribution curve lies within 1 standard deviation of the mean. An area of 95% is encompassed by 2 standard deviations. This means that there is a 65% probability (or about a two in three chance) that the true value will lie within x Is, and a 95% chance (19 out of 20) that it will lie within x 2s. It follows that the standard deviation of a set of observations is a good measure of the likely error associated with the mean value. A quoted error of 2s around the mean is likely to capture the true value on 19 out of 20 occasions. 

Just as an aside, look back at the formula for the 95 per cent confidence interval. Where does the 1.96 come from It comes from the normal distribution 1.96 is the number of standard deviations you need to move out to, to capture 95 per cent of the values in the population. The reason we get the so-called 95 per cent coverage for the confidence interval is directly linked to this property of the normal distribution. 

A curve with the shape given by Eq. 18-2 is called a normal (or Gaussian) distribution. Usually it is denoted as p (x) where x is the spatial coordinate and a is the standard deviation which characterizes the width of the distribution along the x-axis. The mathematical definition and properties of the normal distribution are presented in Box 18.2. 

The highest point of the Normal curve occurs for the mean of the population. The properties of the Normal distribution ensure that this point is also the median value and the mode. 

This would be described as a 2.5 sigma process, depicted in Exhibit 37.3. Using the properties of the normal distribution, we would predict that 0.6 percent (derived from a standard table) of the total population of customers would experience response times in excess of 120 seconds. 

It follows from the properties of the normal distribution that the portion of the river lying within one standard deviation (cr) upstream or downstream on either side of the point of maximum chemical concentration includes 68% of the chemical mass in the river. It also follows that the chemical concentration one standard deviation from the point of maximum concentration equals the maximum concentration multiplied by 0.61 (see Fig. 2-4, lower panel). 

These properties of the normal distribution are of great importance, as they are used to calculate confidence intervals and limits (see below). 

In [9], s has replaced cr as expected, and the numbers 1.96 or 3, z-values derived from the properties of the normal distribution, are replaced by the statistic t. This varies with the sample size n, becoming larger as n decreases to take account of the greater unreliability of s as an estimator of cr. The value of t, readily obtained from statistical tables using the data for ( — 1) degrees of freedom, also depends on the confidence level required 95%, 99.7%, etc.). 

Figure 2.4 Properties of the normal distribution [i) approximately 68% of values lie within 1 a of the mean [ii] approximately 95% of values lie within 2a of the mean (iii] approximately 99.7% of values lie within +3(t of the mean.

Common properties of the normal distribution are summarised in Table 2.1, while Fig. 2.2 gives a probability plot of the normal distribution. 

A wide variety of pattern tests (also called zone rules) can be developed based on the IID and normal distribution assumptions and the properties of the normal distribution. For example, the following excerpts from the Western Electric Rules (Western Electric Company, 1956 Montgomery and Runger, 2007) indicate that the process is out of control if one or more of the following conditions occur ... 

These examples demonstrate that for complicated products or processes, 3a quality is no longer adequate, and there is no place for failure. These considerations and economic pressures have motivated the development of the six sigma approach (Pande et al., 2000). The statistical motivation for this approach is based on the properties of the normal distribution. Suppose that a product quality variable x is normally distributed, N x, a ). As indicated on the left portion of Fig. 21.7, if the product specifications are x 6a, the product will meet the specifications 99.999998% of the time. Thus, on average, there will only be two defective products for every billion produced. Now suppose that the process operation changes so that the mean value is shifted from x = x to either x = [x + 1.5a or X = [X - 1.5a, as shown on the right side of Fig. 21.7. Then the product specifications will still be satisfied 99.99966% of the time, which corresponds to 3.4 defective products per million produced. 

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