True standard deviation

The standard deviation. s° for the sample corresponds to the true standard deviation O for the whole population in the same way that the mean x of the sample corresponds to the arithmetic average [L for the whole population. Equation (9-70) can be written more compactly as... 

The relates to the faet that this is not the true standard deviation, but an estimate to measure the potential shift in the distribution. 

The in equation 4.19 relates to the fact that this is not the true standard deviation, but an estimate to measure the process shift (or drift) in the distribution over the expected duration of production. Equation 4.20 is the best estimate for the standard deviation of the distribution as determined by CA with no process shift. 

The true standard deviation Ox is expected inside the confidence interval CI(5 , ) = /Vi. .. /V with a total error probability 2 p (in connection with F and x P taken to be one-sided). 

I. A measure, symbolized by square root of the variance. Hence, it is used to describe the distribution about a mean value. For a normal distribution curve centered on some mean value, fjt, multiples of the standard deviation provides information on what percentage of the values lie within na of that mean. Thus, 68.3% of the values lie within one standard deviation of the mean, 95.5% within 2 cr, and 99.7% within 3 cr. 2. The corresponding statistic, 5, used to estimate the true standard deviation cr = (2(Xi - x) )/(n - 1). See Statistics (A Primer)... 

Repeating a routine analysis over and over again for a period of time (perhaps sometimes years) and assembling the results into a data set that is free of bias and determinate errors create a basis for calculating a standard deviation that approaches o, the true standard deviation. The 2o theoretically associated with 95.5% of the values (Section 4.3), or the 3o associated with 99.7% of the values then comes close to reality. If a given analysis result on a given day is then within 2o, it is a signal that "all is well" and the process or procedure is considered to be under what is called statistical control. If a process or procedure is under statistical control, then only 4.5% of the points (about 1 of every 20) would be outside the 2a limits and only... 

People often ask why divide by — 1 rather than Well, the answer is a fairly technical one. It can be shown mathematically that dividing by n gives a quantity that, on average, underestimates the true standard deviation, particularly in small samples and dividing by n — 1 rather than n corrects for this underestimation. Of course for a large sample size it makes very little difference, dividing by 99 is much the same as dividing by 100. 

Remember, x and s are quantities that we calculate from our data while p. and a are theoretical quantities (parameters) that are unknown to us but nonetheless exist in the context of the broader population from which the sample (and therefore the data) is taken. If we had access to every single subject in the population then yes we could compute p. but this is never going to be the case. We can also think of p. and a as the true mean and true standard deviation respectively in the complete population. 

In addition, the standard deviation as calculated from a sample may sometimes be used as an estimate of the true standard deviation of a method or process. In these situations, tt is found that the standard deviation of a small sample tends to underestimate the true standard deviation. This bias can be compensated for by using one less than the number of observations as divisor of the sum of the squared deviations as given above. 

The range and standard deviation are two simple statistics for expressing the amount of variability or scatter of the four potencies. The range is easier to compute because it is the difference between the maximum and minimum values. Using R for the range, R = 46.2 - 43.8 = 2.4. The standard deviation, symbolized by s, is not as easy to compute, and its formula is presented later. For the four potency values, s = 1.021. The value s is an estimate of variability, of the assay-measuring process. The true standard deviation is noted by a. 

There are a variety of ways in which to characterize the variability of data. One of the more useful quantities, although not the simplest, is the true standard deviation o, which is defined as the square root of the sum of the squares of the deviations of the data points from the true mean divided by the number of... 

Units for the standard deviation are the same as for the individual observation. Because the true mean is practically never known, the true standard deviation is generally a theoretical quantity. However, o- may be approximated by the estimated standard deviation, j(jc), where... 

Squaring the true standard deviation gives a term called the true vuriunce, a2. It can be shown that the standard deviation of means a, calculated for samples taken from the total population of data, will have a true standard deviation equal to a/ n, where n is the sample size. In other words, the spread of these means is less than the spread of the overall data around the group mean. 

In the same manner, squaring the estimated standard deviation results in an estimated vuriunce s2(x). An estimated standard deviation of the mean has the same relation to the estimated standard deviation of the population as the true standard deviation of the mean, that is, x(x) is given by s(x)/ fn. ... 

For a normal distribution, the true standard deviation is approximately 1.25 times the mean deviation. 

Usefulness of the normal distribution curve lies in the fact that from two parameters, the true mean p. and the true standard deviation true mean determines the value on which the bell-shaped curve is centered, and most probability concentrated on values near the mean. It is impossible to find the exact value of the true mean from information provided by a sample. But an interval within which the true mean most likely lies can be found with a definite probability, for example, 0.95 or 0.99. The 95 percent confidence level indicates that while the true mean may or may not lie within the specified interval, the odds are 19 to 1 that it does.f Assuming a normal distribution, the 95 percent limits are x 1.96 where a is the true standard deviation of the sample mean. Thus, if a process gave results that were known to fit a normal distribution curve having a mean of 11.0 and a standard deviation of 0.1, it would be clear firm Fig. 17-1 that there is only a 5 percent chance of a result falling outside the range of 10.804 and 11.196. 

In terms of the previously mentioned normal distribution, the probability that a randomly selected observation x from a total population of data will be within so many units of the true mean p can be calculated. However, this leads to an integral which is difficult to evaluate. To overcome this difficulty, tables have been developed in terms of p Znormal distribution under study is known and assuming that the difference between the sample x and the true mean p is only the result of chance and that the individual observations are normally distributed, then a confidence interval in estimating p can be determined. This measure was referred to previously as the confidence level. 

If the true standard deviation is not known, a corresponding confidence interval can still be determined. However, this estimate must utilize the t-distri-bution instead of the Z-distribution since the t-concept includes the additional variation introduced by the estimate of standard deviation. In this case the rearranged t-equation is used. 

There is no rigorous mathematical method for combining the individual precisions, but the variances of the individual measurements may be combined. The precision, at the 95 percent confidence level, is defined as approximately twice the true standard deviation. Since the latter is the square root of the true variance, this permits evaluation of the precision for the combined operation. 

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