Variance population

The values of x and s vary from sample set to sample set. However, as N increases, they may be expected to become more and more stable. Their limiting values, for very large N, are numbers characteristic of the frequency distribution, and are referred to as the population mean and the population variance, respectively. 

The fact that each sample variance is related to its own population variance means that the sample variance being used for the calculation need not come from the same population. This is a significant departure from the assumptions inherent in the z, r, and statistics. 

Population Variances Are Unequal When population variances are unequal, an approximate t quantity can be used ... 

This states that the sample standard deviation will be at least 72 percent and not more than 128 percent of the population variance 90 percent of the time. Conversely, 10 percent of the time the standard deviation will underestimate or overestimate the population standard deviation by the corresponding amount. Even for samples as large as 25, the relative reliability of a sample standard deviation is poor. 

Confidence Interval for a Variance The chi-square distribution can be used to derive a confidence interval for a population variance <7 when the parent population is normally distributed. For a 100(1 — Ot) percent confidence intei val... 

The statistical measures can be calculated using most scientific calculators, but confusion can arise if the calculator offers the choice between dividing the sum of squares by N or by W — 1 . If the object is to simply calculate the variance of a set of data, divide by N . If, on the other hand, a sample set of data is being used to estimate the properties of a supposed population, division of the sum of squares by W — r gives a better estimate of the population variance. The reason is that the sample mean is unlikely to coincide exactly with the (unknown) true population mean and so the sum of squares about the sample mean will be less than the true sum of squares about the population mean. This is compensated for by using the divisor W — 1 . Obviously, this becomes important with smaller samples. 

The component-of-varlance analysis Is based upon the premise that the total variance for a particular population of samples Is composed of the variance from each of the Identified sources of error plus an error term which Is the sample-to-sample variance. The total population variance Is usually unknown therefore. It must be estimated from a set of samples collected from the population. The total variance of this set of samples Is estimated from the summation of the sum of squares (SS) for each of the Identified components of variance plus a residual error or error SS. For example ... 

Welch BL (1937) The significance of the difference between two means when the population variances are unequal. Biometrika 29 350... 

Let 1, x2,..., xn be a random sample of N observations from an unknown distribution with mean fi and variance o2. It can be demonstrated that the sample variance V, given by equation A.8, is an unbiased estimator of the population variance a2. 

If it were possible to know the whole population of responses, then a mean and variance could be calculated for it. These descriptors are known as the population mean (p) and the population variance (o ), respectively. For a given population, there can be only one value of p and one value of o. The population mean and the population variance are usually unknown ideal descriptors of a complete population about which only partial information is available. 

Now consider a small sample (n = 9, say) drawn from an infinite population. The responses in this sample can be used to calculate the sample mean, y and the sample variance, s. It is highly improbable that the sample mean will equal exactly the population mean (yj = p), or that the sample variance will equal exactly the population variance (s = cr ). It is true that the sample mean will be approximately equal to the population mean, and that the sample variance will be approximately equal to the population variance. It is also true (as would be expected) that as the number of responses in the sample increases, the closer the sample mean approximates the population mean, and the closer the sample variance approximates the population variance. The sample mean, yy, is said to be an estimate of the population mean, p, and the sample variance, is said to be an estimate of the population variance,... 

The term r-i, is not a parameter of the model but is a single value sampled from the population of possible deviations [Natrella (1963)]. The magnitude of r-i, might be used to provide an estimate of a parameter associated with that population of residuals, the population variance of residuals, aj. The population standard deviation of residuals is a,. The estimates of these two parameters are designated s] and s respectively [Neter, Wasserman, and Kutner (1990)]. If DF, is the number of degrees of freedom associated with the residuals, then... 

Equations 5.28 and 5.30 provide a general matrix approach to the calculation of the sum of squares of residuals. This sum of squares, SS divided by its associated number of degrees of freedom, DF, is the sample estimate, s, of the population variance of residuals, CJ. ... 

We believe such simulations should be analyzed using state-population variances. This approach, after all, is insensitive to the origins of the analyzed "trajectories" and any internal time correlations or lack thereof. No method that relies explicitly or implicitly on time correlations would be appropriate. 

The criterion of mean-unbiasedness seems to be occasionally overemphasized. For example, the bias of an MLE may be mentioned in such a way as to suggest that it is an important drawback, without mention of other statistical performance criteria. Particularly for small samples, precision may be a more important consideration than bias, for purposes of an estimate that is likely to be close to the true value. It can happen that an attempt to correct bias results in lowered precision. An insistence that all estimators be UB would conflict with another valuable criterion, namely parameter invariance (Casella and Berger 1990). Consider the estimation of variance. As remarked in Sokal and Rohlf (1995), the familiar sample variance (usually denoted i ) is UB for the population variance (a ). However, the sample standard deviation (s = l is not UB for the corresponding parameter o. That unbiasedness cannot be eliminated for all transformations of a parameter simply results from the fact that the mean of a nonlinearly transformed variable does not generally equal the result of applying the transformation to the mean of the original variable. It seems that it would rarely be reasonable to argue that bias is important in one scale, and unimportant in any other scale. 

Figure 21.1 Cumulative frequency plot illustration 25th, 75th percentiles, and the interquartile range. estimate of the population variance. Population means and variances are by convention denoted by the Greek letters p and o2, respectively, while the corresponding sample parameters are denoted by X and s2.

In the analysis of variance, the comparisons are made using the a values for the one-tailed situation. In comparing two observed variances, the one-tailed test is used when we are asking whether the population variance represented by Sf is larger than that represented by Stwo-tailed test when we are asking, Are they equal ... 

The purpose of the F test is to answer the question of whether data with two different sample variances might have come from a single population.The test does not tell one what the population variance might be. Given a value for s, the population variance, a sample variance (s from n measurements) might be tested against it using a chi-square test ... 

If the population variances are considered not to be equal, the t value is calculated by... 

Now consider a small sample (n = 9, say) drawn from an infinite population. The responses in this sample can be used to calculate the sample mean, r,. and the sample variance, s2. It is highly improbable that the sample mean will equal exactly the population mean (y,=ji), or that the sample variance will equal exactly the population variance (s1 = a1). It is true that the sample mean will be approximately... 

We would have ten different values of sample variance. It can be shown that these values would have a mean value nearly equal to the population variance Ox. Similarly, the mean of the sample means will be nearly equal to the population mean [t. Strictly speaking, our ten groups will not give us exact values for Ox and p. To obtain these, we would have to take an infinite number of groups, and hence our sample would include the entire infinite population, which is defined in statistics as Glivenko s theorem [3]. 

We can now see that our sample means in the ten groups scatter around the population mean. The mean of the ten group-means is 4.58, which is close to the population mean. The two would be identical if we had an infinite number of groups. Similarly, the sample variances scatter around the population variance, and their mean of 7.69 is dose to the population variance. 

The population variance of the random variable X is defined as the expected value of the square of the difference between a value of X and the mean ... 

The expected value of the sample variance is found to be the population variance ... 

The definition of sample variance with an (n-1) in the denominator leads to an unbiased estimate of the population variance, as shown above. Sometimes the sample variance is defined as the biased variance ... 

From the reactor data given earlier, 32, 55, 58, 59, 59, 60, 63, 63, 63, 63, 67 determine an interval estimate of yield, presuming that the population variance is ax=81. 

This procedure is really equivalent to our earlier method of hypothesis testing, as an inspection of Eqs. (1.47)—(1.52) shows. In this part and the previous one, we have outlined the principles of statistical tests and estimates. In several examples, we have made tests on the mean, assuming that the population variance is known. This is rarely the case in experimental work. Usually we must use the sample variance, which we can calculate from the data. The resulting test statistic is not distributed normally, as we shall see in the next part of this chapter. 

We have also seen that X is an unbiased, efficient, consistent estimate of p, if the sample is from an underlying normal population. If the underlying population deviates substantially from normality, the mean may not be the efficient estimate and some other measure of location such as the median may be preferable. We have previously illustrated a simple test on the mean with an underlying normal population of known variance. We shall review this case briefly, applying it to tests between two means, and then proceed to tests where the population variance is unknown. 

Usually when we have collected some data and wish to use them for tests or estimations, we have no idea of the numerical value of the population variance. As a result, the tests requiring known variance cannot be used. Instead, we calculate the sample... 

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