Unbiased Estimate Of Population Variance

Biased Versus Unbiased Estimates
of Population Parameters

Before we talked nearly biased samples, which were samples that conspicuously did non represent the population of interest. Now we are going to talk about a unlike kind of bias.

You learned earlier that one can retrieve of statistical procedures as a way of cartoon conclusions about population parameters on the basis of sample statistics. Y'all also learned in the section on annotation that the distinction between population parameters and sample statistics is so important that nosotros use dissimilar letters to refer to them. We utilize Greek letters to refer to population parameters and Roman letters to refer to sample statistics. Now we are going to pull all of those concepts together and brand a very important point about the estimation of population parameters.

We define a statistic as an unbiased estimate of a population parameter if the statistic tends to requite values that tend to be neither consistently loftier nor consistently low. They may not be exactly right, considering after all they are simply an judge, but they have no systematic source of bias. If you lot compute the sample hateful using the formula beneath, you will get an unbiased guess of the population hateful, which uses the identical formula.

Saying that the sample mean is an unbiased approximate of the population mean just means that there is no systematic distortion that will tend to go far either overestimate or underestimate the population parameter.

We run into a problem when we piece of work with the variance, although information technology is a problem that is easily fixed. The formula for the population variance is shown below. If we use that same formula for computing the sample variance, we will become a perfectly fine index of variability, which is equal to the average squared departure from the hateful. That formula is also shown below.

Unfortunately, the formula for the sample variance shown above is a biased estimate of the population variance. Information technology tends to underestimate the population variance.

Fortunately, it is possible to make up one's mind how much bias at that place is and suit the equation to right for the bias. The equation beneath, in which you divide by N-one instead of North, provides an unbiased approximate of the population variance. For that reason, it is the equation that statisticians apply when calculating the variance. After all, nigh all statistics are used to brand judgments well-nigh the population on the ground of a sample. And then it makes sense to use unbiased estimates of population parameters.

If N is pocket-size, the corporeality of bias in the biased approximate of variance equation can exist big. For instance, if N is v, the degree of bias is 25%. But as N increases, the caste of bias decreases. For instance, if N is 100, the amount of bias is merely about ane%. But whatever bias is unacceptable, then we volition always exist using the unbiased estimate of variance that divides the sum of squared differences from the mean past N-1.

The mathematics of why dividing past N-1 provides an unbiased estimate of the population variance is across the level of this text. Yet, it has to do with the fact that nosotros are forced to estimate the population mean in lodge to compute the sample variance. The sample hateful may exist an unbiased judge of the population hateful, just information technology will never be a perfect estimate of that mean. The slight error that is introduced by having to estimate the population mean is the source of the bias.

Every bit you will learn when nosotros discuss sampling distributions, the larger the sample, the more accurate our guess of the population hateful. That is why the correction for bias in the variance formula has niggling outcome when the sample size is large, but a much larger result when the sample size is pocket-sized.

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