Illuminating Statistical Analysis Using Scenarios and Simulations

21Variance of Evidence

Statistical Scenario—Variance of Evidence

When variance decreases, will the sample mean distribution get more compact like it did with sample proportions?

The original histogram, reproduced in Figure 21.1, shows the distribution of random sample means for a sample size of 50 when data values are 1–7.

Figure 21.1

Figure 21.2 is a histogram of sample means with random samples of the same size but the data only take on the values 3–5. The resulting distribution is more compact.

Figure 21.2

Analogous to what we saw with sample proportions:

When variance is lower, sample means tend to be closer to the population mean.

Lower variance → less uncertainty

Recall that a key statistic to summarize variety is appropriately named the variance. And for a sample it is sample variance. Expressed in words, the sample variance of scaled data is the average squared distance of the sample values from the sample mean. It captures how spread out a sample's data values are.

To get the sample variance, we do the following: For each sample value, subtract the sample mean from it and square the difference; add the result to a running total. Note that we have to square the distances to make them all positive, otherwise the positive and negative distances would ...