Standardizing Sums

Standardizing sums by dividing them by the sample size is common in statistics, as later chapters will continue to demonstrate. To understand the need for this type of division, imagine two samples that each sum to 100. The first sample contains 10 cases and the second one contains 50. Calculate the mean for each of these samples, and explain why they are different even though the sums are the same.

There is a bit of a hiccup, however: The variance formula for samples tends to produce an estimate of the population variance that is downwardly biased (i.e., too small) because samples are littler than populations and therefore typically contain less variability. This problem is especially evident in very small samples, such as when N < 50.

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The way to correct for this bias in sample-based estimates of population variances is to subtract 1.00 from the sample size in the formula for the variance. Reducing the sample size by 1.00 shrinks the denominator and thereby slightly increases the number you get when you perform the division. The sample variance is

symbolized s2, whereas the population variance is symbolized σ2, which is a lowercase Greek sigma. The reason for the exponent is that, as described earlier, the variance is made up of squared deviation scores. The

symbol s2 communicates the squared nature of this statistic.

We can now assemble the entire s2 formula and compute the variance of the juvenile homicide arrest data. The formula for the sample variance is

Plugging in the numbers and solving yields

The variance of the juvenile homicide arrest data is 44.86. This is the average squared deviation from the mean in this data set.

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