The formulas on the previous page for variance and standard deviation are appropriate when we have all the data we are interested in. That is not usually the case. For example, suppose we wanted to know how many hours of television the average American watched each week. There is obviously no way we ask every person in the United States how many hours of television they watched. Instead, we would ask a random sample of people how many hours they watched and then use the mean of that sample of data to estimate the mean for the entire population. The mean for the entire population is usually represented by . So far, so good.

We would also want to estimate the spread of the number of hours of TV watched. The difficulty, as we will demonstrate on the next page, is that the standard deviation of the sample calculated (as on the previous page) slightly underestimates the true population standard deviation . The difficulty arises because when we calculate squared errors, we would really like to be subtracting the true population mean , instead of our sample mean . We know that the sample mean makes the sum of squared errors as small as possible for that particular sample. Hence, the true sum of squared errors if we were able to use the population mean would necessarily be slightly larger. It turns out that dividing the sum of squares by (n-1) instead of by n is exactly the correction needed to remove the bias. That is,

provides the most accurate estimate of the population variance. And the square root of variance calculated in this way provides the best estimate of the population standard deviation. So, if our goal is to estimate the true spread in the population, the above formula is the one to use.