Why n 1 is used in sample variance

We see that the biased measure of variance is indeed biased. The average variance is lower than the true variance indicated by the dashed line , for each sample size. We also see that the unbiased variance is indeed unbiased. On average, the sample variance matches that of the population variance. The results of using the biased measure of variance reveals several clues for understanding the solution to the bias. We see that the amount of bias is larger when the sample size of the samples is smaller.

So the solution should be a function of sample size, such that the required correction will be smaller as the sample size increases. Ideally we would estimate the variance of the sample by subtracting each value from the population mean. This is where the bias comes in. In fact, the mean of a sample minimizes the sum of squared deviations from the mean. This means that the sum of deviations from the sample mean is always smaller than the sum of deviations from the population mean.

The only exception to that is when the sample mean happens to be the population mean. Below are two graphs.

In each graph I show 10 data points that represent our population. I also highlight two data points from this population, which represents our sample. In the left graph I show the deviations from the sample mean and in the right graph the deviations from the population mean.

We see that in the left graph the sum of squared deviations is much smaller than in the right graph. The sum is smaller when using the sample mean compared to using the population mean. This is true for any sample you draw from the population again, except when the sample mean happens to be the same as the population mean.

The difference is small now, but using the sample mean still results in a smaller sum compared to using the population mean. In short, the source of the bias comes from using the sample mean instead of the population mean. The sample mean is always guaranteed to be in the middle of the observed data, thereby reducing the variance, and creating an underestimation.

Now that we know that the bias is caused by using the sample mean, we can figure out how to solve the problem. Looking at the previous graphs, we see that if the sample mean is far from the population mean, the sample variance is smaller and the bias is large. So every data point we add up. So this is the i-th data point, so x sub 1 plus x sub 2 all the way to x sub capital N.

And then we divide by the total number of data points we have. Well, how do we calculate the sample mean? Well, the sample mean-- we do a very similar thing with the sample. And we denote it with a x with a bar over it. And that's going to be taking every data point in the sample, so going up to a lower case n, adding them up --so these are the sum of all the data points in our sample-- and then dividing by the number of data points that we actually had. Now, the other thing that we're trying to calculate for the population, which was a parameter, and then we'll also try to calculate it for the sample and estimate it for the population, was the variance, which was a measure of how dispersed or how much of the data points vary from the mean.

So let's write variance right over here. And how do we denote any calculate variance for a population? Well, for population, we'd say that the variance --we use a Greek letter sigma squared-- is equal to-- and you can view it as the mean of the squared distances from the population mean. But what we do is we take, for each data point, so i equal 1 all the way to n, we take that data point, subtract from it the population mean.

So if you want to calculate this, you'd want to figure this out. Well, that's one way to do it. We'll see there's other ways to do it, where you can calculate them at the same time. But the easiest or the most intuitive is to calculate this first, then for each of the data points take the data point and subtract it from that, subtract the mean from that, square it, and then divide by the total number of data points you have.

Now, we get to the interesting part-- sample variance. There's are several ways-- where when people talk about sample variance, there's several tools in their toolkits or there's several ways to calculate it. One way is the biased sample variance, the non unbiased estimator of the population variance. And that's denoted, usually denoted, by s with a subscript n. And what is the biased estimator, how we calculate it? Well, we would calculate it very similar to how we calculated the variance right over here.

But what we would do it for our sample, not our population. So for every data point in our sample --so we have n of them-- we take that data point. And from it, we subtract our sample mean. We subtract our sample mean, square it, and then divide by the number of data points that we have. But we already talked about it in the last video. How would we find-- what is our best unbiased estimate of the population variance?

This is usually what we're trying to get at. We're trying to find an unbiased estimate of the population variance. Well, in the last video, we talked about that, if we want to have an unbiased estimate --and here, in this video, I want to give you a sense of the intuition why. We would take the sum. So we're going to go through every data point in our sample. We're going to take that data point, subtract from it the sample mean, square that.

But instead of dividing by n, we divide by n minus 1. We're dividing by a smaller number. And when you divide by a smaller number, you're going to get a larger value. So this is going to be larger. This is going to be smaller. Let's think about what a larger vs. If the sample variance is larger than there is a greater chance that it captures the true population variance. Because we are trying to reveal information about a population by calculating the variance from a sample set we probably do not want to underestimate the variance.

There was a good post here on CV that will give you some good insight. Hope this helps! Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group.

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