Why does standard deviation use n 1

An unbiased estimator is one whose expectation tends to the true expectation. The sample mean is an unbiased estimator. To see why:. Suppose that you have a random phenomenon. Oddly, the variance would be null with only one sample. This makes no sense. The illusion of a zero-squared-error can only be counterbalanced by dividing by the number of points minus the number of dofs. This issue is particularly sensitive when dealing with very small experimental datasets. Generally using "n" in the denominator gives smaller values than the population variance which is what we want to estimate.

This especially happens if the small samples are taken. If you are looking for an intuitive explanation, you should let your students see the reason for themselves by actually taking samples! Watch this, it precisely answers your question. There is one constraint which is that the sum of the deviations is zero. I think it's worth pointing out the connection to Bayesian estimation. You want to draw conclusions about the population. The Bayesian approach would be to evaluate the posterior predictive distribution over the sample, which is a generalized Student's T distribution the origin of the T-test.

The generalized Student's T distribution has three parameters and makes use of all three of your statistics. If you decide to throw out some information, you can further approximate your data using a two-parameter normal distribution as described in your question. From a Bayesian standpoint, you can imagine that uncertainty in the hyperparameters of the model distributions over the mean and variance cause the variance of the posterior predictive to be greater than the population variance.

I'm jumping VERY late into this, but would like to offer an answer that is possibly more intuitive than others, albeit incomplete.

The non-bold numeric cells shows the squared difference. My goodness it's getting complicated! I thought the simple answer was You just don't have enough data outside to ensure you get all the data points you need randomly. The n-1 helps expand toward the "real" standard deviation. Sign up to join this community. The best answers are voted up and rise to the top.

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Learn more. Ask Question. Asked 11 years ago. Active 10 months ago. Viewed k times. Improve this question. Tal Galili Tal Galili You ask them "why this? Watch this, it precisely answers you question. Add a comment. Active Oldest Votes. Improve this answer. Michael Lew Michael Lew In essence, the correction is n-1 rather than n-2 etc because the n-1 correction gives results that are very close to what we need.

More exact corrections are shown here: en. What if it overestimates? Show 1 more comment. Dror Atariah 2 2 silver badges 15 15 bronze badges. Why is it that the total variance of the population would be the sum of the variance of the sample from the sample mean and the variance of the sample mean itself?

How come we sum the variances? See here for intuition and proof. Show 4 more comments. I have to teach the students with the n-1 correction, so dividing in n alone is not an option. As written before me, to mention the connection to the second moment is not an option. Although to mention how the mean was already estimated thereby leaving us with less "data" for the sd - that's important. Regarding the bias of the sd - I remembered encountering it - thanks for driving that point home.

In other words, I interpreted "intuitive" in your question to mean intuitive to you. Thank you for the vote of confidence :. The loose of the degree of freedom for the estimation of the expectancy is one that I was thinking of using in class.

But combining it with some of the other answers given in this thread will be useful to me, and I hope others in the future. Show 3 more comments. You know non-mathers like us can't tell. I did say gradually. Mooncrater 2 2 gold badges 8 8 silver badges 19 19 bronze badges. Any way to sum-up the intuition, or is that not likely to be possible? I'm not sure it's really practical to use this approach with your students unless you adopt it for the entire course though.

Mark L. Stone Mark L. Stone I am unhappy to see the downvotes and can only guess that they are responding to the last sentence, which could easily be seen as attacking the O. Richard Hansen Richard Hansen 1 1 silver badge 3 3 bronze badges. Dilip Sarwate Dilip Sarwate Ben Ben B Student B Student.

Even though the equation is interesting, I don't get how it could be used to teach n-1 intuitively? But why n-1? If you knew the sample mean, and all but one of the values, you could calculate what that last value must be. Statisticians say there are n-1 degrees of freedom.

Statistics books often show two equations to compute the SD, one using n, and the other using n-1, in the denominator. Some calculators have two buttons. The n-1 equation is used in the common situation where you are analyzing a sample of data and wish to make more general conclusions.

The SD computed this way with n-1 in the denominator is your best guess for the value of the SD in the overall population. If you simply want to quantify the variation in a particular set of data, and don't plan to extrapolate to make wider conclusions, then you can compute the SD using n in the denominator.

The resulting SD is the SD of those particular values. It makes no sense to compute the SD this way if you want to estimate the SD of the population from which those points were drawn. It only makes sense to use n in the denominator when there is no sampling from a population, there is no desire to make general conclusions.

The goal of science is always to generalize, so the equation with n in the denominator should not be used. The only example I can think of where it might make sense is in quantifying the variation among exam scores. But much better would be to show a scatterplot of every score, or a frequency distribution histogram. Analyze, graph and present your scientific work easily with GraphPad Prism. No coding required.