Degrees of freedom for standard deviation of sample

would someone please explain why the degrees of freedom for a random sample is n-1 instead of n ?

I'm looking for an explanation that is intuitive and easily understood by a high school student. The short answer is that dividing by n returns a biased approximation of the population standard deviation (which is usually what we are trying to estimate from our sample.) Such a calculation for sample standard deviation will be biased low (i.e. an underestimate) relative to the population standard deviation.

Dividing by n-1 makes the sample variance an unbiased estimator, and the sample standard deviation a less biased estimator (this bias is still an issue while n is small.)

Wikipedia provides some details here.

• See also stats.stackexchange.com/q/11707/16974 for more technical explanations (which is not what the OP wanted, but for those interested in theoretical underpinnings) May 6 '13 at 1:14
• I'd suggest avoiding talking about bias in the standard deviation as a motivation; the $n-1$ denominator is related to unbiasing the variance. Consequently - by the Jensen inequality - that guarantees bias in the standard deviation. May 6 '13 at 3:31
• Thank you both for the help! I'll definitely take a look at the information linked above later this week. May 6 '13 at 4:15
• @Glen_b Fair point -- most of the (applied biostats) textbooks I've seen simply say that using n-1 gives a better estimate of variance, and leave it at that. May 6 '13 at 4:22
• Interesting though, dividing by $n$ gives a smaller mean square error $E [(s^2 - \sigma^2) ^2]$, and dividing by $n+1$ smaller still! Also, you don't get an unbiased estimate for $\sigma$ by dividing by $n-1$ (which is what $s^2$ is used for more likely) May 24 '16 at 10:30

Version 1 ("they complain that the real answer is too hard, so here's some handwaving for non-mathematically-inclined students"):

If you knew the real distribution average, you could construct a sum of squares of independent identically distributed standard normal variables $$\xi_i$$, similar to $$S^2$$, and it would've been $$\chi_n^2$$-distributed: $$\frac{(n-1)\hat{S}^2}{\sigma^2} = \sum \limits_{i=1}^{n} (\frac{\xi_i-\mu}{\sigma})^2 \sim \chi^2_n$$ with $$n$$ degrees of freedom.

However, you need to know the real distribution mean $$\mu$$ and distribution variance $$\sigma^2$$ to achieve the desired sum, and they are not available in real-life small sample situation.

As you don't know the real distribution mean $$\mu$$, you have to approximate it with sample average $$\bar{\xi}=\frac{\sum \limits_{i=1}^n \xi_i}{n}$$.

But this takes away one degree of freedom (if you know the sample mean, then only $$\xi_i$$ from $$1$$ to $$n-1$$ can take arbitrary values, but the $$n$$th has to be $$\xi_n = \hat{\xi} - \sum \limits_{i=1}^{n-1}\xi_i$$).

So your real $$S^2$$ loses one degree of freedom: $$\frac{(n-1)S^2}{\sigma^2} = \sum \limits_{i=1}^{n} (\frac{\xi_i-\bar{\xi}}{\sigma})^2 \sim \chi^2_{n-1}$$ Oh, and here's a cute kitten for you.

Version 2 ("the real answer, if you do care, as short as I can make it and with as little background required, as possible"):

1. If you had sum of squares of i.i.d. standard normal variables $$\xi_i$$, it would've been $$\chi_n^2$$-distributed: $$\sum \limits_{i=1}^{n} (\frac{\xi_i-\mu}{\sigma})^2 \sim \chi^2_n$$ with $$n$$ degrees of freedom.

2. However, you need to know the real distribution mean $$\mu$$ and distribution variance $$\sigma^2$$ to achieve the desired sum that are not available in real-life small sample situation. Your best bet is to use sample average $$\bar{\xi}=\frac{\sum \limits_{i=1}^n \xi_i}{n}$$ and sample variance $$S^2 = \frac{1}{n-1}\sum \limits_{i=1}^n (\xi_i-\bar{\xi})^2$$ instead. If you substituted sample mean for the distribution mean, notice that the squares of normal random variables in the sum $$\sum \limits_{i=1}^n (\frac{\xi_i-\bar{\xi}}{\sigma})^2$$ stop being independent (e.g. if $$n=2$$ two normal variables $$\frac{\xi_1-\hat{\xi}}{\sigma}$$ and $$\frac{\xi_2-\bar{\xi}}{\sigma}$$ are exact opposites of each other, if $$\frac{\xi_1-\bar{\xi}}{\sigma} = k$$, you know that $$\frac{\xi_2-\bar{\xi}}{\sigma} = -k$$). Thus you can no longer assume that $$\sum \limits_{i=1}^n (\frac{\xi_i-\bar{\xi}}{\sigma})^2$$ is chi-square distributed.

3. Let us find another way to achieve chi-square-distributed random variables. You can express your true sum of independent random variables through the approximate one (add and subtract the sample mean to the numerator of each term of your sum and rearrange):

$$\sum \limits_{i=1}^n \frac{(\xi_i - \mu)^2}{\sigma^2} = \sum \limits_{i=1}^n \frac{(\xi_i - \bar{\xi} + \bar{\xi} - \mu)^2}{\sigma^2} = \sum \limits_{i=1}^n (\frac{(\xi_i-\bar{\xi})^2}{\sigma^2} + \underbrace{2 \frac{(\xi_i - \bar{\xi})(\bar{\xi} - \mu)}{\sigma^2}}_{0 \text{ due to }\sum \limits_{i=1}^n (\xi_i - \bar{\xi}) = 0} + \frac{(\bar{\xi} - \mu)^2}{\sigma^2})$$

$$\sum \limits_{i=1}^n \frac{(\xi_i - \mu)^2}{\sigma^2} = (n-1)\frac{S^2}{\sigma^2} + n\frac{(\bar{\xi} - \mu)^2}{\sigma^2}$$, where $$\sum \limits_{i=1}^n \frac{(\xi_i - \mu)^2}{\sigma^2} \sim \chi^2_n$$, $$n\frac{(\bar{\xi} - \mu)^2}{\sigma^2} \sim \chi^2_1$$.

1. By Cochran's theorem $$S^2$$ is independent of $$\bar{\xi}$$, thus, $$(n-1)\frac{S^2}{\sigma^2}$$ is independent of $$n\frac{(\bar{\xi} - \mu)^2}{\sigma^2}$$. For independent random variables we can apply convolution formula to their probability density functions: $$f_{\eta+\psi}(t) = \int \limits_{s=-\infty}^{\infty}f_{\eta}(t-s)f_{\psi}(s)ds$$.

$$f_{\sum \limits_{i=1}^n \frac{(\xi_i - \mu)^2}{\sigma^2}}(t) = \int \limits_{s=-\infty}^{\infty}f_{(n-1)\frac{S^2}{\sigma^2}}(t-s) \cdot f_{n\frac{(\bar{\xi} - \mu)^2}{\sigma^2}}(s)ds$$

1. Substitute pdf of chi-square distributions with n and 1 degrees of freedom into the last formula:

$$\frac{t^{n/2-1}e^{-t/2}}{\Gamma(n/2)2^{n/2}} = \int \limits_{s=-\infty}^{\infty} f_{(n-1)\frac{S^2}{\sigma^2}}(t-s) \cdot \frac{s^{1/2-1}e^{-s/2}}{\Gamma(1/2)2^{1/2}} ds$$.

Suppose that $$f_{(n-1)\frac{S^2}{\sigma^2}}(t-s) \sim \chi_{n-1}^2$$, then $$f_{(n-1)\frac{S^2}{\sigma^2}}(t-s) = \frac{(t-s)^{(n-1)/2-1}e^{-(t-s)/2}}{\Gamma((n-1)/2)2^{(n-1)/2}}$$. Substitute it into the integral:

$$\frac{t^{n/2-1}e^{-t/2}}{\Gamma(n/2)2^{n/2}} = \int \limits_{s=-\infty}^{\infty} \frac{(t-s)^{(n-1)/2-1}e^{-(t-s)/2}}{\Gamma((n-1)/2)2^{(n-1)/2}} \cdot \frac{s^{1/2-1}e^{-s/2}}{\Gamma(1/2)2^{1/2}} ds = \frac{e^{-t/2}}{2^{n/2} \Gamma((n-1)/2) \Gamma(1/2)} \int \limits_{s=-\infty}^{\infty} (t-s)^{(n-1)/2-1}s^{1/2-1}ds =$$

$$= \frac{e^{-t/2}}{2^{n/2}\Gamma(n/2)} \int \limits_{s=-\infty}^{\infty} \frac{\Gamma(n/2)}{\Gamma((n-1)/2) \Gamma(1/2)} (t-s)^{(n-1)/2-1}s^{1/2-1}ds$$.

Do a variable substitution $$s = tu$$, $$ds = tdu$$:

$$\frac{e^{-t/2}}{2^{n/2}\Gamma(n/2)} \int \limits_{s=-\infty}^{\infty} \frac{\Gamma(n/2)}{\Gamma((n-1)/2) \Gamma(1/2)} (t-tu)^{(n-1)/2-1}(tu)^{1/2-1}tdu = \frac{t^{n/2-1}e^{-t/2}}{\Gamma(n/2)2^{n/2}} \underbrace{\int \limits_{s=-\infty}^{\infty} (1-u)^{(n-1)/2-1}u^{(1/2-1)}du}_{\beta-\text{distribution integral } =1}$$.

Thus, we've shown that $$\chi^2_{n-1}(x)$$ fits as the distribution of $$f_{(n-1)\frac{S^2}{\sigma^2}}(x)$$.

• Unfortunately I have to downvote this answer, as the mathematical complexity involved is unlikely to be accessible and easily understood by a high school student who perhaps just had their first classes in statistics. Jun 30 '21 at 9:51
• @B.Liu Thanks for your suggestion. I added a "simplified" version for students, who don't really care. I don't know a way to prove this statement without getting one's hands dirty. Note that I did my best to avoid using complicated mathematical tools, such as moment-generating functions/cumulants or characteristic functions/Fourier transform. To the best of my knowledge this proof is as simple as it gets. Jun 30 '21 at 10:29
• I have retracted my downvote, but my concern remains. I don't believe every answer requires a proof, but feel there is no problem whatsoever for opinions to diverge on that. Jul 7 '21 at 11:30