Teaching Explorations and Resources | Zhu Xuelin

Why n − 1 in Sample Variance?

Sat, 19 Sep 2026 00:00:00 +0000

Adapted from a student’s question in NUS ST2132.

Suppose our population consists of one million people who voted for Trump in the 2024 election. We want to know: roughly how old are they on average? And is their age distribution fairly concentrated, or is there a wide mix of younger and older voters?

In statistical terms, we are interested in the population mean $\mu$ and variance $\sigma^2$. But we can’t possibly ask all one million people, so here’s what we do:

We use the numbers calculated from our sample—say, $\bar x=45$ and $s^2=10$—to estimate the mean $\mu$ and variance $\sigma^2$ of this population of one million people. Now here comes the question:

Why is the denominator in $S^2$ equal to $5-1$? Or more generally, why define

\[ S^2=\frac{1}{\color{red}{\boldsymbol{n-1}}}\sum_{i=1}^{n}(X_i-\bar X)^2? \]

To make the picture easier to draw, let’s represent the population with 15 points (we’re not drawing a million dots!). We sample five people, as shown below:

The population’s average age $\mu$ is 39, and the ages are quite spread out. Now that we have sampled $(X_1,\ldots,X_5)$, we can naturally look at how far each person in our sample is from the true population mean:

\[ \begin{pmatrix} X_1-\mu\\ X_2-\mu\\ \vdots\\ X_5-\mu \end{pmatrix} \xrightarrow{\text{square, then average}} \frac{(X_1-\mu)^2+\cdots+(X_5-\mu)^2}{5}, \]

and use this to estimate the population variance $\sigma^2$. This makes perfect sense: if we knew $\mu$, dividing by $n$ (five here) would be absolutely fine.

But here’s the trouble: we don’t know $\mu$! It’s the mean of all one million people, after all. So a natural thought is: we’re already using $\bar X$ to estimate $\mu$, so why not replace $\mu$ with $\bar X$ here too? That gives us our first step:

\[ \frac{1}{n}\sum_{i=1}^{n}(X_i-\mu)^2 \;\xrightarrow[\text{replace }\mu\text{ with }\bar X]{\text{Revision}}\; \underset{\star}{\underline{\frac{1}{n}\sum_{i=1}^{n}(X_i-\bar X)^2}}. \]

A small problem has quietly slipped in. We wanted to measure “how far the sample is from the population’s average age $\mu$.” But $\bar X$ isn’t the population’s centre: it is chosen after seeing our sample $X_1,\ldots,X_5$. In our example, $\bar X=45$. Look back at the picture: it’s pretty close to the five sampled people, isn’t it? It makes us think, “Hmm, these people—the solid blue dots ●—don’t seem all that spread out.” Yet in this example, the population, including the hollow grey dots ○, is much more spread out than our sample.

So think of it this way: the true mean $\mu$ stays put, while the sample mean $\bar X$ “chases after our sample.” If we happen to sample a younger group today, $\bar X$ moves towards the younger ages. If we happen to sample an older group tomorrow, $\bar X$ moves towards the older ages instead. So using $\underline{\star}$ to estimate the population’s spread tends to underestimate it, on average. This brings us to the next step:

\[ \underset{\star}{\underline{\frac{1}{n}\sum_{i=1}^{n}(X_i-\bar X)^2}} \;\xrightarrow[\text{divide by }n-1\text{ instead}]{\text{Revision}}\; \frac{1}{n-1}\sum_{i=1}^{n}(X_i-\bar X)^2. \]

For our five-person sample, we go from dividing by five to dividing by four. This makes the estimate a little bigger, to offset the underestimation caused by using $\bar X$—and that answers our opening question.

A small remark. How much does $\underline{\star}$ underestimate the variance, exactly? We can describe this using the mathematical language of ST2131: expectation. With independent, identically distributed observations of finite variance, a little calculation gives:

\[ \begin{aligned} E\!\left[\underline{\star}\right] &=E\!\left[\frac{1}{n}\sum_{i=1}^{n}(X_i-\bar X)^2\right]\\ &=\frac{1}{n}\sum_{i=1}^{n}E\!\left[(X_i-\bar X)^2\right]\\ &=\frac{1}{n}\sum_{i=1}^{n}E\!\left[\big[(X_i-\mu)+(\mu-\bar X)\big]^2\right]\\ &=\cdots=\frac{n-1}{n}\times\sigma^2. \end{aligned} \]

Don’t worry—try expanding the square, treating each expression in parentheses as one piece!

This means that if we repeat the “sample just five people” experiment many times, $\underline{\star}$ averages out to $4/5=80\%$ of the true variance. If we repeat the “sample 100 people” experiment, it averages out to $99/100=99\%$. So as the sample size gets larger, dividing by $n$ or $n-1$ makes less and less difference.

Let’s try sampling!

This time, let’s sample from $N(0,10^2)$, so the true variance is 100. Choose a sample size, draw ten samples, and see where the two estimates end up on average.

Sample size 0 / 10 samples

— Population: N(0, 10²) ● Current sample

Divide by n —

Divide by n − 1 —

One column for each sample
Estimate	1st	2nd	3rd	4th	5th	6th	7th	8th	9th	10th
÷ n
÷ (n − 1)

Click Sample to draw your first sample.

How Many Cups of Coffee Can I Buy with $10?

Sat, 25 Jan 2025 00:00:00 +0000

Imagine this: I’m a coffee lover with only 10 dollars to spend. But here’s the twist—each cup of coffee costs a random price between 0 and 10 dollars. So, on average, how many cups can I actually buy?

If you’re curious about the answer, check out the video!

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Buying Coffee

For a theoretical justification (a little probability theory is needed), you can watch the video (in Chinese) below:

LaTeX & Keynote Templates for Statistics

Sat, 25 Jan 2025 00:00:00 +0000

Feel free to download these resources for study or reference. I hope they assist in your journey to understanding the language of data and science. These notes are for informational purposes only and reflect my personal understanding. While I strive for accuracy, I make no guarantees regarding the completeness or reliability of the content.

By using these notes, you acknowledge that any reliance on the material is at your own risk. I am not responsible for any errors, omissions, or outcomes resulting from the use of this information. For official guidance or advice, please refer to relevant textbooks, authorities, or professionals in the respective field.

All notes are generated using this custom $\LaTeX$ template, designed specifically for statistics students with helpful shortcuts. The videos in this section are produced with Keynote, the MacOS presentation software, using this custom template. I would be very happy if they could be helpful in your homeworks or presentations!

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An Example of Keynote Template

Statistical Theory

Lecture 0: Preliminaries in Integration and Conditioning

What Is the Sample Space in Statistics?

Wed, 13 Nov 2024 00:00:00 +0000

In statistics, we often work with independently and identically distributed (iid) samples. For instance, consider a set of observations: $ (X_1, X_2,\dots,X_{10}). $ However, what is the underlying sample space (a foundational concept from probability) associated with these samples?

Specifically, let’s take an element $ \omega\in\Omega, $ where this $ \omega $ represent an individual like Alice. How, then, could we use $ X_1 $ to denote Alice and, simultaneously, $ X_2 $ to denote another individual, say Bob?

This question once confused me for years. If you are having the same trouble, check out the video!

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Sample Space

My Teaching Philosophy: Bringing Statistics to Life

Thu, 10 Oct 2024 00:00:00 +0000

One of my favorite professors, (蔡敬衡), once said: “True discoveries or creations are not made by statisticians themselves, but by other scientists. Yet, almost all of them rely on an indispensable tool: statistics.” I like this word and believe that the power of statistics lies not only in its mathematical rigor but in its ability to help us make sense of the world. I find much joy in sharing statistical concepts with others—whether with friends, peers, roommates, or family—and I am committed to promoting the broad application of statistical methods across diverse fields.

One of the most effective ways I’ve found to teach statistics is through visualization. By turning abstract statistical concepts into visual representations, complex ideas become more intuitive and easier to grasp. Visuals can break down barriers for people unfamiliar with mathematics, allowing them to engage with and understand key statistical ideas. For example, the state space decomposition of a discrete Markov chain—a concept that can seem abstract at first—becomes far more tangible with visualization:


Math Statement	Visualized Interpretation

Similarly, we can visualize the idea of support vector machines (SVM), another widely used statistical method, to highlight how these algorithms create decision boundaries in classification problems:


Math Statement	Visualized Example

In my philosophy of teaching and using statistics, the key is to simplify without losing the depth of the subject, by making concepts more accessible through visualization and emphasizing the practical impact of statistics.