Why We Use n-1 Instead of n When Calculating Variance

When you calculate variance, you take the squared differences from the mean, and average them out. But, unlike how we've been taught in school, when dealing with a sample, you divide the total by $(n - 1)$ elements, not $n$ . Also known as Bessel's Correction, it is known for providing a more accurate result of your mean deviation, by reducing the bias in the data. I don't know for you, but I found it puzzling, so rather than just taking it at face value, I wanted to make sure I understood the reasoning for this particularity.

The Intuition

When you calculate the sample mean deviation (variance), you’re using the data itself to define the center. This creates a bias—your sample mean is tailor-made to be as close as possible to your data points. Thus, it inherently makes deviations look smaller than they truly are.

Think of it this way:
Your sample mean is a compromiser. It's the value closest to every element of your sample, but not of the true population. It sits where it does to minimize the drama (i.e., squared differences). If you had the true population mean (which you never do), deviations from it would be larger.

So if you divide by $n - 1$ instead of $n$ , your result will be bigger, thus closer to the population mean.

Example Time

You’re estimating Pokémon card counts in a school. You sample three kids:

Friend A: 50 cards
Friend B: 60 cards
Friend C: 70 cards

Sample mean = 60. Deviations: -10, 0, +10. Squared: 100, 0, 100.

If you divide by $n$ (3):

Variance = \frac{100 + 0 + 100}{3} \approx 66.67

But this represents only the variance for your sample, i.e. three friends. In reality, say the total population is the entire school. There may be a kid with 40 cards, another with 75 cards.
Inevitably, the true population variance (if you could measure everyone) will always be higher. In this case, if you add the two other kids:

Deviations: -20, -10, 0, +10, +15. Squared: 400, 100, 0, 100, 225.

Variance = \frac{400 + 100 + 0 + 100 + 225}{5} = 165

You can see that it's quite different from the result you get when dividing the sample mean by $n$ .
Now, if you divide by $n - 1$ (2):

Variance = \frac{200}{2} = 100

Sure, it's not the real variance either, but it's definitely closer to it than the initial result.

The Takeaway

Use $n - 1$ because your sample mean is biased. It’s not about being fancy—it’s about correcting for the fact that when you’re making an estimation using a sample from a population, your mean deviation will always be smaller than the one for the entire population.