Variance formulas (sample vs population)

Variance is one of the most important “spread” statistics in math and data analysis. It answers a simple question: How much do these numbers vary? If two datasets have the same average, the one with higher variance is the one with bigger swings. That matters in school (statistics homework), in science (measurement noise), and in finance (risk).

There are two common formulas: population variance and sample variance. They look almost identical, but they divide by a different number at the end. That small change makes a big difference in interpretation.

For a dataset of n values: x₁, x₂, …, x_n, the mean (average) is:

• Mean: x̄ = (x₁ + x₂ + … + x_n) / n

Next, compute each deviation from the mean: (x_i − x̄). If you summed deviations directly, positives and negatives would cancel out, often producing 0. That would be useless as a measure of spread. So variance squares deviations to keep everything positive and to penalize bigger gaps more strongly.

The “sum of squared deviations” is:

• SS (sum of squares): Σ(x_i − x̄)²

Here’s where population vs sample splits:

• Population variance (σ²): σ² = Σ(x_i − μ)² / n
• Sample variance (s²): s² = Σ(x_i − x̄)² / (n − 1)

In the population formula, μ is the true population mean (not just the average of your list). In the sample formula, x̄ is your sample mean, and dividing by (n − 1) is called Bessel’s correction. It slightly increases the result to correct for the fact that a sample tends to underestimate the true population variance when you use the sample mean.

Variance is in “squared units” (if your data is in dollars, variance is dollars²). That’s why standard deviation is so popular: it converts spread back into the original units.

• Standard deviation: σ = √(σ²) or s = √(s²)

In practice: use variance for formulas (like in probability and machine learning), and use standard deviation for human intuition (because it’s easier to interpret).

Worked examples (with real numbers)

Let’s walk through a classic dataset: 2, 4, 4, 4, 5, 5, 7, 9. This is common in textbooks because it produces clean steps.

• Step 1: Mean: (2+4+4+4+5+5+7+9) / 8 = 40 / 8 = 5
• Step 2: Deviations: −3, −1, −1, −1, 0, 0, 2, 4
• Step 3: Squares: 9, 1, 1, 1, 0, 0, 4, 16 → sum = 32
• Step 4: Divide by n: 32 / 8 = 4 (population variance)
• Std dev: √4 = 2

Same dataset, but treat it as a sample. Only the final division changes:

• Sum of squares: 32
• Divide by (n − 1): 32 / 7 = 4.571428…
• Sample std dev: √4.571428… ≈ 2.138

Both results describe the same spread, but the sample version is slightly larger. That’s intentional: sample variance compensates for using the sample mean rather than the true population mean.

If all numbers are identical (like 10, 10, 10, 10), then every deviation from the mean is 0. Sum of squares is 0 → variance is 0 → standard deviation is 0. That’s the “no spread” case.

What this variance calculator is doing under the hood

This page calculates variance in a browser-friendly, copy-paste-first way. Here’s the exact flow:

The calculator accepts numbers separated by commas, spaces, tabs, or new lines. It also ignores empty items. If you paste something like “3, 4, five, 6”, it will flag it, because “five” isn’t a number. That’s important for trust: a single typo can throw off variance drastically.

Once the list is clean, we compute the mean x̄ by summing all values and dividing by n. The mean is the anchor point of the variance calculation.

For every value x_i, the calculator computes a deviation from the mean (x_i − x̄), then squares it. Squaring has two benefits: it avoids negative cancellations and gives extra weight to outliers.

Next, it adds all the squared deviations to get Σ(x_i − x̄)². Then it divides:

• Population: divide by n
• Sample: divide by (n − 1)

Finally, it takes the square root of variance to compute standard deviation. This is usually the number people want to interpret (spread in original units).

The colored meter is not a scientific scale—variance doesn’t have a universal “max.” Instead, we map a quick preview using the coefficient of variation idea: std dev divided by the absolute mean (when mean isn’t near zero). This gives a rough, share-friendly “how spread out is it compared to the mean?” signal.

Everything runs locally in your browser, and “Save Result” stores a small history in localStorage so you can compare multiple datasets without creating an account.

Variance Calculator FAQs

• What’s the difference between variance and standard deviation?

  Variance is the average squared distance from the mean. Standard deviation is the square root of variance, so it’s in the same units as your original data. Most people interpret standard deviation more easily.

• When should I use sample variance vs population variance?

  Use sample variance when your dataset is a sample drawn from a larger population (surveys, experiments, batches). Use population variance only when you have the complete population. If you’re unsure, sample variance is usually correct for statistics problems.

• Why does sample variance divide by (n − 1)?

  Because using the sample mean makes the spread look slightly smaller on average. Dividing by (n − 1) corrects this bias and produces an “unbiased” estimator of population variance under common assumptions.

• Can variance be negative?

  No. Variance is based on squared values, so it’s always 0 or positive. If you ever see a negative variance, it’s a calculation or rounding error.

• Why is variance in squared units?

  Squaring deviations is what keeps them positive and emphasizes outliers. The tradeoff is the unit becomes squared. That’s why standard deviation (square root) is often preferred for interpretation.

• Does rounding affect the answer?

  The calculator computes using full precision first, then rounds for display. If you round too aggressively, you may hide important differences—especially in homework where exact steps matter. Use 4–6 decimals for safety.

• What if my mean is 0?

  Variance still works normally. The “spread meter” uses a rough normalization by mean for a share-friendly preview, and if the mean is near 0 it switches to a safer fallback so it doesn’t explode.