Estimated Population Variance Calculator

This calculator estimates the population variance from a sample of numbers. You enter your sample values, and it returns the estimated population variance (the average squared deviation from the mean, using n−1 in the denominator). It also shows the sample standard deviation, which is the square root of that variance and is often the number people actually report.

Formula

The estimated population variance is written as s² (or σ̂²) and is calculated with the formula:

s² = Σ (xᵢ − x̄)² / (n − 1)

Where:

• xáµ¢ = each individual value in the sample
• xÌ„ = the sample mean (average of all values)
• n = the number of values in the sample
• Σ = sum over all values

The result is in the squared units of your data. If your data is in meters, the variance is in square meters. The standard deviation (√s²) is in the original units, which is why it is often preferred for reporting.

Worked Example

Using the default values from the calculator: 4, 8, 15, 16, 23, 42.

1. Find the mean: (4 + 8 + 15 + 16 + 23 + 42) / 6 = 108 / 6 = 18
2. Subtract the mean from each value and square the result:
   (4−18)² = 196, (8−18)² = 100, (15−18)² = 9, (16−18)² = 4, (23−18)² = 25, (42−18)² = 576
3. Sum the squared differences: 196 + 100 + 9 + 4 + 25 + 576 = 910
4. Divide by (n − 1) = 5: 910 / 5 = 182

So the estimated population variance is 182, and the estimated standard deviation is √182 ≈ 13.49. The calculator rounds to two decimal places, giving 182.00 and 13.49.

Why Use n−1 Instead of n?

This is the most common question about variance. If you divide by n, you get the variance of the sample itself. But if you intend to estimate the variance of the larger population from which the sample was drawn, dividing by n gives a value that is on average too small. This bias is called the sample variance bias.

Using n−1 corrects for the fact that you have already used the data to estimate the mean. Once you know the mean, only n−1 of the deviations are “free� to vary — the last one is determined by the others. This is why n−1 is called the degrees of freedom. The correction makes s² an unbiased estimator of the population variance: if you took many random samples and averaged their s² values, the average would approach the true population variance.

For a concrete example: with the sample 2, 4, 6, the mean is 4. Dividing by n gives (4+0+4)/3 = 2.67. Dividing by n−1 gives (4+0+4)/2 = 4. If the true population variance were 4, the first method would underestimate it on average.

When to Use This Calculator vs. the Population Variance

Use the estimated population variance (this calculator) when your data is a sample — a subset of a larger group — and you want to infer the spread of the whole group. Typical examples:

• Measuring the heights of 30 randomly chosen adults to estimate the variance of all adults in a city
• Testing the weights of 10 batches of a product to estimate variability across all batches
• Recording daily temperatures for one week to estimate the variance of the whole month

Use the population variance (dividing by n) only when you have every single member of the group. Examples: all students in a class, all measurements from a complete census, or all products from a single production run that you fully inspected. In those cases, there is no estimation — you are describing the data you already have.

Many textbooks call the sample-based version “sample variance� and the complete-data version “population variance.� The key is not the word “sample� but whether you are estimating or describing.

Common Mistakes and How to Avoid Them

• Dividing by n instead of n−1: If you are estimating a population parameter, use n−1. If you are describing the sample itself, use n. The calculator uses n−1 because the page title asks for an estimate.
• Mixing up variance and standard deviation: Variance is in squared units; standard deviation is in the original units. If your data is in dollars, variance is in dollars², which has no intuitive meaning.
• Using the wrong mean: The formula requires the arithmetic mean, not the median or mode. A single typo in the mean changes every squared deviation.
• Rounding too early: If you round the mean to a whole number before computing deviations, your variance will be noticeably off. Keep full precision until the final step.
• Ignoring outliers: Variance is extremely sensitive to extreme values because deviations are squared. One outlier can dominate the result. Check your data for entry errors before trusting the output.

Limits of the Formula

The formula assumes your sample is representative of the population. If the sample is biased (e.g., all values from one subgroup), the estimate will be wrong regardless of the math. It also assumes the data is numerical and measured on an interval or ratio scale — you cannot compute variance for categorical data like colors or names.

For small samples (n < 10), the estimate can be quite unstable. The variance of a variance is large when n is small. For example, with n=3, the estimate can easily be double or half the true value. With n=30 or more, the estimate becomes much more reliable.

The formula also assumes the population variance is finite. Some heavy-tailed distributions (like the Cauchy distribution) have infinite variance, in which case this formula will not converge to anything meaningful no matter how large the sample.

Frequently Asked Questions

What is the difference between sample variance and estimated population variance?

They are the same thing. “Sample variance� usually means the variance computed from a sample using n−1, which is an estimate of the population variance. Some textbooks call it “sample variance� because it is computed from a sample, and “population variance� when dividing by n. The distinction is whether you are estimating (use n−1) or describing a complete set (use n).

Why is variance squared? Why not just use the average distance from the mean?

The average absolute deviation (mean absolute deviation) exists, but variance is preferred because it is mathematically tractable. Squaring makes the function differentiable, which is essential for calculus-based statistics like regression and ANOVA. It also gives more weight to large deviations, which is often desirable. The standard deviation (√variance) brings the units back to the original scale.

Can variance be negative?

No. Variance is the sum of squared deviations divided by a positive number. Since every squared deviation is zero or positive, the variance is always zero or positive. A variance of zero means all values are identical. Negative variance is impossible.

What does a large variance mean in practice?

A large variance means the data points are spread widely around the mean. For example, test scores with variance 400 (standard deviation 20) are much more spread out than scores with variance 100 (standard deviation 10). In the first case, most students are within 20 points of the mean; in the second, within 10 points. It does not tell you whether the spread is “good� or “bad� — that depends on context.

Should I use the population or sample formula for my data?

Ask yourself: do I have every possible value, or only a subset? If you have all values (e.g., all students in your class), use the population formula (divide by n). If you have a subset and want to infer about the larger group, use the sample formula (divide by n−1). When in doubt, most real-world data is a sample, so the n−1 version is the safer default.

What if I only have the sum and count, not the individual values?

You cannot compute variance from just the sum and count — you also need the sum of squares (Σx²). The formula can be rewritten as s² = (Σx² − (Σx)²/n) / (n−1). If you have Σx², you can use that. If not, you need the raw data or a frequency table.

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in list | Sample values | 4, 8, 15, 16, 23, 42 | 0
out var_est | Estimated population variance | - | pvar | 2
out sd_est | Estimated population standard deviation | - | sqrt(pvar) | 2
out n | Sample size | - | count | 0