Variance Calculator Step by Step

This variance calculator works out the sample or population variance from a list of numbers you enter, giving you the variance value and the standard deviation. You type or paste your data set, choose whether it is a sample or a population, and the calculator returns the variance, standard deviation, and count of values.

Formula

Variance measures how spread out a set of numbers is from their average. For a population (all members of a group), the formula is:

σ² = Σ(xᵢ − μ)² / N

For a sample (a subset of a population), the formula uses N − 1 in the denominator:

s² = Σ(xᵢ − x̄)² / (n − 1)

• xáµ¢ – each individual value in the data set
• μ – population mean (average of all values)
• xÌ„ – sample mean (average of the sampled values)
• N – number of values in the population
• n – number of values in the sample
• Σ – sum over all values

The units of variance are the square of the original units. If your data is in meters, variance is in square meters. The standard deviation (square root of variance) returns to the original units, which is why it is often easier to interpret.

Worked Example

Using the default data set {4, 8, 15, 16, 23, 42} and assuming it is a sample (the default), here is the step-by-step calculation:

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 = (−14)² = 196
   • (8 − 18)² = (−10)² = 100
   • (15 − 18)² = (−3)² = 9
   • (16 − 18)² = (−2)² = 4
   • (23 − 18)² = 5² = 25
   • (42 − 18)² = 24² = 576
3. Sum the squared differences: 196 + 100 + 9 + 4 + 25 + 576 = 910
4. Divide by n − 1 (sample): 910 / 5 = 182
5. Variance = 182, standard deviation = √182 ≈ 13.4907

The calculator rounds to 4 decimal places, giving variance = 182.0000 and standard deviation = 13.4907. If you switch to population, the denominator becomes 6, giving variance ≈ 151.6667 and standard deviation ≈ 12.3150.

When to Use Sample vs Population Variance

This is the most common source of confusion. Use population variance (σ²) when your data includes every member of the group you are studying. For example, the test scores of all 30 students in a class, or the heights of all players on a team. Use sample variance (s²) when your data is a random sample from a larger group, such as 50 voters out of a city’s population, or 10 bags of chips from a production line.

The sample formula divides by n − 1 instead of n. This is called Bessel’s correction. It makes the sample variance a better estimate of the true population variance, because a sample tends to be less spread out than the whole population. If you divide by n, you would slightly underestimate the population variance on average.

When in doubt, ask whether you could, in principle, collect data from every member of the group. If not, you are dealing with a sample.

How to Read the Result

A variance of 0 means all values are identical. Larger variance means more spread. However, the variance is in squared units, which can be hard to interpret directly. For example, if your data is in dollars, variance is in square dollars – not a meaningful unit. That is why the standard deviation is often reported alongside variance. The standard deviation tells you the typical distance of a value from the mean, in the original units.

For the default sample, the mean is 18 and the standard deviation is about 13.49. This means most values are roughly 13.5 units away from the mean, though the actual distances range from 2 to 24.

Variance is sensitive to outliers. A single very large or very small value can inflate the variance dramatically. For instance, in the default data set, the value 42 contributes 576 to the sum of squared differences – more than half of the total 910. If you removed it, the sample variance would drop to about 78.5.

Common Mistakes and How to Avoid Them

• Using n instead of n − 1 for a sample: This is the most frequent error. Always check whether your data is a sample or a population before choosing the formula.
• Mixing up variance and standard deviation: Variance is the squared value; standard deviation is the square root. They are not interchangeable.
• Forgetting to square the differences: If you sum the raw differences from the mean, you always get zero. Squaring makes all values positive and gives more weight to large deviations.
• Unit errors: Variance is in squared units. If you compare variances from data in meters and centimeters, the latter will be 10,000 times larger even if the spread is identical.
• Rounding too early: Keep full precision during intermediate steps. Round only the final result to avoid accumulated errors.
• Using variance for ordinal or categorical data: Variance only makes sense for numerical data where the mean is meaningful. For categories like “red, blue, greenâ€� or rankings like “first, second, third,â€� variance is not appropriate.

Limits of the Variance Formula

Variance assumes your data is numerical and measured on an interval or ratio scale. It does not work for nominal categories or ordinal rankings. It also assumes the mean is a meaningful measure of center, which it is not for highly skewed distributions. For skewed data, the median and interquartile range might be more informative.

Variance is also sensitive to outliers, as noted above. A single extreme value can dominate the calculation. If your data has obvious outliers, consider reporting the median and IQR alongside variance, or use a robust measure like the median absolute deviation.

Finally, variance describes spread around the mean, but it does not tell you the shape of the distribution. Two data sets can have the same variance but very different distributions – one symmetric, one skewed, one bimodal. Always plot your data to understand its shape.

Frequently Asked Questions

Why is sample variance divided by n−1 instead of n?

Because a sample typically has less spread than the full population. Dividing by n would make the variance too small on average. The n−1 correction (Bessel’s correction) makes the sample variance an unbiased estimate of the population variance. In other words, if you took many random samples and calculated the sample variance each time, the average of those variances would equal the true population variance.

What is the difference between variance and standard deviation?

Variance is the average of the squared differences from the mean. Standard deviation is the square root of the variance. Standard deviation is in the same units as the original data, making it easier to interpret. For example, if heights are in centimeters, variance is in square centimeters, but standard deviation is in centimeters.

Can variance be negative?

No. Variance is the sum of squared differences divided by a positive number. Squared values are always zero or positive, so variance is always zero or positive. A variance of zero means all values are identical.

How do I calculate variance from a frequency table?

If you have values with frequencies (e.g., 5 appears 3 times, 7 appears 2 times), you multiply each squared difference by its frequency, sum those products, then divide by the total count (or n−1 for a sample). The formula becomes Σ fᵢ(xᵢ − μ)² / N, where fᵢ is the frequency of value xᵢ.

What does a high variance mean in real life?

A high variance means the values are spread widely around the mean. For example, if you are comparing the daily sales of two stores, the one with higher variance has more unpredictable sales – some days very high, some days very low. Low variance means consistent values. In finance, higher variance often means higher risk, but this calculator does not provide investment advice.

Should I use variance or standard deviation for reporting?

Most researchers report standard deviation because it is in the same units as the data. Variance is useful in mathematical contexts, such as ANOVA or when combining variances. If you are reporting to a general audience, standard deviation is usually clearer.

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list x | Data values | 4, 8, 15, 16, 23, 42
sel type | Data type | Sample=1, Population=2
out var | Variance | - | (type=1 ? (sum((x - mean(x))^2) / (count(x) - 1)) : (sum((x - mean(x))^2) / count(x))) | 4
out sd | Standard deviation | - | sqrt(var) | 4
out cnt | Count | - | count(x) | 0