Variance Calculator

A variance calculator measures how far a set of numbers spreads out around its own average, by averaging the squared distance of each value from the mean. For the data set 12, 15, 11, 18, 14 the mean is 14, the sum of squared deviations is 30, the population variance is 6 and the sample variance is 7.5. This page returns both instantly, with every intermediate step shown.

Three neighboring tools answer closely related questions. The Standard Deviation Calculator gives the square root of the same figure when you want the spread back in the original units, the Average Calculator handles the mean, median and mode on their own, and the Z-Score Calculator converts one individual value into standard deviations above or below the mean. Use this page when the variance itself is what your homework, lab report or quality check asks for.

What variance actually measures

Every data set has a center and a spread. The mean describes the center. Variance describes the spread, and it does so in a very specific way: it asks how far each value sits from the mean, squares that distance so that values below the mean count just as much as values above it, and then averages those squares.

Squaring is not an arbitrary choice. If you simply averaged the raw deviations they would always cancel to exactly zero, because the mean is by definition the balance point of the data. Squaring removes the sign, and it also gives extra weight to values that sit far out, which matches the intuition that one wildly different observation makes a data set more variable than several slightly different ones.

The result is a single non-negative number. Zero means every value is identical. A small variance means the values huddle near the mean. A large variance means they are scattered. Because the deviations were squared, the variance is expressed in squared units, which is the one awkward feature of the measure and the reason the standard deviation exists.

The variance formula

Both versions share the same numerator, the sum of squared deviations from the mean. Only the divisor changes:

Here x is each individual value, μ is the population mean, x̄ is the sample mean, N is the population size and n is the sample size. There is also an algebraically identical shortcut that avoids listing the deviations one by one:

The shortcut needs only two running totals, the sum of the values and the sum of their squares, which is why it is the version built into hand calculators and spreadsheets. The calculator above shows it as an independent check on the deviation table.

Worked example: 12, 15, 11, 18, 14

Five measurements, worked all the way through. Step 1, the mean. The values add to 12 + 15 + 11 + 18 + 14 = 70, and 70 ÷ 5 = 14. Step 2, the deviations. Subtract 14 from each value. Step 3, square and add. The squared deviations total 30:

Step 4, divide. As a population, 30 ÷ 5 = 6. As a sample, 30 ÷ 4 = 7.5. Step 5, square root for the standard deviation. √6 ≈ 2.4495 and √7.5 ≈ 2.7386. Notice two details in the table: the deviations column sums to exactly zero, which is a good arithmetic check, and the shortcut formula reproduces the same numerator, since 1,010 − 70² ÷ 5 = 1,010 − 980 = 30.

Second worked example: six test scores

Take six exam scores: 46, 69, 32, 60, 52, 41. They add to 300, so the mean is 300 ÷ 6 = 50. The deviations are −4, 19, −18, 10, 2 and −9, and their squares are 16, 361, 324, 100, 4 and 81, which add to 886. Treating the six students as the whole class (a population), the variance is 886 ÷ 6 ≈ 147.6667 and the standard deviation is about 12.1518 points. Treating them as a sample of a much larger cohort, the variance is 886 ÷ 5 = 177.2 and the standard deviation is about 13.3116 points.

The shortcut confirms the numerator: the squares of the scores add to 15,886, and (300)² ÷ 6 = 90,000 ÷ 6 = 15,000, so the sum of squares is 15,886 − 15,000 = 886. Note how large the variance looks next to the standard deviation. A spread of roughly 12 points on a 100-point exam is perfectly ordinary, but stated as a variance of 147.67 squared points it sounds dramatic. That mismatch is presentation, not mathematics.

Sample and population variance side by side

The table below runs five data sets through both formulas, computed with the definitions above. The two figures always move together, and the sample value is always the larger of the two unless the spread is exactly zero.

Bessel's correction and why the gap shrinks

Dividing by n − 1 rather than n multiplies the answer by exactly N ÷ (N − 1). That factor is large for tiny data sets and almost invisible for big ones, which is why the sample-versus-population choice matters enormously in a classroom exercise and hardly at all in a data set of a few hundred rows.

With two observations the sample variance is double the population variance. With a hundred it is one percent larger, a difference that rarely changes any decision. The correction still belongs in the formula, because being unbiased on average is what makes the estimate trustworthy when many small samples are combined.

How to use this variance calculator

1. Paste your numbers. Commas, spaces, semicolons and line breaks all work, so a column copied straight out of a spreadsheet is accepted as it is. Decimals and negative values are fine.
2. Pick sample or population. Choose Sample when your figures are a subset of a larger group, and Population when they cover the entire group. If you are unsure, read the section below on which one to use.
3. Set the decimal places. Two decimals suit a report, six help when you are checking a hand calculation digit by digit. Rounding happens only on display, never inside the arithmetic.
4. Read the headline figure. The gradient card shows the variance for the mode you selected, with the matching standard deviation, the mean, the count, the sum of squares and the divisor actually used.
5. Check the working. The step-by-step block, the deviations table and the raw-score shortcut all recompute the same result along different routes, so you can compare each line with your own paperwork.

Anything that is not a number is skipped and reported in an amber notice, so a stray header row or a unit label pasted along with the data will not silently corrupt the result.

Which one do you need, s² or σ²?

The question is not about the size of the data set, it is about what the data represents. Ask yourself whether the numbers in front of you are everything you care about, or a slice of something larger.

• Population (σ²) when the data is complete: the final grades of every student in one class when the class itself is the subject, the weights of all twelve parts in a single batch, the monthly totals for all twelve months of a finished year.
• Sample (s²) when the data is a subset used to say something about a bigger group: forty survey responses standing in for a city, ten measured parts standing in for a production run, one week of readings standing in for a whole season.
• When in doubt, use the sample formula. Most real data is a sample, most software defaults to s², and the sample figure is the more conservative of the two because it is slightly larger.

Who this calculator is for

• Students working through a statistics assignment who need the answer and the intermediate steps that earn the marks.
• Teachers building worksheets and answer keys, who want a fast second opinion on a set of hand-computed values.
• Lab and quality staff checking whether a run of measurements is tighter or looser than the previous run.
• Analysts and researchers who have a short column of numbers and do not want to open a spreadsheet or a statistics package for it.
• Anyone comparing two data sets that share the same units and want a single number for which one is steadier.

Key terms explained

• Mean (x̄ or μ): the arithmetic average, the sum divided by the count. Every deviation is measured from it.
• Deviation (x − x̄): how far one value sits from the mean, positive above and negative below. The deviations always sum to zero.
• Sum of squares: the total of the squared deviations, written Σ(x − x̄)². It is the numerator both formulas share.
• Degrees of freedom: the n − 1 in the sample denominator. Once the mean is fixed, only n − 1 of the deviations can vary freely, because the last one is forced by the requirement that they sum to zero.
• Bessel's correction: the name for using n − 1 instead of n so that the sample variance is an unbiased estimate of the population variance.
• Coefficient of variation: the standard deviation divided by the mean, shown as a percentage. It compares the relative spread of data sets with different units or very different averages.
• Mean absolute deviation: the average of the unsigned deviations. It is easier to explain than variance but has worse mathematical behavior, which is why variance dominates statistics.

What changes the result the most

Two rules make variance predictable. Shifting does nothing: add the same constant to every value and the spread is untouched, so 12, 15, 11, 18, 14 and 112, 115, 111, 118, 114 both have a sample variance of 7.5. Scaling squares: multiply every value by k and the variance is multiplied by k². Doubling the data set raises the sample variance from 7.5 to 30, tripling it raises it to 67.5, and multiplying by ten raises it to 750, while the standard deviation only doubles, triples and multiplies by ten.

The third and largest influence is a single distant value. Because deviations are squared before they are averaged, one outlier can dominate the entire sum of squares:

Moving one value from 14 to 70 leaves four numbers untouched yet multiplies the variance by more than eighty. Before you report a variance, look at the deviations table and check that the biggest contributor is a real observation rather than a typing slip or a misplaced decimal point.

Variance or standard deviation: which to report

They carry identical information, since each is a simple function of the other, but they are used in different places. Report the standard deviation when a human has to interpret the number, because it shares the units of the data: a class average of 50 points with a standard deviation of 13 points is immediately meaningful. Work with the variance when the mathematics demands it, because variances of independent quantities add together while standard deviations do not. That additivity is what makes variance the building block of analysis of variance, regression, portfolio mathematics and error propagation.

A practical rule: compute the variance, then quote the standard deviation. This calculator gives you both at once, along with the coefficient of variation for cases where you need to compare spreads measured on different scales.

Tips for getting a trustworthy number

• Check that the deviations sum to zero. If they do not, the mean was computed or copied incorrectly, and everything downstream is wrong.
• Do not round the mean before squaring. Rounding 14.333 to 14.3 in a five-value data set can move the variance by a full percent. Keep full precision until the final display.
• Verify with the shortcut formula. Computing the sum of squares twice, once through the deviations and once through Σx² − (Σx)² ÷ N, catches most arithmetic slips.
• Repeat values must be entered every time. A value that occurs four times counts four times, so 4, 4, 4, 4 is not the same input as a single 4.
• Watch for mixed units. Three lengths in inches and two in feet produce a variance that means nothing. Convert first with the Unit Converter.
• Say which formula you used. A variance reported without stating sample or population is ambiguous, and on small data sets the two answers differ noticeably.

Limitations and assumptions

• It expects a raw list of values. Grouped or frequency tables must be expanded first, repeating each value as many times as its frequency.
• Every value is weighted equally. There is no weighted variance, so unequal sample weights are not supported.
• It describes one variable at a time. Covariance and correlation between two variables are a different calculation.
• Variance is a descriptive figure. It does not tell you whether a difference between two data sets is statistically significant, which needs a formal test.
• It is sensitive to outliers by design. For data with extreme values, robust measures such as the interquartile range often describe the spread better.
• Ordinary floating-point arithmetic is used, so extremely large values combined with extremely small differences can lose a little precision in the last digits.

How it compares to related calculators

This page answers the question "how spread out is this data, in squared units?" If your question is slightly different, a sister tool fits better:

• For the same spread in the original units, use the Standard Deviation Calculator.
• For the mean, median and mode on their own, use the Average Calculator.
• For a wider summary of one data set at once, use the Statistics Calculator.
• To see how unusual one individual value is, use the Z-Score Calculator.
• For the long-run average of a random outcome, use the Expected Value Calculator.
• To compare a measured value against a known true value, use the Percent Error Calculator.

Sources

Variance is a mathematical definition rather than a regulated figure, so no agency publishes a value to look up. Every number on this page follows directly from the definitions above, and each table entry was computed from those formulas.

• Deterministic definitions used here: mean = Σx ÷ N; population variance = Σ(x − μ)² ÷ N; sample variance = Σ(x − x̄)² ÷ (n − 1); standard deviation = the square root of the variance. No source is required for these, and no external data is used.
• National Institute of Standards and Technology (NIST) - NIST/SEMATECH e-Handbook of Statistical Methods: measures of scale, for the standard definitions of variance and standard deviation.
• National Institute of Standards and Technology (NIST) - NIST Special Publication 811, Guide for the Use of the International System of Units, for unit conventions when your data carries physical units and the variance is reported in squared units.