Discrete Variance Calculator: Formula & Examples

This calculator computes the variance of a discrete data set, showing both the sample variance (n-1 denominator) and the population variance (n denominator). You enter a comma-separated list of numbers, and the tool returns the variance, standard deviation, and count, with the sample variance shown as the headline result.

Formula

For a discrete data set with n values x₁, x₂, …, x_n, the variance measures how spread out the numbers are around the mean.

Population variance (when the data covers the entire group):

σ² = Σ(x_i − μ)² / n

Sample variance (when the data is a sample from a larger group):

s² = Σ(x_i − x̄)² / (n − 1)

Where:

• x_i = each individual value
• μ = population mean = (Σx_i) / n
• xÌ„ = sample mean (same formula as μ, but used for a sample)
• n = number of values
• σ² = population variance (sigma squared)
• s² = sample variance

The standard deviation is the square root of the variance: σ = √σ² or s = √s². It is in the same units as the original data, while variance is in squared units.

Worked Example

Using the default data set: 4, 8, 15, 16, 23, 42.

1. Count: n = 6
2. Mean: (4 + 8 + 15 + 16 + 23 + 42) / 6 = 108 / 6 = 18
3. Squared deviations from the mean:
   • (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 = (5)² = 25
   • (42 − 18)² = (24)² = 576
4. Sum of squared deviations: 196 + 100 + 9 + 4 + 25 + 576 = 910
5. Population variance: 910 / 6 ≈ 151.6667
6. Sample variance: 910 / (6 − 1) = 910 / 5 = 182
7. Population standard deviation: √151.6667 ≈ 12.3149
8. Sample standard deviation: √182 ≈ 13.4907

The calculator shows sample variance = 182, sample SD ≈ 13.4907, population variance ≈ 151.6667, population SD ≈ 12.3149, matching the hand calculation.

When to Use Which Variance

Choosing between sample and population variance depends on what your data represents:

• Population variance – use when your list contains 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 5 starting players on a team.
• Sample variance – use when your list is a random subset of a larger group. For example, the heights of 50 randomly chosen adults from a city, used to estimate the variance of all adults in that city.

The sample variance uses n − 1 (called Bessel’s correction) because dividing by n tends to underestimate the true population variance when applied to a sample. The correction makes the sample variance an unbiased estimator of the population variance.

How to Read the Result

The variance itself is in squared units. If your data is in dollars, variance is in dollars squared – which is hard to interpret directly. That is why the standard deviation (square root of variance) is often reported alongside it: it is in the original units.

A larger variance means the data points are more spread out from the mean. A variance of zero means all values are identical. Variance is always non-negative because it is a sum of squared deviations.

For a quick sense of spread: about 68% of data in a normal distribution lies within one standard deviation of the mean, 95% within two, and 99.7% within three. This rule of thumb works best for roughly bell-shaped distributions, not for skewed or multi-modal data.

Common Mistakes and Pitfalls

• Using n instead of n − 1 for a sample – this makes your variance too small, especially with small sample sizes. For n = 10, the difference is 11%; for n = 100, it is about 1%.
• Mixing up units – if your data is in meters, variance is in square meters. Convert all values to the same unit before calculating.
• Rounding too early – keep full precision during intermediate steps. Round only the final answer. Rounding the mean to 2 decimal places before computing deviations can change the variance noticeably.
• Forgetting that variance is not robust – a single outlier can inflate variance dramatically. For example, changing 42 to 420 in the default data set raises the sample variance from 182 to about 26,000.
• Treating ordinal or categorical data as numeric – variance only makes sense for interval or ratio data (e.g., temperature, height, time). It does not apply to categories like “redâ€�, “blueâ€�, “greenâ€�.

Alternatives and Related Measures

Variance is one of several ways to describe spread. Depending on your goal, one of these may be more appropriate:

• Standard deviation – the square root of variance; same units as the data. This is usually the preferred measure for reporting.
• Range – maximum minus minimum. Simple but sensitive to outliers.
• Interquartile range (IQR) – the difference between the 75th and 25th percentiles. Robust to outliers and good for skewed data.
• Mean absolute deviation (MAD) – the average absolute distance from the mean. Less sensitive to outliers than variance, but less common.
• Coefficient of variation (CV) – standard deviation divided by the mean. Useful for comparing variability between data sets with different means or units.

For grouped data (e.g., values in intervals like 0–10, 10–20), you would use the midpoint of each interval and the frequency to compute a weighted variance. This calculator expects raw individual values.

Limitations of the Formula

The variance formula assumes your data is numeric and measured on a scale where equal intervals mean equal differences. It does not work well for:

• Data with extreme outliers – variance can be distorted; consider using IQR or median absolute deviation instead.
• Skewed distributions – variance alone does not capture asymmetry; pair it with a histogram or skewness measure.
• Data with a natural floor or ceiling (e.g., percentages capped at 100) – variance may be artificially limited.
• Very small samples (n < 5) – variance estimates are unstable; treat results with caution.

Also note that variance is not a measure of “average distance from the mean� in the intuitive sense. The mean absolute deviation is that. Variance gives more weight to large deviations because they are squared.

Frequently Asked Questions

What is the difference between sample variance and population variance?

Population variance (σ²) divides the sum of squared deviations by n, the total number of values. Sample variance (s²) divides by n − 1. Use population variance when your data includes every member of the group; use sample variance when your data is a random sample from a larger group. The n − 1 correction makes the sample variance an unbiased estimate of the population variance.

Why is variance in squared units?

Variance is the average of squared deviations from the mean. Because you square each deviation, the units are also squared. For example, if your data is in kilograms, variance is in kg². That is why the standard deviation, which is the square root of variance, is often reported – it returns to the original units (kg).

Can variance be negative?

No. Variance is the sum of squared deviations, and a square of any real number is always non-negative. The smallest possible variance is 0, which occurs when all values in the data set are identical. A negative variance indicates an error in calculation.

How do I know if my variance is large or small?

Variance has no absolute scale – it depends on the units and the mean of your data. A variance of 100 for heights in centimeters is small, but a variance of 100 for salaries in thousands of dollars is large. Compare the standard deviation to the mean (coefficient of variation) or compare variances from similar data sets. There is no universal “good� or “bad� variance.

What is the shortcut formula for variance?

The shortcut formula is: σ² = (Σx² / n) − μ². That is, the mean of the squares minus the square of the mean. For sample variance, use s² = (Σx² − n·x̄²) / (n − 1). This formula is mathematically equivalent to the definition but can be less numerically stable with very large numbers, so the direct method is often preferred.

Does variance work for data with negative numbers?

Yes. Variance uses squared deviations, so negative values are handled naturally. For example, the data set −5, 0, 5 has a mean of 0 and a population variance of (25 + 0 + 25)/3 ≈ 16.67. Negative values do not affect the calculation in any special way.

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in list | Data values | 4, 8, 15, 16, 23, 42 | 0
out var_s | Sample variance | - | var(list) | 4
out sd_s | Sample standard deviation | - | sd(list) | 4
out var_p | Population variance | - | pvar(list) | 4
out sd_p | Population standard deviation | - | psd(list) | 4