What Is Variance?
A frequency table records each distinct value once, along with how many times it occurs. To find its variance you weight every squared deviation from the mean by that frequency, then divide by N or N − 1.
Enter the values in one box and their frequencies in the other. The calculator returns the mean, the sample variance, the population variance and both standard deviations, plus a table with f·x and f(x − mean)² for each row.
For grouped data such as class intervals, enter the class midpoints as the values. The answer is then an estimate, because the exact values inside each class are not known.
How to Use the Variance Calculator
- Enter the distinct values in the first box.
- Enter the matching frequencies in the second box.
- Pick the sample or population result, depending on whether the table is a sample or the whole group.
Variance Formula
Mean
Sample variance
Population variance
Shortcut
Variance Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| x | A distinct value, or a class midpoint | |
| f | Frequency of that value | Whole number, 0 or more |
| N | Total frequency, Σf | |
| x̄ or μ | Mean, Σfx ÷ N | |
| s² and σ² | Sample and population variance | Divide by N − 1 or N |
Variance Worked Example
With the example values loaded in the calculator above, the steps are:
- N = Σf = 15 and Σfx = 76.
- Mean = 76 ÷ 15 = 5.066667.
- Σf(x − mean)² = 78.93333 (see the table).
- Sample variance = 78.93333 ÷ (15 − 1) = 5.638095.
- Population variance = 78.93333 ÷ 15 = 5.262222.
Result: Sample variance (s²) = 5.638095; Population variance (σ²) = 5.262222.
Worked example
Values 2, 4, 6, 8, 10 with frequencies 3, 5, 4, 2, 1 give N = 15 and Σfx = 6 + 20 + 24 + 16 + 10 = 76, so the mean is 5.0667.
Σfx² = 12 + 80 + 144 + 128 + 100 = 464, so Σf(x − mean)² = 464 − 76² ÷ 15 = 78.9333. Dividing by 14 gives the sample variance 5.6381, and dividing by 15 gives the population variance 5.2622.
Grouped data and class midpoints
If your data are in classes such as 0–10 and 10–20, use the midpoint of each class as x. The result is an estimate, because the true values inside each class are unknown.
Where Variance Is Used
Frequency table homework
Check each row of f·x and f(x − mean)² against your own working.
Grouped data
Estimate variance and SD from class midpoints and counts.
Survey results
Summarise spread when responses are already tallied by value.
Common Variance Mistakes to Avoid
- Check that Σf equals the number of observations you started with.
- Use the sample formula unless the table describes every member of the group.
Variance Terms and Definitions
- Frequency
- How many times a value occurs.
- Class midpoint
- The centre of a class interval, used as its value in grouped data.
Variance FAQs
How do you find variance from a frequency table?
Find the mean with Σfx ÷ Σf, then add f × (x − mean)² for each row and divide by N for a population or N − 1 for a sample.
What is N in a frequency table?
N is the sum of the frequencies, which is the total number of observations.
Is the result exact for grouped data?
No. With class midpoints it is an estimate of the true variance.
Can a frequency be 0?
Yes, it simply adds nothing. Negative or fractional frequencies are not allowed here.
How is the standard deviation found?
Take the square root of the variance. Both versions are shown.
Does it match the variance of the raw list?
Yes. Entering each value as many times as its frequency in a normal variance calculator gives the same answer.