Population Variance from a Frequency Table: A Small Worked Calculation

For a complete population with values x and frequencies f, variance is Σf(x − μ)²/N, where N = Σf and μ = Σfx/N. It measures average squared distance from the mean.

Choose the correct divisor first

This guide treats the listed observations as the entire population of interest. Divide the squared-deviation total by N. If a question instead asks for sample variance as an estimate from a sample, the usual divisor is n − 1. These are different calculations. Write “population” or “sample” next to the formula before inserting numbers.

A frequency tells you how many copies of a value occur. It must multiply both the value contribution to the mean and the squared-deviation contribution to variance. Counting rows instead of observations gives the wrong denominator whenever a frequency exceeds one.

Worked example

A small collection contains value 2 once, value 4 twice, and value 6 once. The population size is 1 + 2 + 1 = 4. Its total is 2 × 1 + 4 × 2 + 6 × 1 = 16, so μ = 16/4 = 4.

Value x	Frequency f	x − μ	f(x − μ)²
2	1	−2	4
4	2	0	0
6	1	2	4

The final column totals 8. Population variance is 8/4 = 2. Population standard deviation is √2, approximately 1.414. If the observations are lengths in centimetres, variance has units cm², while standard deviation has units cm.

Why the deviations must be squared

For this data, ordinary signed deviations total −2 + 0 + 0 + 2 = 0. Adding them cannot reveal spread because negative and positive differences cancel. Squaring makes their contributions nonnegative. It also gives larger distances more influence. Therefore a single far-away value can strongly affect variance.

For a quick independent check, expand the data to 2, 4, 4, 6 and calculate directly. You can also use Σfx²/N − μ²: here (4 + 32 + 36)/4 − 16 = 18 − 16 = 2. Retain exact values until the end rather than rounding the mean in intermediate rows.

Common mistake

Do not subtract the mean after squaring each value unless you are deliberately using the expanded formula. The direct formula requires the square of the whole difference, (x − μ)². Also remember that standard deviation is the square root of variance, not variance divided by two.

Rows nahi, frequencies ka total gino; phir squared distance ka average lo.

Try three questions, then check the working

1. Find population variance of 1, 1, 3, 3.

   The mean is 2. Every squared deviation is 1, so the total is 4. Divide by four observations: variance = 1.

2. Values 5 and 9 each occur once. Find population variance.

   The mean is 7. Squared deviations are 4 and 4. Their average is 8/2 = 4; standard deviation is 2.

3. All six observations equal 7. What is the variance?

   The mean is 7 and every deviation is zero. Their squared total is zero, so population variance is 0/6 = 0.

Build the next step

• Mean from a Frequency Table
• Statistics Class 10 Notes: Mean, Median, Mode Formulas and Solved Examples

Stuck on one step? Open the DoubtPay study tool and ask for that step to be explained. Check the current credit price before using a paid answer.

Reference and authorship

The examples and practice questions here are original, with calculations checked separately. Definitions and related background: OpenStax: Measures of the Spread of the Data. Prepared with AI assistance and editorial arithmetic review; no textbook affiliation is claimed.