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Statistics questions in JEE are usually short, and that makes them easy marks to lose. A question might give the mean and variance of ten numbers, say one value was copied wrongly, and ask for the corrected variance. If you try to rebuild the data, you run out of time. If you work with sums, it takes four lines. This post covers the formulas, the two shortcuts that matter and the standard question types, with every number checked.
What does the syllabus cover?
The JEE (Main) 2026 syllabus lists measures of dispersion and the "calculation of mean, median, mode of grouped and ungrouped data, calculation of standard deviation, variance and mean deviation for grouped and ungrouped data". The CBSE 2026-27 curriculum puts range, mean deviation, variance and standard deviation in Class 11, which is assessed by your school rather than by a board exam. NCERT's Class 11 chapter on statistics defines variance as the mean of the squares of the deviations from the mean, dividing by n. Check the documents for your own exam year.
How do you calculate variance and standard deviation?
For n values with mean x̄:
Variance σ² = Σ(x − x̄)²/n.
Shortcut: σ² = Σx²/n − x̄², the mean of the squares minus the square of the mean.
Standard deviation σ = √σ², in the same units as the data.
Mean deviation about the mean = Σ|x − x̄|/n.
Example. Data: 2, 4, 4, 4, 5, 5, 7, 9.
Mean = 40/8 = 5.
Deviations: −3, −1, −1, −1, 0, 0, 2, 4. Squares: 9, 1, 1, 1, 0, 0, 4, 16, which add to 32.
Variance = 32/8 = 4, so σ = 2.
Shortcut check: Σx² = 4 + 16 + 16 + 16 + 25 + 25 + 49 + 81 = 232. Then 232/8 − 5² = 29 − 25 = 4.
Mean deviation: |deviations| add to 3 + 1 + 1 + 1 + 0 + 0 + 2 + 4 = 12, so it is 12/8 = 1.5.
What happens when you add to or scale every value?
If every value x becomes ax + b:
The mean becomes a·x̄ + b.
The variance becomes a²σ². Adding b changes nothing, because every deviation stays the same.
The standard deviation becomes |a|σ.
Example. Multiply each value above by 3 and add 2. The new mean is 3(5) + 2 = 17, the new variance is 9 × 4 = 36, and the new standard deviation is 6.
How do you correct a wrongly copied value?
Work with Σx and Σx², never with the data itself.
Example. Ten values have mean 10 and variance 9. Later it is found that one value, 8, was copied as 18. Find the correct mean and variance.
Σx = 10 × 10 = 100. From σ² = Σx²/n − x̄², Σx² = 10 × (9 + 100) = 1090.
Correct the sums: Σx = 100 − 18 + 8 = 90, and Σx² = 1090 − 324 + 64 = 830.
Correct mean = 90/10 = 9.
Correct variance = 830/10 − 9² = 83 − 81 = 2.
If you ever get a negative variance at step 4, you have made an arithmetic slip, because a variance cannot be negative.
How do you handle grouped data?
Use frequencies as weights: x̄ = Σfx/N and σ² = Σfx²/N − x̄², where N = Σf. For class intervals, use each class mark (the midpoint) as x.
Example. Values 2, 4, 6, 8 with frequencies 3, 5, 8, 4, so N = 20.
Σfx = 6 + 20 + 48 + 32 = 106, so x̄ = 106/20 = 5.3.
Σfx² = 12 + 80 + 288 + 256 = 636, so σ² = 636/20 − 5.3² = 31.8 − 28.09 = 3.71, and σ ≈ 1.93.
Check with deviations: 3(10.89) + 5(1.69) + 8(0.49) + 4(7.29) = 32.67 + 8.45 + 3.92 + 29.16 = 74.2, and 74.2/20 = 3.71.
Why do students misjudge spread?
In interviews with 12 introductory statistics students, delMas and Liu (2005) watched them rearrange histogram bars to make the standard deviation as large or as small as possible. Many began with "one-dimensional" ideas that did not consider variation about the mean, and moved towards seeing how both distance from the mean and frequency affect the standard deviation. The practical habit: before calculating, look at the data and predict which set is more spread out around its mean.
Which mistakes cost the most marks?
Squaring the mean instead of averaging the squares. Σx²/n and x̄² are different numbers.
Dividing by n − 1. NCERT and JEE use n unless the question says otherwise.
Forgetting a² when scaling, or thinking that adding a constant changes the variance.
Correcting Σx but not Σx² in a wrongly copied value question.
Reporting the standard deviation when the question asked for variance, or the other way round.
A practice routine
Two-method drill (weekly): for five small data sets, find the variance by deviations and by the shortcut. They must agree.
Sums-only problems (twice a week): three correction or combination questions using only n, Σx and Σx².
Predict first: before each calculation, write which data set you expect to have the larger standard deviation, then check.
Connect topics: variance shows up again with random variables. For the people behind the subject, read about P. C. Mahalanobis and C. R. Rao.
If the idea of spread about the mean still feels vague, the Fermi Concept Library covers every concept in physics, chemistry, biology, mathematics and computing, each with its own lesson, so you can go back to the definition before the formulas.
Today, take the data 2, 4, 4, 4, 5, 5, 7, 9, double every value and add 1, and predict the new variance before you calculate it.
