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"On rechecking, it was found that one observation was wrong." This sentence turns up in board papers and JEE Main, and the one-value version is in NCERT. Variations are where students lose time: two wrong values, a value to be removed, two values missing, two groups combined. All of them yield to one method. This post sets it out and works four types with checked numbers.
Where is this in the syllabus?
The JEE (Main) 2026 syllabus lists measures of dispersion and the "calculation of standard deviation, variance and mean deviation for grouped and ungrouped data". The CBSE 2026-27 curriculum places variance and standard deviation in Class 11. NCERT's Class 11 chapter on statistics (reprint 2026-27) defines variance by dividing by n and works a correction in Example 16. Check the documents for your exam year.
What is the method?
Work only with three numbers: n, Σx and Σx².
Rebuild the sums. Σx = n × mean. From σ² = Σx²/n − x̄², Σx² = n(σ² + x̄²).
Correct the sums. Subtract wrong values, add right ones, and change n if values are removed or added.
Recompute. New mean = Σx/n; new variance = Σx²/n − (new mean)².
NCERT's Example 16 has 100 observations with mean 40 and standard deviation 5.1, where 50 was used instead of 40. The sums are Σx = 4000 and Σx² = 100(26.01 + 1600) = 162,601. Correcting gives Σx = 3990 and Σx² = 161,701, so the mean is 39.9 and the variance is 1617.01 − 1592.01 = 25, a standard deviation of 5.
Type 1: two values misread
Twenty observations have mean 15 and standard deviation 4. The values 25 and 12 should have been 15 and 22.
Σx = 20 × 15 = 300; Σx² = 20(16 + 225) = 4820.
New Σx = 300 − 25 − 12 + 15 + 22 = 300, so the mean stays 15.
New Σx² = 4820 − 625 − 144 + 225 + 484 = 4760.
Variance = 4760/20 − 15² = 238 − 225 = 13, and σ = √13 ≈ 3.61.
The mean did not change but the spread did. Do not stop when the mean comes out the same.
Type 2: a value removed
Twenty-five observations have mean 40 and variance 36. The value 52 was included by mistake.
Σx = 1000; Σx² = 25(36 + 1600) = 40,900.
Remove 52: n = 24, Σx = 948, Σx² = 40,900 − 2704 = 38,196.
Mean = 948/24 = 39.5.
Variance = 38,196/24 − 39.5² = 1591.5 − 1560.25 = 31.25, so σ ≈ 5.59.
Removing a value above the mean lowers both the mean and the spread here, which is a useful sense check.
Type 3: two values missing
Five observations have mean 5 and variance 4. Three of them are 2, 4 and 6. Find the other two, a and b.
Σx = 25, so a + b = 25 − 12 = 13.
Σx² = 5(4 + 25) = 145, so a² + b² = 145 − (4 + 16 + 36) = 89.
ab = ((a + b)² − (a² + b²))/2 = (169 − 89)/2 = 40.
a and b are the roots of t² − 13t + 40 = 0, so they are 5 and 8.
Check: the mean of 2, 4, 6, 5, 8 is 25/5 = 5, and the squares add to 145, giving 145/5 − 25 = 4.
Type 4: two groups combined
Group A has 40 values with mean 50 and variance 16; group B has 60 values with mean 55 and variance 25.
Combined mean = (40 × 50 + 60 × 55)/100 = 5300/100 = 53.
ΣA x² = 40(16 + 2500) = 100,640 and ΣB x² = 60(25 + 3025) = 183,000.
Combined variance = 283,640/100 − 53² = 2836.4 − 2809 = 27.4, so σ ≈ 5.23.
The combined variance is larger than either group's because the two means differ.
Which mistakes cost marks?
Correcting Σx but not Σx².
Using σ instead of σ² when rebuilding Σx². Square the standard deviation first.
Keeping the old n after removing or adding values.
Squaring the old mean in the last step instead of the corrected one.
Dividing by n − 1. NCERT and JEE divide by n unless told otherwise.
Ignoring a negative variance. It cannot happen, so find the slip.
For the basics behind these steps, see our post on mean, variance and standard deviation.
A practice routine
Mixed practice prepares you for the version you have not seen. In Rohrer and Taylor's (2007) experiment, students who practised one problem type at a time scored 89% in practice and the mixed group 60%, but a week later the mixed group scored 63% on the test and the blocked group 20%.
Sums first (twice a week): for four questions, write n, Σx and Σx² before anything else.
One of each: a set with all four types, shuffled.
Sense check: predict whether the mean and spread will rise or fall before calculating.
Reverse it: make up a data set, compute its statistics, introduce an error, and correct it back.
If a step in a correction question will not come out, the Fermi Problem Library has a step-by-step solver that explains the reasoning rather than printing an answer.
Today, redo NCERT's Example 16 from memory, then change it so two values were wrong and solve the new version.
