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JEE Mains Maths · Statistics

Correcting Data and Combining Groups

Fixing the mean and variance after a wrongly recorded value is corrected or dropped, and finding the mean and variance of two groups put together.

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Why this matters

Twenty-three PYQs, thirteen of them multiple choice, and two from 2026. Fifteen correct one or two wrongly read values, or omit them, and ask for the new mean, variance or standard deviation; eight combine two groups with known sizes, means and variances, sometimes after shifting each group. Two ideas cover the page.

Concept 1 of 2: Correcting a wrong value

A wrong value sits inside both totals. Rebuild ∑x\sum x and ∑x2\sum x^2 from the wrong mean and variance, take the wrong value out of each, put the right one in, and recompute. The variance changes too, so it must be rebuilt, not carried over.

Definition

  • Recorded totals: ∑x=nxˉ\sum x=n\bar x, ∑x2=n(σ2+xˉ2)\sum x^2=n(\sigma^2+\bar x^2).
  • Wrong ww, correct cc: ∑x→∑x−w+c\sum x\to\sum x-w+c, ∑x2→∑x2−w2+c2\sum x^2\to\sum x^2-w^2+c^2.
  • Omitting a value: subtract it and reduce nn by one.
  • Then σ2=1n∑x2−xˉ2\sigma^2=\frac{1}{n}\sum x^2-\bar x^2 with the corrected totals.

Replace in both totals

∑x→∑x−w+c,∑x2→∑x2−w2+c2\sum x\to\sum x-w+c,\qquad \sum x^2\to\sum x^2-w^2+c^2

Worked example

Ten observations have mean 8 and variance 6. A value 13 was recorded as 3. Find the correct mean and variance.
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2026 · 24 Jan 2026 Shift 1 · Q68Moderate

Example 1 · Statistics · Correcting Data and Combining Groups

The mean and variance of a data of 10 observations are 10 and 2, respectively. If an observation α\alpha in this data is replaced by β\beta, then the mean and variance become 10.1 and 1.99, respectively. Then α+β\alpha + \beta equals.

Rebuild the totals from the wrong data

The given mean and variance belong to the wrong data, so ∑x2=n(σ2+xˉ2)\sum x^2=n(\sigma^2+\bar x^2) must use the recorded mean. Correct the totals after that, never the mean first.
Drill 14 more on correcting a wrong value

Concept 2 of 2: Combining two groups

Two groups put together simply add their totals: ∑x\sum x and ∑x2\sum x^2 add. In formula form, the combined variance is the weighted average of the group variances plus the spread of the group means about the combined mean. If a group is shifted first, only its mean moves; its variance stays.

Definition

  • xˉ=n1xˉ1+n2xˉ2n1+n2\bar x=\frac{n_1\bar x_1+n_2\bar x_2}{n_1+n_2}.
  • σ2=n1(σ12+d12)+n2(σ22+d22)n1+n2\sigma^2=\frac{n_1(\sigma_1^2+d_1^2)+n_2(\sigma_2^2+d_2^2)}{n_1+n_2}, where di=xˉi−xˉd_i=\bar x_i-\bar x.
  • The same as the formula below, which needs no combined mean.
  • A shifted group: shift its mean, keep its variance.

Combined variance

σ2=n1σ12+n2σ22n1+n2+n1n2(xˉ1−xˉ2)2(n1+n2)2\sigma^2=\frac{n_1\sigma_1^2+n_2\sigma_2^2}{n_1+n_2}+\frac{n_1n_2\left(\bar x_1-\bar x_2\right)^2}{\left(n_1+n_2\right)^2}

Worked example

Group A has 5 values with mean 10 and variance 4. Group B has 15 values with mean 14 and variance 8. Find the mean and variance of all 20.
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2026 · 8 Apr 2026 Shift 2 · Q57Moderate

Example 2 · Statistics · Correcting Data and Combining Groups

A set of four observations has mean 1 and variance 13. Another set of six observations has mean 2 and variance 1. Then, the variance of all these 10 observations is equal to :

Group variances do not simply average

Averaging σ12\sigma_1^2 and σ22\sigma_2^2 misses the gap between the group means. The extra term n1n2N2(xˉ1−xˉ2)2\frac{n_1n_2}{N^2}(\bar x_1-\bar x_2)^2 is zero only when the means are equal.
Drill 7 more on combining two groups

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

  • Correcting a wrong value

    Replace in both totals

    ∑x→∑x−w+c,∑x2→∑x2−w2+c2\sum x\to\sum x-w+c,\qquad \sum x^2\to\sum x^2-w^2+c^2
  • Combining two groups

    Combined variance

    σ2=n1σ12+n2σ22n1+n2+n1n2(xˉ1−xˉ2)2(n1+n2)2\sigma^2=\frac{n_1\sigma_1^2+n_2\sigma_2^2}{n_1+n_2}+\frac{n_1n_2\left(\bar x_1-\bar x_2\right)^2}{\left(n_1+n_2\right)^2}

Watch out for (2)

Test yourself on Statistics

20 past JEE Mains questions from this chapter, timed at 48 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

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