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JEE Mains Maths · Statistics

Variance from Sums and Shifts

Getting the mean and variance from given sums — of the values, of shifted squares or of pairwise products — and from a shift or scale of the data.

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Why this matters

Seventeen PYQs, ten of them multiple choice, and three from 2026. Nine turn given sums, such as Σ(x − a) and Σ(x − b)² or the pairwise products, into the mean and variance; eight shift or scale the data, or need the variance of an arithmetic progression. Two ideas cover the page.

Concept 1 of 2: Mean and variance from sums

Everything here comes from three numbers: nn, ∑xi\sum x_i and ∑xi2\sum x_i^2. Any given sum, such as ∑(xi−a)\sum(x_i-a) or ∑(xi−a)2\sum(x_i-a)^2, expands into those two sums. A shift by aa does not change the variance, so you can also work with di=xi−ad_i=x_i-a directly. Pairwise products come in through the square of the total.

Definition

  • xˉ=1n∑xi\bar x=\frac{1}{n}\sum x_i, σ2=1n∑xi2−xˉ2\sigma^2=\frac{1}{n}\sum x_i^2-\bar x^2.
  • ∑(xi−a)=∑xi−na\sum(x_i-a)=\sum x_i-na.
  • ∑(xi−a)2=∑xi2−2a∑xi+na2\sum(x_i-a)^2=\sum x_i^2-2a\sum x_i+na^2.
  • With di=xi−ad_i=x_i-a: σ2=1n∑di2−dˉ 2\sigma^2=\frac{1}{n}\sum d_i^2-\bar d^{\,2}.
  • (∑xi)2=∑xi2+2∑i<jxixj\left(\sum x_i\right)^2=\sum x_i^2+2\sum_{i<j}x_ix_j.

Variance from sums

σ2=∑xi2n−(∑xin)2\sigma^2=\frac{\sum x_i^2}{n}-\left(\frac{\sum x_i}{n}\right)^2

Worked example

For 10 observations, ∑(xi−3)=20\sum(x_i-3)=20 and ∑(xi−3)2=90\sum(x_i-3)^2=90. Find the mean and variance.
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2026 · 6 Apr 2026 Shift 1 · Q59Moderate

Example 1 · Statistics · Variance from Sums and Shifts

A data consists of 20 observations x1,x2,…..,x20x_{1},x_{2},\ldots..,x_{20}. If ∑i=120(xi+5)2=2500\sum_{i= 1}^{20} \left( x_{i}+ 5 \right)^{2}= 2500 and ∑i=120(xi−5)2=100\sum_{i= 1}^{20} \left( x_{i}- 5 \right)^{2}= 100, then the ratio of mean to standard deviation of this data is :

Squares about a is not the variance

1n∑(xi−a)2\frac{1}{n}\sum(x_i-a)^2 is the variance only when aa is the mean. For any other aa, subtract (1n∑(xi−a))2\left(\frac{1}{n}\sum(x_i-a)\right)^2.
Drill 8 more on mean and variance from sums

Concept 2 of 2: Shifts, scales and arithmetic progressions

Adding bb to every value moves the mean by bb and leaves the spread alone. Multiplying by aa multiplies the mean by aa, the standard deviation by ∣a∣|a| and the variance by a2a^2. Values in an A.P. are a scaled and shifted copy of 1,2,…,n1,2,\ldots,n, so their variance has a closed form.

Definition

  • yi=axi+by_i=ax_i+b: yˉ=axˉ+b\bar y=a\bar x+b, σy=∣a∣ σx\sigma_y=|a|\,\sigma_x.
  • Adding a constant: the variance does not change.
  • 1,2,…,n1,2,\ldots,n: σ2=n2−112\sigma^2=\frac{n^2-1}{12}.
  • An A.P. of nn terms with difference dd: σ2=d2(n2−1)12\sigma^2=\frac{d^2(n^2-1)}{12}.

Linear change of data

yi=axi+b ⇒ yˉ=axˉ+b,σy2=a2σx2y_i=ax_i+b\ \Rightarrow\ \bar y=a\bar x+b,\quad \sigma_y^2=a^2\sigma_x^2

Worked example

Each of the values 1,2,…,91,2,\ldots,9 is changed to 3x−23x-2. Find the mean and variance of the new values.
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2026 · 24 Jan 2026 Shift 2 · Q64Moderate

Example 2 · Statistics · Variance from Sums and Shifts

Let X={x∈N:1≤x≤19}X = \{ x \in N:1 \leq x \leq 19\} and for some a,b∈Ra,b \in R, Y={ax+b:x∈X}Y = \{ ax + b:x \in X\}. If the mean and variance of the elements of Y are 30 and 750, respectively, then the sum of all possible values of bb is

The sign of a is lost

The variance is multiplied by a2a^2, so aa and −a-a give the same spread. A question that fixes the variance usually has two values of aa, and so two values of bb.
Drill 7 more on shifts, scales and arithmetic progressions

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)

  • Mean and variance from sums

    Variance from sums

    σ2=∑xi2n−(∑xin)2\sigma^2=\frac{\sum x_i^2}{n}-\left(\frac{\sum x_i}{n}\right)^2
  • Shifts, scales and arithmetic progressions

    Linear change of data

    yi=axi+b ⇒ yˉ=axˉ+b,σy2=a2σx2y_i=ax_i+b\ \Rightarrow\ \bar y=a\bar x+b,\quad \sigma_y^2=a^2\sigma_x^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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