PYQ Vault

JEE Mains Maths · Formula sheet

Statistics formulas

10 formulas and 10 common traps for JEE Mains Maths Statistics, grouped by subtopic.

Full notes with worked examples

Variance from Sums and Shifts

Learn this subtopic in the notes

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

Common traps

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.

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.

Finding Unknown Observations

Learn this subtopic in the notes

Two unknowns from the mean and variance

The pair from its sum and sum of squares

(a−b)2=2(a2+b2)−(a+b)2(a-b)^2=2\left(a^2+b^2\right)-(a+b)^2

A part of the data

Totals of the whole

∑xi=nxˉ,∑xi2=n(σ2+xˉ2)\sum x_i=n\bar x,\qquad \sum x_i^2=n\left(\sigma^2+\bar x^2\right)

Common traps

Keep the square of the mean

The variance gives ∑xi2=n(σ2+xˉ2)\sum x_i^2=n(\sigma^2+\bar x^2), not nσ2n\sigma^2. Dropping the xˉ2\bar x^2 term gives an a2+b2a^2+b^2 that is too small, often negative.

Divide by the new count

After removing values, the mean and variance of the rest use the new count mm, not nn. Only the totals carry over.

Correcting Data and Combining Groups

Learn this subtopic in the notes

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}

Common traps

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.

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.

Mean Deviation and Median

Learn this subtopic in the notes

Mean deviation about the mean

MD(xˉ)=1n∑i=1n∣xi−xˉ∣\text{MD}(\bar x)=\frac{1}{n}\sum_{i=1}^{n}\left|x_i-\bar x\right|

Mean deviation about the median or the mode

Mean deviation about the median

MD(M)=1n∑i=1n∣xi−M∣\text{MD}(M)=\frac{1}{n}\sum_{i=1}^{n}\left|x_i-M\right|

Common traps

The mean deviation is not the SD

1n∑∣xi−xˉ∣\frac{1}{n}\sum|x_i-\bar x| and 1n∑(xi−xˉ)2\sqrt{\frac{1}{n}\sum(x_i-\bar x)^2} are different numbers, and the MD is never larger than the SD. Do not square the distances or take a root.

Sort before you pick the median

The median is the middle of the sorted list, not of the list as printed. With unknown values, first decide where they fall in the order; each case can give a different answer.

Frequency Distributions

Learn this subtopic in the notes

Mean and variance of a frequency table

Frequency table

xˉ=∑fixi∑fi,σ2=∑fixi2∑fi−xˉ2\bar x=\frac{\sum f_ix_i}{\sum f_i},\qquad \sigma^2=\frac{\sum f_ix_i^2}{\sum f_i}-\bar x^2

Median and mean deviation of a table

Grouped median

M=l+N2−Ff×hM=l+\frac{\frac{N}{2}-F}{f}\times h

Common traps

The unknown frequency is in N too

When a frequency is kk, the total N=∑fiN=\sum f_i also contains kk. Clear the fraction before solving; if the equation is a quadratic, keep only the root that is a whole, non-negative frequency.

F stops before the median class

In the grouped median, FF is the cumulative frequency of the classes before the median class, not up to and including it. Using the larger figure puts the median too low.

More JEE Mains Maths formula sheets