# Statistics

> Statistics for JEE Mathematics; mean, median, mode, variance, standard deviation and effect of change of origin and scale with tips.

- Canonical URL: https://prepelephant.com/topics/jee/mathematics/statistics
- Exam / course: JEE · Subject: Mathematics
- Publisher: PrepElephant (https://prepelephant.com) — Prepared and reviewed by the PrepElephant Academic Review Team
- First published: 2026-10-02
- Last updated: 2026-10-02
- How to cite: "Statistics", PrepElephant, https://prepelephant.com/topics/jee/mathematics/statistics

## Direct answer

For ungrouped data x1, x2, ..., xn, the mean is the sum of observations divided by n, the variance is sigma^2 = (sum of squared deviations)/n = (sum of squares)/n - (mean)^2, and the standard deviation sigma is its square root. The median is the middle value of the ordered data and the mode the most frequent value; JEE questions compute these directly and test how a shift or scaling of the data changes them.

## What you must remember

- Central values: mean = (Σxi)/n; median = middle value for odd n, average of the two middle values for even n, after ordering; mode = most frequent observation.
- Variance and SD: sigma^2 = Σxi^2/n - (Σxi/n)^2, computed from raw sums — faster and safer than forming deviations; sigma = sqrt(variance) is never negative and is zero only when all observations are equal.
- Change of origin: adding a constant to every observation leaves the variance, standard deviation and mean deviation unchanged — only the mean shifts.
- Change of scale: multiplying every observation by k multiplies the mean by k, the standard deviation by |k| and the variance by k^2.
- Empirical relation: for moderately skewed data, mode is approximately 3 median - 2 mean.
- Comparison measures: range = largest minus smallest; coefficient of variation = (sigma/mean) × 100, used to compare relative spread between data sets with different units or magnitudes.
- Mean deviation: (Σ|xi - mean|)/n — the average absolute deviation, computable about the median as well.

## Common confusion

The division convention: the variance here divides by n, the population form that NCERT and JEE use for ungrouped data — mixing in n - 1 from statistics courses elsewhere shifts every answer. The second surprise is invariance under shifting: students expect the spread to move when the data does, but adding a constant translates the whole distribution and leaves every deviation untouched. Also keep the units straight — variance carries squared units, and only the standard deviation returns to the data's units.

## Exam-focused takeaway

Statistics is a JEE Main topic: direct computation of mean, variance and standard deviation for small data sets, the transformation rules (add k, multiply by k) stated as numerical answers, and the empirical relation or coefficient of variation as one-liners. The chapter is short and the questions predictable, which makes arithmetic discipline the whole game — squaring, summing and subtracting without slips banks the marks. Variance of a discrete random variable appears in probability contexts and follows the same pattern with probability weights.

## Frequently asked questions

### What happens to the variance when each observation is increased by 5?

Nothing — deviations from the mean are unchanged, so the variance, standard deviation and mean deviation all stay put while the mean rises by 5.

### What happens when each observation is doubled?

The mean doubles, the standard deviation doubles and the variance becomes four times; in general multiplying by k scales sigma by |k| and sigma^2 by k^2.

### What is the coefficient of variation?

(Standard deviation / mean) × 100 — a percentage measure of relative spread that lets you compare variability across data sets of different size or units.

### What is the empirical relation between mean, median and mode?

Mode is approximately 3 median - 2 mean for moderately skewed unimodal data; it is a working approximation, not an identity.

### Which measure of central tendency is most affected by extreme values?

The mean, since every observation enters its sum; the median, being positional, resists outliers, which is why skewed data is often summarised by the median.