Formula Sheet — Statistics

Card 1: Mean deviation

Data type Formula
Range Maximum −- Minimum
Ungrouped, about aa (mean or median) M.D.(a)=1n∑∣xi−a∣\mathrm{M.D.}(a) = \frac{1}{n}\sum \lvert x_i - a\rvert
Grouped (N=∑fiN = \sum f_i) M.D.(a)=1N∑fi∣xi−a∣\mathrm{M.D.}(a) = \frac{1}{N}\sum f_i\lvert x_i - a\rvert
Grouped median M=l+N/2−Cf×hM = l + \frac{N/2 - C}{f} \times h

Card 2: Variance and standard deviation

Data type Formula
Ungrouped σ2=1n∑(xi−xˉ)2\sigma^2 = \frac{1}{n}\sum(x_i - \bar{x})^2
Computing identity σ2=∑xi2n−xˉ2\sigma^2 = \frac{\sum x_i^2}{n} - \bar{x}^2; ∑xi2=n(σ2+xˉ2)\sum x_i^2 = n(\sigma^2 + \bar{x}^2)
Frequency data σ2=1N∑fi(xi−xˉ)2\sigma^2 = \frac{1}{N}\sum f_i(x_i - \bar{x})^2
Shortcut (yi=xi−Ahy_i = \frac{x_i - A}{h}) xˉ=A+hyˉ\bar{x} = A + h\bar{y}; σ2=h2N2[N∑fiyi2−(∑fiyi)2]\sigma^2 = \frac{h^2}{N^2}\left[N\sum f_iy_i^2 - (\sum f_iy_i)^2\right]
First nn naturals mean n+12\frac{n+1}{2}, variance n2−112\frac{n^2-1}{12}
Standard deviation σ=+σ2\sigma = +\sqrt{\sigma^2}, in the data's own units

Card 3: JEE Corner

Tool Statement
Linear transformation y=ax+by = ax + b yˉ=axˉ+b\bar{y} = a\bar{x} + b; σy=∣a∣σx\sigma_y = \lvert a\rvert\sigma_x
Shift only variance unchanged
Scale only variance × k2\times\, k^2
Coefficient of variation C.V.=σxˉ×100\mathrm{C.V.} = \frac{\sigma}{\bar{x}} \times 100; lower = more consistent
Combined mean n1xˉ1+n2xˉ2n1+n2\frac{n_1\bar{x}_1 + n_2\bar{x}_2}{n_1 + n_2}
Combined variance n1(σ12+d12)+n2(σ22+d22)n1+n2\frac{n_1(\sigma_1^2 + d_1^2) + n_2(\sigma_2^2 + d_2^2)}{n_1 + n_2}, di=xˉi−xˉd_i = \bar{x}_i - \bar{x}
Basic inequality ∑xi2n≥xˉ2\frac{\sum x_i^2}{n} \ge \bar{x}^2, equality only for constant data

Last-Minute Mistake Checklist

  1. Deviations about the mean sum to zero — that is why mean deviation needs ∣⋅∣|\cdot| and variance needs squares; never average raw deviations.
  2. Variance vs SD: variance is in squared units, SD in original units — answer the one the question asked and say which is which.
  3. The median for grouped data comes from the formula M=l+N/2−CfhM = l + \frac{N/2 - C}{f}h; picking the midpoint of the median class instead is a classic error.
  4. Gapped classes need the half-unit continuity correction before midpoints and medians (16-20 becomes 15.5-20.5).
  5. In the shortcut method, multiply back: σx=hσy\sigma_x = h\sigma_y, and xˉ=A+hyˉ\bar{x} = A + h\bar{y} — forgetting hh (or adding AA to σ\sigma) wrecks the answer.
  6. Shift never changes variance; scale changes it by k2k^2 — and the SD takes ∣k∣|k|, so a negative multiplier still gives a positive SD.
  7. ∑xi2=n(σ2+xˉ2)\sum x_i^2 = n(\sigma^2 + \bar{x}^2) is the entry point for missing-observation and corrected-statistics problems — recover both sums before touching anything.
  8. Correcting a wrong entry changes ∑xi\sum x_i by (right −- wrong) and ∑xi2\sum x_i^2 by (right2^2 −- wrong2^2); omitting an entry also reduces nn.
  9. Combined mean is weighted, not the plain average of the two means — sizes matter.
  10. C.V. needs xˉ≠0\bar{x} \ne 0 and is the only fair consistency comparison when means or units differ; with equal means, comparing SDs suffices.

How this chapter flows on: Probability (next chapter) closes Class 11; in Class 12, random variables re-import mean and variance as expectation and spread of distributions — with these formulas assumed fluent.