Lesson 11 of 11 · 15 min

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Dispersion and mean deviation, full chapter
Embibe: Achieve CBSE Class 11&12 · Hinglish · Whole chapter · Open on YouTube

Must-know facts

17 facts

1Range = maximum − minimum; it uses only the two extreme values.
2Sum of deviations from the mean is always 0: Σ(xᵢ − x̄) = 0.
3M.D.(a) = (1/n) Σ|xᵢ − a|; for frequencies (1/N) Σ fᵢ|xᵢ − a| with N = Σfᵢ.
4The sum of absolute deviations is least about the median, so M.D.(M) ≤ M.D.(x̄).
5Discrete median: the value whose cumulative frequency first reaches N/2.
6Continuous median = l + ((N/2 − C)/f) × h.
7Class data use mid-points; make gapped classes continuous by ±0.5 first.
8Step-deviation mean: x̄ = a + h (Σfᵢdᵢ)/N with dᵢ = (xᵢ − a)/h.
9Variance σ² = (1/n) Σ(xᵢ − x̄)² = (Σxᵢ²)/n − x̄²; σ is its positive square root.
10Frequency form: σ² = (1/N²)[N Σfᵢxᵢ² − (Σfᵢxᵢ)²].
11Shortcut: σₓ² = (h²/N²)[N Σfᵢyᵢ² − (Σfᵢyᵢ)²], so σₓ = h σᵧ.
12Adding a constant: mean shifts, variance and σ unchanged.
13Multiplying by k: mean × k, variance × k², σ × |k|.
14Wrong entry: correct Σx and Σx² = n(σ² + x̄²), then recompute.
15Missing two values: mean gives x + y, variance gives x² + y², then (x − y)² = 2(x² + y²) − (x + y)².
16σ = 0 exactly when every observation equals the mean.
17NCERT results: batsmen with mean 53 and ranges 117 and 14; 6, 8, …, 24 has variance 33; Ex. results σ ≈ 6.77 and 14.18.

Common traps

Where marks are lost

Averaging the signed deviations to measure spread.

Deviations from the mean always sum to 0; take absolute values or squares first.

Dividing Σfᵢ|xᵢ − x̄| by the number of distinct values.

Divide by N = Σfᵢ, the total number of observations.

Using the class limits instead of mid-points in grouped data.

Every class is represented by its mid-point xᵢ.

Taking the median class as the one with the largest frequency.

The median class is where the cumulative frequency first reaches N/2.

Forgetting h (or using h instead of h²) in the step-deviation variance.

σₓ² = h² σᵧ²; σₓ = h σᵧ.

Thinking that adding 10 to every observation raises the standard deviation.

A shift moves the mean only; every deviation, and so σ, stays the same.

Multiplying the variance by k when every value is multiplied by k.

Variance scales by k² and σ by |k|.

Reporting the variance when the question asks for standard deviation.

σ is the positive square root of σ², in the data's own unit.

Correcting only Σx after a wrong entry.

Correct both Σx and Σx², then use σ² = Σx²/n − x̄² with the new mean.

Formulas

11 to know

Mean (raw)

x̄ = (1/n) Σ xᵢ

Deviations from x̄ add to 0.

Range

Range = maximum value − minimum value

Uses only the extremes.

Mean deviation (raw)

M.D.(a) = (1/n) Σ |xᵢ − a|

a is usually x̄ or the median M.

Mean deviation (frequency)

M.D.(a) = (1/N) Σ fᵢ |xᵢ − a|, N = Σ fᵢ

Class data use mid-points xᵢ.

Step-deviation mean

x̄ = a + h × (Σ fᵢdᵢ)/N, dᵢ = (xᵢ − a)/h

a is an assumed mean, h the common width.

Median (continuous)

M = l + ((N/2 − C)/f) × h

l, f, h: median class; C: cumulative frequency before it.

Variance (raw)

σ² = (1/n) Σ (xᵢ − x̄)² = (Σ xᵢ²)/n − x̄²

In squared units.

Standard deviation

σ = √[(1/n) Σ (xᵢ − x̄)²]

Positive root; same unit as the data.

Variance (frequency)

σ² = (1/N) Σ fᵢ (xᵢ − x̄)² = (1/N²)[N Σ fᵢxᵢ² − (Σ fᵢxᵢ)²]

N = Σ fᵢ.

Shortcut method

σₓ² = (h²/N²)[N Σ fᵢyᵢ² − (Σ fᵢyᵢ)²], yᵢ = (xᵢ − A)/h

x̄ = A + h ȳ and σₓ = h σᵧ.

Shift and scale

y = kx + a ⇒ ȳ = k x̄ + a, σᵧ² = k² σₓ², σᵧ = |k| σₓ

Adding a changes nothing in the spread.

Key terms

13 terms

Measure of central tendency: A single value (mean, median or mode) around which the data are centred.
Dispersion: How spread out or scattered the observations are around a central value.
Range: The difference between the largest and smallest observations.
Deviation: The difference x − a between an observation and a fixed value a.
Mean deviation: The average distance of the observations from a central value, using absolute deviations.
Discrete frequency distribution: A table of distinct values with the number of times each occurs.
Continuous frequency distribution: Data grouped into gap-free class intervals, each with its frequency.
Cumulative frequency: The running total of frequencies up to and including a value or class.
Median class: The class whose cumulative frequency first equals or exceeds N/2.
Assumed mean: A convenient value near the centre from which deviations are measured to ease arithmetic.
Step deviation: A deviation from the assumed mean divided by a common factor h.
Variance: The mean of the squared deviations from the mean, written σ².
Standard deviation: The positive square root of the variance, written σ, in the data's unit.