Lesson 11 of 11 · 15 min
Chapter review
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Dispersion and mean deviation, full chapter
Must-know facts
17 facts
- 01Range = maximum − minimum; it uses only the two extreme values.
- 02Sum of deviations from the mean is always 0: Σ(xᵢ − x̄) = 0.
- 03M.D.(a) = (1/n) Σ|xᵢ − a|; for frequencies (1/N) Σ fᵢ|xᵢ − a| with N = Σfᵢ.
- 04The sum of absolute deviations is least about the median, so M.D.(M) ≤ M.D.(x̄).
- 05Discrete median: the value whose cumulative frequency first reaches N/2.
- 06Continuous median = l + ((N/2 − C)/f) × h.
- 07Class data use mid-points; make gapped classes continuous by ±0.5 first.
- 08Step-deviation mean: x̄ = a + h (Σfᵢdᵢ)/N with dᵢ = (xᵢ − a)/h.
- 09Variance 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
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.