Measures of Dispersion
1. Range
Range (R):
R = L - S
Where L = largest value, S = smallest value.
Coefficient of Range (CR):
CR = (L - S) / (L + S)
2. Quartiles
Interquartile Range (IQR): IQR = Q3 - Q1
Quartile Deviation (QD): QD = (Q3 - Q1) / 2
Coefficient of Quartile Deviation (CQD): CQD = (Q3 - Q1) / (Q3 + Q1)
3. Mean Deviation (M.D.) / Average Deviation
Measures the average absolute deviation from a central value.
| Series Type |
From Mean (X̄) |
From Median (Md) |
From Mode (Mo) |
| Individual |
M.D.X̄ = ∑|X - X̄| / n |
M.D.Md = ∑|X - Md| / n |
M.D.Mo = ∑|X - Mo| / n |
| Discrete |
M.D.X̄ = ∑f|X - X̄| / N |
M.D.Md = ∑f|X - Md| / N |
M.D.Mo = ∑f|X - Mo| / N |
| Continuous |
M.D.X̄ = ∑f|m - X̄| / N |
M.D.Md = ∑f|m - Md| / N |
M.D.Mo = ∑f|m - Mo| / N |
Coefficient of Mean Deviation (CMD): CMD = M.D. / (Mean / Median / Mode)
4. Standard Deviation (σ)
Measures spread of data around the mean.
| Method |
Individual Series |
Discrete Series |
Continuous Series |
| Actual Mean / Direct Method |
σ = √(∑(X - X̄)2 / n) |
σ = √(∑f(X - X̄)2 / N) |
σ = √(∑f(m - X̄)2 / N) |
| Assumed Mean / Shortcut Method |
σ = √( (∑d2 / n) - ( (∑d) / n )2 ) |
σ = √( (∑fd2 / N) - ( (∑fd) / N )2 ) |
σ = √( (∑fd2 / N) - ( (∑fd) / N )2 ) |
| Step Deviation Method |
— |
— |
σ = √( (∑fd'2 / N) - ( (∑fd') / N )2 ) × h |
Where: d = X - A (deviation from assumed mean), d' = (X - A) / h (step deviation), h = class width
Coefficient of Standard Deviation (CSD): CSD = σ / X̄
Coefficient of Variation (CV): CV = (σ / X̄) × 100%
5. Variance
Variance (σ2): Square of standard deviation.
σ2 = ∑(X - X̄)2 / n (or N / f)
Standard Deviation is the square root of variance: σ = √(σ2)
6. Combined Standard Deviation
For combining two datasets with different means and SDs:
σ12 = √( ( n1(σ12 + d12) + n2(σ22 + d22) ) / (n1 + n2) )
Where: d1 = X̄1 - X̄12, d2 = X̄2 - X̄12
X̄12 = combined mean of both datasets.