Measures of Dispersion: Range, Quartile Deviation, Mean Deviation and Standard Deviation

Overview

Measures of central tendency (mean/median/mode) provide a typical value, but they do not show how much the data varies. Two data sets can have the same mean but very different spread. Dispersion measures the degree of variability around the average. In business, dispersion is used to study risk, quality variation, wage differences, price fluctuations and sales stability. This topic explains key measures: Range, Quartile Deviation, Mean Deviation and Standard Deviation, along with CV for comparison.

Dispersion: meaning and need

Dispersion refers to the extent to which observations scatter around a central value.

Need/importance:

• tells consistency/stability of data
• compares variability between series
• helps measure risk and uncertainty
• supports quality control and performance analysis

Range: meaning and formula

Range is the difference between the largest and smallest observations. $R = L - S$ Coefficient of range: $\\text{Coeff. of Range} = \\frac{L - S}{L + S}$

Quartiles and quartile deviation (QD)

Quartiles divide ordered data into four equal parts:

• $Q_1$: first quartile (25th percentile)
• $Q_3$: third quartile (75th percentile)

Quartile deviation (semi-interquartile range): $QD = \\frac{Q_3 - Q_1}{2}$ Coefficient of QD: $\\text{Coeff. of QD} = \\frac{Q_3 - Q_1}{Q_3 + Q_1}$

Mean deviation (MD)

Mean deviation is the average of absolute deviations of observations from a central value (usually mean or median).

Individual series (about mean): $MD = \\frac{\\sum |x - \\bar{x}|}{n}$ Coefficient of MD (about mean): $\\text{Coeff. of MD} = \\frac{MD}{\\bar{x}}$

Standard deviation (SD) and variance

Standard deviation is the square root of the average of squared deviations from the mean. It is the most widely used measure of dispersion.

Individual series: $\\sigma = \\sqrt{\\frac{\\sum (x - \\bar{x})^2}{n}}$ Variance: $\\sigma^2 = \\frac{\\sum (x - \\bar{x})^2}{n}$ For discrete frequency series: $\\sigma = \\sqrt{\\frac{\\sum f(x - \\bar{x})^2}{\\sum f}}$

Coefficient of variation (CV)

CV compares variability between different series: $CV = \\frac{\\sigma}{\\bar{x}} \\times 100$ Lower CV = more consistency (less relative variation).

Comparison of dispersion measures (table)

Short worked examples (exam style)

Example 1 (Range)

Data: 5, 8, 10, 12, 15
Range $R = 15 - 5 = 10$

Example 2 (SD – very small set)

Data: 2, 4, 6
Mean $\\bar{x} = (2+4+6)/3 = 4$
Deviations: -2, 0, +2; squares: 4, 0, 4
Variance $\\sigma^2 = (4+0+4)/3 = 8/3$
SD $\\sigma = \\sqrt{8/3}$

Uses of dispersion in business

• compare stability of sales across months (lower SD = more stable)
• measure risk in returns and costs
• study wage inequality within departments
• quality control: variation in product weights/defects

Common exam mistakes

• confusing SD with variance (SD is square root)
• using negative deviations in MD (MD uses absolute values)
• forgetting coefficient formulas (range/QD/CV)

Exam tips (how to write 3/5 marks)

• For 3 marks: definition + formula + one small example.
• For 5 marks: explain 2–3 measures + comparison table + CV concept + conclusion.

Key Terms

• Dispersion — spread of data around average.
• Range — L − S.
• QD — (Q3 − Q1)/2.
• MD — mean of absolute deviations.
• SD — square root of variance.
• CV — (SD/mean) × 100.

Quick Recap (1-minute revision)

• Dispersion shows variability and risk.
• Range uses extremes; SD uses all values (best).
• CV compares consistency across series (lower CV = more stable).