Dispersion: range, quartile deviation and mean deviation

1. Meaning of dispersion

An average tells us the centre of a series but not how the items are spread around it. Two series can have the same mean and look quite different: 48, 50, 52 and 10, 50, 90 both average 50. Dispersion measures the extent to which items vary from the central value. Bowley called it “the measure of the variation of the items”; Spiegel, “the degree to which numerical data tend to spread about an average value.”

Objects: to judge how far the average is representative; to compare the variability (consistency, uniformity) of two or more series; to control variation (quality control); and as a basis for further analysis (correlation, regression, skewness).

Absolute measures are in the units of the data (rupees, kg) and cannot compare series in different units. Relative measures (coefficients) are pure numbers and can.

A good measure should be rigidly defined, simple, based on all items, capable of algebraic treatment, and not unduly affected by extreme values or sampling fluctuations. Standard deviation meets most of these; it is in the next note. The Lorenz curve is the graphic method.

2. Range

Range is the difference between the largest (L) and the smallest (S) value. For a grouped table, L = upper limit of the highest class and S = lower limit of the lowest class.

Worked (June 2025). Marks 10–20 to 60–70 (f = 5, 8, 10, 7, 12, 9). L = 70, S = 10. Range = 70 − 10 = 60. Coefficient of range = 60 / 80 = 0.75. The frequencies play no part, which is the main weakness of the measure.

Merits (January 2026 short note): simplest to understand and calculate; gives a quick idea of spread; used in quality control (range charts), in stock-market reporting (daily high and low), and in weather reports (maximum and minimum temperature). Demerits: depends only on two extreme items; affected by a single unusual value; cannot be found for open-end classes; tells nothing about the distribution of items between the extremes.

3. Quartile deviation (semi-interquartile range)

The inter-quartile range Q₃ − Q₁ covers the middle 50% of items; half of it is the quartile deviation.

Worked (January 2026 weights). 90–100: 20, 100–110: 25, 110–120: 40, 120–130: 35, 130–140: 30; N = 150; c.f. 20, 45, 85, 120, 150.

• Q₁: N/4 = 37.5, class 100–110: Q₁ = 100 + (37.5 − 20)/25 × 10 = 100 + 7 = 107.
• Q₃: 3N/4 = 112.5, class 120–130: Q₃ = 120 + (112.5 − 85)/35 × 10 = 120 + 7.857 = 127.857.
• QD = (127.857 − 107)/2 = 10.43; coefficient = 20.857 / 234.857 = 0.089.

Merits (January 2025 short note): easy to compute; not affected by extreme values; the only measure usable with open-end classes; good for skewed series. Demerits: ignores the top and bottom 25% of items; not capable of algebraic treatment; affected by sampling fluctuations; it is a positional measure, not a true deviation from an average.

4. Mean deviation (average deviation)

Mean deviation is the arithmetic mean of the absolute deviations (signs ignored) of the items from an average, usually the median (where it is least) or the mean.

MD = Σf|x − A| / N, where A is the median or mean.

Worked (August 2024). Runs scored 5, 10, 15 … 40 in 16, 32, 36, 44, 28, 18, 12, 14 matches. Find MD and its coefficient from mean and median.

N = 200; Σfx = 80 + 320 + 540 + 880 + 700 + 540 + 420 + 560 = 4,040; mean = 20.2. c.f. 16, 48, 84, 128 …; the (N + 1)/2 = 100.5th item lies at c.f. 128, so median = 20.

• MD from mean = 1,531.2 / 200 = 7.656; coefficient = 7.656 / 20.2 = 0.379.
• MD from median = 1,520 / 200 = 7.60; coefficient = 7.60 / 20 = 0.38.

The MD about the median (7.60) is smaller than about the mean (7.656), as theory says it should be.

Merits (January 2026 short note): based on all items; simple to understand; less affected by extreme values than SD; useful for small samples and in reports meant for the general public, where a simple idea of average spread is enough. Demerits: ignoring signs is mathematically illogical; not capable of further algebraic treatment; not well defined, since it may be taken from mean, median or mode; it is rarely used in advanced work.

How this is examined

“Define dispersion. Explain the various measures of dispersion”: January 2025 (ten marks), June 2025 (six marks) and the 100-mark January 2026 paper (with merits and demerits). Range and its coefficient: June 2025. “Define range. What are the merits of range?” and “mean deviation, merits and demerits”: January 2026. Quartile deviation: January 2025. Mean deviation from mean and median: August 2024. For the essay, use the table in section 1 as your skeleton and add merits and demerits under each measure.