Measures of dispersion | ISC Class 11 Economics Notes

This note covers measures of dispersion in statistics: what dispersion means and why averages alone are not enough, the absolute and relative measures of variation, and the calculation of range, quartile deviation, mean deviation, standard deviation, variance and the coefficient of variation, with worked numericals, comparison tables and exam-style practice for ISC Class 11 Economics.

What is dispersion and why does it matter?

Dispersion (also called variation or spread) measures the extent to which the values of a variable differ from one another or from their central value. Two series can share the same average and yet be very different in character: one may be tightly clustered around the mean, the other widely scattered.

Definition: Dispersion is the degree to which numerical data tend to spread about an average value; the measures of dispersion quantify this spread in a single number.

Consider two students' marks out of 100 in five tests. Student A scores 60, 62, 58, 61, 59; Student B scores 40, 80, 55, 70, 55. Both have a mean of 60, but A is consistent and B is erratic. A single average hides this difference; a measure of dispersion reveals it.

Dispersion matters for three reasons. First, it tells us how representative an average is: a small dispersion means the average describes the data well, a large dispersion means it does not. Second, it is needed for comparison of two or more series, for example comparing the variability of wages in two industries or the consistency of two bowlers. Third, it underlies many later statistical tools such as correlation, regression and quality control.

What are the absolute and relative measures of dispersion?

Measures of dispersion fall into two broad classes. Absolute measures are expressed in the same unit as the original data (rupees, kilograms, marks). They are useful for studying the spread of a single series but cannot be used to compare series measured in different units. Relative measures are pure numbers, free of units, obtained by dividing an absolute measure by an average. They are used to compare the variability of two or more series.

The syllabus names four methods of studying variation: range, standard deviation, quartile deviation and mean (or average) deviation, together with the coefficient of variation. Each is treated in turn below.

How is the range calculated?

The range is the simplest measure of dispersion. It is the difference between the largest and the smallest value in a series.

Range = L − S, where L is the largest value and S is the smallest value.

For a continuous frequency distribution, the range is the upper limit of the highest class minus the lower limit of the lowest class. The coefficient of range expresses the range as a fraction of the sum of the two extremes:

Coefficient of range = (L − S) ÷ (L + S)

Worked example 1. The daily wages (in ₹) of seven workers are 120, 150, 180, 200, 250, 300, 400. Find the range and the coefficient of range.

Answer: L = 400, S = 120. Range = 400 − 120 = ₹280. Coefficient of range = (400 − 120) ÷ (400 + 120) = 280 ÷ 520 = 0.538 (approximately).

The range has real merits: it is easy to understand and quick to compute, and it is useful where extreme values matter, such as in weather forecasting (highest and lowest temperatures) or in quality control of a batch. Its limitations are serious, however. It uses only two observations and ignores all the rest, so a single outlier can distort it completely. It tells us nothing about the shape of the distribution between the extremes, and it is not based on all items, so it is not a reliable measure for detailed analysis.

How is quartile deviation calculated?

The quartile deviation (also called the semi-interquartile range) measures spread around the median. It uses the first quartile Q₁ (the value below which 25 per cent of the observations lie) and the third quartile Q₃ (the value below which 75 per cent lie). The median is Q₂.

Quartile deviation = (Q₃ − Q₁) ÷ 2

Coefficient of quartile deviation = (Q₃ − Q₁) ÷ (Q₃ + Q₁)

For a discrete or continuous series, the quartiles are located using the cumulative frequency. For a continuous series, the class in which the cumulative frequency first reaches N/4 gives Q₁ and the class reaching 3N/4 gives Q₃. The interpolation formula is:

Q₁ = L₁ + [(N/4 − cf) ÷ f] × h, where L₁ is the lower limit of the Q₁ class, cf is the cumulative frequency of the class before it, f is the frequency of the Q₁ class and h is the class width. The same formula with 3N/4 in place of N/4 gives Q₃.

Worked example 2. For the data 10, 12, 15, 18, 20, 22, 25, 28, 30, 35 (ten items), find Q₁, Q₃, the quartile deviation and its coefficient.

Answer: N = 10. Q₁ is the value of the (N+1)/4 = 2.75th item = 12 + 0.75 × (15 − 12) = 14.25. Q₃ is the value of the 3(N+1)/4 = 8.25th item = 28 + 0.25 × (30 − 28) = 28.5. Quartile deviation = (28.5 − 14.25) ÷ 2 = 14.25 ÷ 2 = 7.125. Coefficient = (28.5 − 14.25) ÷ (28.5 + 14.25) = 14.25 ÷ 42.75 = 0.333 (approximately).

Worked example 3. Find the quartile deviation and its coefficient for a distribution with classes 0 to 10, 10 to 20, 20 to 30, 30 to 40 and 40 to 50, whose frequencies are 5, 8, 12, 10 and 5.

Answer: N = 40, and the cumulative frequencies are 5, 13, 25, 35 and 40. N/4 = 10, which first falls within the cumulative frequency of 13, so the Q₁ class is 10 to 20: Q₁ = 10 + [(10 − 5) ÷ 8] × 10 = 10 + 6.25 = 16.25. 3N/4 = 30, which falls in the 30 to 40 class: Q₃ = 30 + [(30 − 25) ÷ 10] × 10 = 35. Quartile deviation = (35 − 16.25) ÷ 2 = 9.375. Coefficient of quartile deviation = (35 − 16.25) ÷ (35 + 16.25) = 18.75 ÷ 51.25 = 0.366 (approximately).

Quartile deviation is better than the range because it is based on the middle 50 per cent of the data and is not affected by extreme values. It is useful when the data are open-ended at both ends. Its limitations are that it still ignores half the observations (the lowest 25 per cent and the highest 25 per cent), it is not based on all items, and it is not suitable for algebraic treatment.

How is mean deviation calculated?

The mean deviation (average deviation) is the arithmetic mean of the absolute deviations of the items from a central value, usually the mean, median or mode. Taking absolute values (ignoring signs) avoids the problem that the signed deviations from the mean always sum to zero.

Mean deviation from mean = Σ|x − x̄| ÷ N, where x̄ is the arithmetic mean and N is the number of items.

For a frequency distribution, the formula becomes MD = Σf|x − x̄| ÷ Σf. The coefficient of mean deviation divides the mean deviation by the average from which it was computed, for example MD ÷ x̄ when deviations are taken from the mean.

Worked example 4. Find the mean deviation from the mean and the coefficient of mean deviation for the data 4, 6, 8, 10, 12.

Answer: Mean x̄ = (4 + 6 + 8 + 10 + 12) ÷ 5 = 40 ÷ 5 = 8. Deviations |x − x̄| are 4, 2, 0, 2, 4, whose sum is 12. Mean deviation = 12 ÷ 5 = 2.4. Coefficient of mean deviation = 2.4 ÷ 8 = 0.3.

Mean deviation has the merit of being based on all the observations, and it is simple to understand. It is less affected by extreme values than the range or standard deviation. Its main limitation is that it ignores the signs of the deviations, which makes it unsuitable for further algebraic work; it also varies according to whether deviations are taken from the mean, median or mode.

How is standard deviation calculated?

The standard deviation (SD, denoted σ) is the most important measure of dispersion. It is the square root of the arithmetic mean of the squared deviations from the mean. Squaring removes the sign problem, and taking the square root returns the measure to the original units.

Standard deviation = √(Σ(x − x̄)² ÷ N) for individual observations, and σ = √(Σf(x − x̄)² ÷ Σf) for a frequency distribution.

The square of the standard deviation is the variance: Variance = σ². A short-cut formula, useful when the mean is not a whole number, is σ = √[(Σx² ÷ N) − x̄²].

Worked example 5. Calculate the standard deviation and variance for the data 2, 4, 6, 8, 10.

Answer: Mean x̄ = 30 ÷ 5 = 6. Deviations are −4, −2, 0, 2, 4; their squares are 16, 4, 0, 4, 16, summing to 40. Variance = 40 ÷ 5 = 8. Standard deviation = √8 = 2.83 (approximately).

Worked example 6. Find the standard deviation and the coefficient of variation for the discrete series x = 2, 4, 6, 8, 10 with frequencies f = 1, 4, 6, 4, 1.

Answer: N = Σf = 16 and Σfx = 2 + 16 + 36 + 32 + 10 = 96, so x̄ = 96 ÷ 16 = 6. The deviations from the mean are −4, −2, 0, 2 and 4, their squares are 16, 4, 0, 4 and 16, and the products f(x − x̄)² are 16, 16, 0, 16 and 16, summing to 64. σ = √(64 ÷ 16) = √4 = 2. Coefficient of variation = (2 ÷ 6) × 100 = 33.33 per cent.

Standard deviation is based on all the observations and is capable of algebraic treatment, which makes it the foundation of correlation, regression and sampling theory. It is also less affected by sampling fluctuations than other measures. Its limitations are that it is difficult to compute, it gives undue weight to extreme values because deviations are squared, and it cannot be used directly to compare series in different units, which is why the coefficient of variation is needed.

What is the coefficient of variation?

The coefficient of variation (CV) is a relative measure of dispersion. It expresses the standard deviation as a percentage of the mean, so that series with different units or very different means can be compared.

Coefficient of variation = (σ ÷ x̄) × 100

A lower CV indicates greater consistency (less relative variability); a higher CV indicates greater variability. The series with the smaller CV is said to be more consistent or more homogeneous.

Worked example 7. Two batsmen have the following scores. Batsman A: mean 40, SD 8. Batsman B: mean 60, SD 15. Which is the more consistent batsman?

Answer: CV of A = (8 ÷ 40) × 100 = 20 per cent. CV of B = (15 ÷ 60) × 100 = 25 per cent. Since A has the lower coefficient of variation, batsman A is more consistent.

The coefficient of variation is widely used to compare the risk of two investments (the one with the lower CV is less risky for a given return) and to compare the variability of prices, wages or production across regions or time periods.

How do the measures of dispersion compare?

The standard deviation is generally regarded as the best absolute measure because it uses every observation and permits algebraic treatment. The coefficient of variation is the best relative measure because it is unit-free. Range and quartile deviation are useful for quick or rough work and for open-ended data, but they are not based on all items.

What are the merits and limitations of the measures of dispersion?

Merits

Each measure has its own merits. The range is simple, quick and useful where extremes matter. Quartile deviation is easy to compute from cumulative frequencies and is unaffected by extreme values. Mean deviation uses all items and is easy to understand. Standard deviation uses all items, is capable of algebraic treatment and forms the basis of advanced statistics. The coefficient of variation allows comparison of series with different units or means.

Limitations

The range ignores most of the data and is distorted by a single outlier. Quartile deviation ignores the outer quarters of the data. Mean deviation ignores signs and so resists algebraic treatment. Standard deviation is harder to compute and over-weights extreme values. No single measure is best for every purpose; the choice depends on the nature of the data and the use to which the result is put.

Glossary

• Dispersion: the extent to which values in a series differ from one another or from their average.
• Range: the difference between the largest and the smallest value in a series.
• Quartile: a value that divides an ordered series into four equal parts; Q₁, Q₂ and Q₃ are the three quartiles.
• Quartile deviation: half the difference between the third and first quartiles.
• Mean deviation: the arithmetic mean of the absolute deviations of items from a central value.
• Standard deviation: the square root of the mean of the squared deviations from the arithmetic mean.
• Variance: the square of the standard deviation.
• Coefficient of variation: the standard deviation expressed as a percentage of the mean.
• Absolute measure: a measure of dispersion expressed in the same unit as the data.
• Relative measure: a unit-free measure obtained by dividing an absolute measure by an average.
• Consistency: the property of a series with low relative variability, shown by a low coefficient of variation.

Common errors and misconceptions

• Misconception: Two series with the same mean have the same dispersion. Correct: Equal means say nothing about spread; the standard deviations or coefficients of variation may differ greatly.
• Misconception: The range is a reliable measure because it uses the extremes. Correct: It uses only two items and is easily distorted by a single outlier.
• Misconception: Mean deviation is found by averaging signed deviations. Correct: Signed deviations from the mean always sum to zero, so absolute values must be used.
• Misconception: Variance and standard deviation are the same. Correct: Variance is the square of the standard deviation and is in squared units.
• Misconception: The series with the larger standard deviation is always more variable. Correct: Only when the means are similar; otherwise the coefficient of variation must be compared.
• Misconception: Quartile deviation uses all the observations. Correct: It uses only the middle 50 per cent of the data.

Exam-style questions with model answers

Q1. Define dispersion and state any two measures of it. [2 marks]

1. Dispersion is the degree to which the values of a variable differ from one another or from their average. Two measures are the range and the standard deviation.

Q2. The marks of five students are 10, 20, 30, 40, 50. Find the range and the coefficient of range. [3 marks]

1. Largest value L = 50, smallest S = 10. Range = 50 − 10 = 40 marks. Coefficient of range = (50 − 10) ÷ (50 + 10) = 40 ÷ 60 = 0.667 (approximately). The range is expressed in marks, while the coefficient is a pure number.

Q3. For the data 5, 10, 15, 20, 25, find the mean deviation from the mean and the coefficient of mean deviation. [4 marks]

1. Mean = 75 ÷ 5 = 15. Deviations |x − 15| are 10, 5, 0, 5, 10, summing to 30. Mean deviation = 30 ÷ 5 = 6. Coefficient = 6 ÷ 15 = 0.4. The mean deviation is in the same units as the data, while the coefficient is unit-free.

Q4. Calculate the standard deviation and variance for the data 3, 6, 9, 12, 15. [4 marks]

1. Mean x̄ = (3 + 6 + 9 + 12 + 15) ÷ 5 = 45 ÷ 5 = 9.
2. The deviations from the mean are −6, −3, 0, 3 and 6, and their squares are 36, 9, 0, 9 and 36.
3. The sum of the squared deviations is 90, so variance = 90 ÷ 5 = 18.
4. Standard deviation = √18 = 4.24 (approximately), in the same units as the data, while the variance is in squared units.

Q5. Two workers A and B have mean daily output of 50 and 60 units with standard deviations of 5 and 9 units respectively. Which worker is more consistent? [5 marks]

1. The two workers have different mean outputs, so their standard deviations alone cannot be compared fairly; the coefficient of variation, a relative measure, is used instead.
2. CV of A = (σ ÷ x̄) × 100 = (5 ÷ 50) × 100 = 10 per cent.
3. CV of B = (9 ÷ 60) × 100 = 15 per cent.
4. A has the lower coefficient of variation, so the output of worker A varies less, relative to its mean, than the output of worker B.
5. Worker A is therefore the more consistent worker, even though worker B produces more on average.

Q6. Explain the merits and limitations of the range as a measure of dispersion. [5 marks]

1. Merits: the range is simple to understand and quick to compute; it is useful where extreme values matter, such as maximum and minimum temperatures; it is easy to find from a frequency distribution using the class limits. Limitations: it uses only the largest and smallest values and ignores all other observations; a single outlier can distort it; it gives no information about the shape of the distribution between the extremes; it is not based on all items and so is not reliable for detailed analysis. For these reasons the range is usually used only for a rough, quick indication of spread.

Q7. Distinguish between absolute and relative measures of dispersion, giving one example of each. [3 marks]

1. Absolute measures are expressed in the same unit as the data and are used to study the spread of a single series; an example is the standard deviation. Relative measures are unit-free numbers obtained by dividing an absolute measure by an average, and are used to compare two or more series; an example is the coefficient of variation.

Key takeaways

• Dispersion measures how far values spread around an average; two series with the same mean can differ greatly in spread.
• Absolute measures carry the unit of the data; relative measures are unit-free and used for comparison.
• Range = L − S; it is quick but uses only two observations and is distorted by outliers.
• Quartile deviation = (Q₃ − Q₁) ÷ 2; it uses the middle 50 per cent and suits open-ended data.
• Mean deviation averages absolute deviations from a central value and uses all items.
• Standard deviation is the square root of the mean of squared deviations and is the best absolute measure.
• Coefficient of variation = (σ ÷ x̄) × 100; the lower the CV, the more consistent the series.
• Variance is the square of the standard deviation and is expressed in squared units.

Test yourself

What is the coefficient of range for the values 20 and 80?

(80 − 20) ÷ (80 + 20) = 60 ÷ 100 = 0.6.

Why are absolute deviations used in mean deviation?

Because signed deviations from the mean always sum to zero, so their average would be zero; absolute values avoid this.

What is the relationship between variance and standard deviation?

Variance is the square of the standard deviation; the standard deviation is the square root of the variance.

Which measure is best for comparing the variability of two series in different units?

The coefficient of variation, because it is a unit-free relative measure.

Find the range of the data 7, 3, 9, 15, 2.

Largest = 15, smallest = 2, so range = 15 − 2 = 13.

What does a low coefficient of variation indicate?

It indicates low relative variability, that is, the series is more consistent or homogeneous.