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Measures of Dispersion

Ch. 8Std 11

Easy Overview

Here's a scenario. Two students, same average score of 75 on five tests. One scored 74, 75, 76, 75, 75 — incredibly consistent. The other scored 50, 100, 60, 90, 75 — wild swings. Same average, but completely different stories. That difference — how spread out the numbers are — is what dispersion measures. Measures of dispersion tell you about the variability or spread within a dataset. Central tendency (mean, median, mode) tells you where the 'centre' is. Dispersion tells you how tightly data clusters around that centre or how loosely it's scattered. Both are needed to understand a dataset fully. The simplest measure is the range: just the difference between the maximum and minimum values. If your data is 2, 5, 7, 10, 12, the range is 12 − 2 = 10. The range is easy to calculate and understand, but it has a serious flaw: it only uses two values from the entire dataset. If there's an outlier, the range can misrepresent the typical spread. For example, data: 1, 45, 47, 48, 50 has range 49, but most values are clustered between 45 and 50. Mean deviation improves on the range by using all the data. Find the mean, find how far each data point is from the mean (absolute difference), then average those distances. It tells you the average distance from the mean. For grouped data (frequency distributions), you use the midpoints of classes and weigh by frequencies. Variance and standard deviation are the most important measures. Variance is the average of squared deviations from the mean. We square the deviations partly because it makes the mathematics nicer (derivatives, further statistics), but also because it gives more weight to larger deviations — a point far from the mean contributes disproportionately. For a population, variance σ² = ∑(xᵢ − μ)²/N. For a sample, we divide by (n − 1) instead of n — this is Bessel's correction, which makes the sample variance an unbiased estimator of the population variance. Standard deviation σ is the square root of variance. It's in the same units as the original data (unlike variance, which is in squared units). This makes it more interpretable. A small standard deviation means the data is clustered closely around the mean. A large one means it's spread out. The coefficient of variation (CV) is used to compare variability across datasets with different units or different means. CV = (σ/μ) × 100%. It's a dimensionless percentage. For example, comparing height variability (in cm) with weight variability (in kg) — different units make standard deviations incomparable, but CV solves that. We'll also study combined standard deviation — what happens when you merge two groups. This is useful when you have statistics for subgroups but need the overall measure. And we'll see how adding a constant or multiplying by a constant affects these measures. Dispersion is the language of reliability, consistency, and risk. In finance, standard deviation measures investment risk. In manufacturing, it measures quality control. In education, it tells you if a test was fair. Master this chapter and you'll see the world in terms of spread, not just average.

Range — quick but rough

Range = Maximum value − Minimum value. It's the simplest measure of dispersion. For the dataset 10, 20, 30, 40, 50, the range is 40. It tells you immediately how wide the spread is. But it completely ignores the internal distribution. Two datasets can have the same range but entirely different shapes: 10, 10, 10, 10, 50 (range 40, mostly left-skewed) vs 10, 20, 30, 40, 50 (range 40, evenly spread). Use range for a quick, rough estimate only.

Quartile deviation (semi-interquartile range)

The quartile deviation is Q = (Q₃ − Q₁)/2, where Q₁ is the lower quartile (25th percentile) and Q₃ is the upper quartile (75th percentile). This measure ignores the extreme values and focuses on the spread of the middle 50% of data. It's more robust than the range because outliers don't affect it. The interquartile range itself (Q₃ − Q₁) is also commonly used. Quartile deviation is especially useful for skewed distributions.

Mean deviation from the mean

Mean deviation (MD) = (1/n) ∑|xᵢ − x̄|, where x̄ is the mean. It's the average absolute distance from the mean. For ungrouped data: MD = ∑|xᵢ − x̄|/n. For grouped data with frequencies: MD = ∑fᵢ|xᵢ − x̄|/∑fᵢ. Mean deviation uses absolute values to handle negative differences. While intuitive, it has mathematical limitations — absolute values don't play well with calculus, which is one reason variance is often preferred.

Mean deviation from the median

You can also compute mean deviation about the median instead of the mean. MD (median) = (1/n) ∑|xᵢ − M|, where M is the median. For symmetric distributions, this equals the mean deviation about the mean. For skewed distributions, the median-based version is often smaller and more representative. An interesting property: the sum of absolute deviations is minimised when taken about the median.

Variance — the average squared deviation

Variance is the most widely used measure of dispersion. For a population: σ² = ∑(xᵢ − μ)²/N. For a sample: s² = ∑(xᵢ − x̄)²/(n − 1). The n−1 denominator is Bessel's correction — it makes s² an unbiased estimator of σ². Variance penalises larger deviations more heavily than mean deviation because it squares them. A single extreme value can dramatically increase the variance.

Standard deviation — variance in the original units

Standard deviation (SD) is the square root of variance: σ = √σ² or s = √s². It brings the measure back to the original units of the data. If the data is in metres, variance is in square metres, but SD is in metres — making it directly interpretable. For roughly bell-shaped distributions, about 68% of data falls within 1σ of the mean, 95% within 2σ, and 99.7% within 3σ (the empirical rule or 68-95-99.7 rule).

Variance and SD for ungrouped data

Ungrouped data is just a list of values. Steps: (1) find the mean x̄, (2) subtract mean from each xᵢ, (3) square each difference, (4) sum the squares, (5) divide by n (or n-1 for sample). For a shortcut formula: σ² = (∑xᵢ²/n) − x̄². This avoids calculating each deviation individually — handy for large datasets. Example: for data 2, 4, 6, 8, 10: mean = 6, ∑x² = 220, σ² = 220/5 − 36 = 44 − 36 = 8, σ = √8 ≈ 2.83.

Variance and SD for grouped frequency data

For a frequency distribution, each value xᵢ has a frequency fᵢ. Formula: σ² = (∑fᵢxᵢ²/∑fᵢ) − (∑fᵢxᵢ/∑fᵢ)². The second term is just the mean squared. Using the step-deviation method (also called the assumed mean method) makes calculation easier when the class intervals are equal. Choose an assumed mean A, define dᵢ = (xᵢ − A)/h (where h is the class width), then σ² = h²[(∑fᵢdᵢ²/∑fᵢ) − (∑fᵢdᵢ/∑fᵢ)²].

Step-deviation method — the shortcut for large data

When you have equal class intervals, the step-deviation method avoids large numbers. Steps: (1) take a guessed mean A (usually the midpoint of the middle class), (2) compute dᵢ = (xᵢ − A)/h, (3) find ∑fᵢdᵢ and ∑fᵢdᵢ², (4) apply the formula. This is the fastest way to compute standard deviation for grouped data with uniform class width.

Combined standard deviation of two groups

When you merge two groups with sizes n₁, n₂, means x̄₁, x̄₂, and SDs σ₁, σ₂, the combined SD is: σ = √[(n₁σ₁² + n₂σ₂² + n₁d₁² + n₂d₂²)/(n₁ + n₂)], where d₁ = x̄₁ − x̄ and d₂ = x̄₂ − x̄ (x̄ is the combined mean). This formula accounts for both the within-group variation (σ₁², σ₂²) and the between-group variation (d₁², d₂²). The combined variance is always at least as large as the individual variances.

Coefficient of variation — comparing apples and oranges

CV = (σ/μ) × 100%. Since it's a percentage, it's unitless. You can compare the variability of heights (in cm) with weights (in kg) using CV. A CV of 5% means the SD is 5% of the mean. Higher CV = more relative variability. In investment, a lower CV indicates a better risk-to-return ratio. In manufacturing, a lower CV indicates more consistent product quality.

Effect of change in origin and scale

Adding a constant to every data point changes the mean but NOT the standard deviation (SD is a measure of spread, not location). So if you add 10 to every value, SD stays the same. Multiplying every value by a constant k multiplies SD by |k| and variance by k². For example, converting from metres to centimetres (×100) multiplies SD by 100. These properties are useful when working with transformed data.

Empirical relationship between measures of dispersion

For moderately skewed distributions (approximately bell-shaped), there's a rough relationship: σ ≈ 1.4826 × MAD (median absolute deviation) and quartile deviation ≈ (2/3)σ. The range is approximately 6σ for normally distributed data. These relationships give you a quick sanity check: if the range divided by 6 is far from the SD, the data might not be normally distributed.

Merits and demerits of different measures

Range: simplest, quickest, but uses only two values and is outlier-sensitive. Quartile deviation: robust, ignores extremes, but ignores half the data's distribution. Mean deviation: uses all data, intuitive, but mathematically inconvenient (absolute values). Variance/SD: uses all data, mathematically tractable (squares allow calculus), but sensitive to outliers and has squared units (SD fixes the unit issue). CV: best for comparing different datasets, but meaningless if mean is near zero.

Applications of dispersion in real life

Dispersion is everywhere. In finance: standard deviation of stock returns measures risk. In quality control: manufacturers monitor the SD of product dimensions to ensure consistency. In education: spread of test scores indicates how well a test differentiates students. In weather: variance in temperature tells you about climate stability. In sports: consistency of a player's performance is measured by how much their scores vary from their average.

Key Points

  • •Range = Max − Min — simple but only uses two values
  • •Quartile deviation = (Q₃ − Q₁)/2 — robust to outliers, based on middle 50%
  • •Mean deviation (about mean) = ∑|xᵢ − x̄|/n — average absolute distance
  • •Mean deviation is minimised when taken from the median
  • •Population variance σ² = ∑(xᵢ − μ)²/N
  • •Sample variance s² = ∑(xᵢ − x̄)²/(n − 1) — Bessel's correction for unbiasedness
  • •Standard deviation = √variance — same units as the original data
  • •For grouped data: variance = (∑fx²/∑f) − (∑fx/∑f)²
  • •Step-deviation method: σ² = h²[(∑fd²/∑f) − (∑fd/∑f)²]
  • •Combined SD accounts for both within-group and between-group variation
  • •Coefficient of variation CV = (σ/μ) × 100% — for comparing datasets
  • •Adding a constant does NOT change SD; multiplying by k multiplies SD by |k|
  • •68-95-99.7 rule: in a normal distribution, data within 1σ, 2σ, 3σ of the mean
  • •SD is more sensitive to outliers than mean deviation due to squaring
  • •CV is meaningless when the mean is close to zero

Practice Questions

  • Find the range, mean deviation from the mean, variance, and standard deviation for the data: 5, 8, 12, 15, 20.
  • Two teams have the same mean score of 65. Team A has SD = 8, Team B has SD = 15. Which team is more consistent, and why?
  • Find the coefficient of variation for the heights: 150, 155, 160, 165, 170 cm. Interpret the result.
  • The mean and standard deviation of 100 items are 50 and 4. If each item is increased by 5, what happens to the mean and standard deviation?
  • Calculate the standard deviation for the following frequency distribution: Class 0-10 (f=3), 10-20 (f=7), 20-30 (f=12), 30-40 (f=8), 40-50 (f=5).
  • The means and SDs of two groups are: Group 1 (n=50, mean=60, SD=8), Group 2 (n=70, mean=70, SD=10). Find the combined mean and combined SD.
  • Explain why variance uses squared deviations instead of absolute deviations. What are the advantages and disadvantages of each?
  • The CV of the first dataset is 12% and the CV of the second is 8%. The means are 100 and 200 respectively. Which dataset has more absolute variability?