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Measures of dispersion: Class 11 Statistics
Range, quartile deviation, mean deviation, variance, standard deviation and the coefficient of variation, with worked calculations.
Key points
Averages show the centre of the data but not how far the values are spread. Two groups can have the same mean and yet differ greatly in spread. A measure of dispersion describes the spread of the values.
The range is the largest value minus the smallest. It is easy to find but depends on two values only, so one extreme value changes it. For a frequency table, the range is the largest class limit minus the smallest.
The quartile deviation is half the difference between the third and first quartiles: QD = (Q3 - Q1) / 2. It ignores the extreme values and describes the spread of the middle half of the data.
The mean deviation is the average of the distances of the values from the mean, taken without regard to sign: MD = sum of |x - mean| / n.
The variance is the average of the squared deviations from the mean. For a population, variance = sum of (x - mean)² / n. The standard deviation is the square root of the variance. It has the same unit as the data and is the most widely used measure of dispersion. For a sample, the divisor n - 1 is used to estimate the population variance.
A short formula is useful for calculation: variance = (sum of x²) / n - (mean)². Find the sum of the squares, divide by n and subtract the square of the mean.
The coefficient of variation compares the spread of data with different units or means: CV = (standard deviation / mean) x 100. A smaller CV shows a more consistent series. The coefficient of dispersion based on the range is (largest - smallest) / (largest + smallest).
When choosing a measure, use the standard deviation for a full description of spread, the quartile deviation when there are extreme values or open ended classes, and the range for a quick idea. Always give the unit with the result.
Study tip: Make a table with columns for x, x - mean and (x - mean)². It keeps the working tidy and makes errors easy to find.
Definitions and flashcards
Say the meaning first, then open the card to check.
Dispersion
The extent to which values are spread about the centre.
Range
Largest value minus smallest value.
Quartile deviation
Half the difference between the third and first quartiles.
Variance
The average of the squared deviations from the mean.
Standard deviation
The square root of the variance.
Coefficient of variation
Standard deviation divided by the mean, times 100.
Formula sheet
| Name | Formula | What it means |
|---|---|---|
| Range | Largest - smallest | The total spread of the data. |
| Quartile deviation | (Q3 - Q1) / 2 | The spread of the middle half. |
| Variance | Sum of (x - mean)² / n | The average squared distance from the mean. |
| Standard deviation | Square root of the variance | The spread in the original unit. |
| Coefficient of variation | (SD / mean) x 100 | A relative measure of spread. |
Short questions with model answers
These are the kind of questions asked for 2, 3 and 5 marks. Try each one before opening the answer.
What is meant by dispersion? (2 marks)
It is the extent to which the values of a set of data are spread about their average.
Why is the standard deviation more useful than the range? (3 marks)
The range uses only two values, the largest and smallest, and is changed by one extreme value. The standard deviation uses every value, so it describes the spread of the whole data and is easy to use in later work.
Define the coefficient of variation. (2 marks)
It is the standard deviation divided by the mean and multiplied by 100. It is used to compare the spread of series with different units or means.
State the formula for the quartile deviation. (2 marks)
Quartile deviation = (Q3 - Q1) / 2.
State two uses of dispersion. (3 marks)
It shows how reliable an average is, because a small spread means that the average represents the data well. It is also used to compare the consistency of two or more groups.
Numericals with worked steps
Find the variance and standard deviation of 2, 4, 4, 4, 5, 5, 7 and 9.
- n = 8 and the sum is 40, so the mean = 40 / 8 = 5.
- Deviations from the mean: -3, -1, -1, -1, 0, 0, 2, 4.
- Squares of the deviations: 9, 1, 1, 1, 0, 0, 4, 16, and their sum is 32.
- Variance = 32 / 8 = 4.
- Standard deviation = square root of 4 = 2.
Answer: Variance = 4 and standard deviation = 2.
A series has a mean of 50 and a standard deviation of 5. Find the coefficient of variation.
- CV = (SD / mean) x 100.
- = (5 / 50) x 100 = 10.
Answer: CV = 10 per cent.
Find the range and the quartile deviation of 3, 5, 7, 8, 9, 11 and 13, taking Q1 = 5 and Q3 = 11.
- Range = 13 - 3 = 10.
- QD = (Q3 - Q1) / 2 = (11 - 5) / 2 = 3.
Answer: Range = 10 and quartile deviation = 3.
Common mistakes
- Taking the square root of the variance without finding the mean first.
- Forgetting to square the deviations, so that they add to zero.
- Using the wrong divisor, n or n - 1, for population or sample.
- Giving the coefficient of variation without multiplying by 100.
- Leaving out the unit of the answer.