Statistics

In this chapter: why central tendency is not enough, range, mean deviation about the mean and the median for ungrouped, discrete and grouped data, variance and standard deviation, the shortcut and step-deviation methods, the effect of changing origin and scale, and the coefficient of variation.

Why we need dispersion

Mean, median and mode tell you where the data is centred. They say nothing about how widely the values are scattered. A measure of dispersion does that. The diagram shows two data sets with the same mean, 50, and very different spread.

Range

Range = largest observation − smallest observation. It is quick but depends only on the two extreme values, so one unusual observation can distort it completely.

Mean deviation

The mean deviation about a point a is the average of the absolute deviations |x_i − a|. The point a is usually the mean x̄ or the median M. Absolute values are used because the plain deviations from the mean always add up to zero.

For grouped data the median is found with M = l + [(N/2 − C)/f] × h, where l is the lower limit of the median class, f its frequency, h its width and C the cumulative frequency of the class before it.

Worked example: Find the mean deviation about the mean of 6, 7, 10, 12, 13, 4, 8, 12.
Solution: Sum = 72, n = 8, so x̄ = 9. The absolute deviations are 3, 2, 1, 3, 4, 5, 1, 3, adding up to 22. M.D.(x̄) = 22/8 = 2.75.

For the median version, first arrange the data in order. For 3, 9, 5, 3, 12, 10, 18, 4, 7, 19, 21 (11 values), the ordered list is 3, 3, 4, 5, 7, 9, 10, 12, 18, 19, 21 and the median is the 6th value, 9. The absolute deviations from 9 add up to 58, so M.D.(M) = 58/11 ≈ 5.27. Mean deviation is smallest when taken about the median.

Variance and standard deviation

Squaring the deviations is the other way to get rid of signs, and it behaves much better in algebra. The variance is the mean of the squared deviations from the mean, and the standard deviation is its positive square root.

The standard deviation has the same unit as the data; the variance has the square of that unit.

Worked example: Find the variance and standard deviation of 6, 8, 10, 12, 14, 16, 18, 20, 22, 24.
Solution: The mean is 150/10 = 15. The deviations are −9, −7, −5, −3, −1, 1, 3, 5, 7, 9, and their squares add up to 2(81 + 49 + 25 + 9 + 1) = 330. Variance = 330/10 = 33 and σ = √33 ≈ 5.74.
Check with the shortcut: ∑x² = 2580, so σ² = 2580/10 − 15² = 258 − 225 = 33.

Worked example: Find the mean, variance and standard deviation of the distribution x_i: 4, 8, 11, 17, 20, 24, 32 with frequencies f_i: 3, 5, 9, 5, 4, 3, 1.
Solution: N = 30 and ∑f_ix_i = 12 + 40 + 99 + 85 + 80 + 72 + 32 = 420, so x̄ = 14. The deviations from 14 are −10, −6, −3, 3, 6, 10, 18, and ∑f_i(x_i − 14)² = 300 + 180 + 81 + 45 + 144 + 300 + 324 = 1374. Variance = 1374/30 = 45.8 and σ = √45.8 ≈ 6.77.

Set out such sums as a table with columns x_i, f_i, f_ix_i, (x_i − x̄)² and f_i(x_i − x̄)². In board exams a clear table earns method marks, and in any exam it makes an arithmetic slip easier to spot.

Step-deviation method

When the values or class marks are large and equally spaced, put y_i = (x_i − A)/h, where A is an assumed mean and h the common width. Then

Effect of change of origin and scale

• Adding or subtracting a constant to every observation shifts the mean by that constant but leaves the variance and SD unchanged.
• Multiplying every observation by k multiplies the mean by k, the SD by |k| and the variance by k².
• So if y = ax + b, then ȳ = a x̄ + b and σ_y = |a| σ_x.

Comparing variability

To compare the spread of two series with different means, use the coefficient of variation:

The series with the greater C.V. is more variable; the one with the smaller C.V. is more consistent. If the means are equal, simply compare the standard deviations.

Common mistakes: (1) Forgetting the modulus in mean deviation, which makes the sum zero about the mean. (2) Finding the median without arranging the data first. (3) Dividing by the number of classes instead of N = ∑f_i. (4) Using the class limits instead of class marks for grouped data. (5) Thinking that adding a constant changes the SD. (6) Giving the SD of −2x as −2σ; the SD is never negative, so it is 2σ. (7) In the shortcut formula, subtracting x̄ instead of x̄².

Exam focus

• Mean deviation about mean and median for ungrouped and frequency data.
• Variance and SD using the shortcut form, often given as ∑x and ∑x².
• Effect of adding a constant or multiplying by a constant on mean, variance and SD.
• Correcting the mean and variance when some observations were recorded wrongly.
• Coefficient of variation to decide which series is more consistent.

Practice questions