Statistics Class 11: notes and important questions

Statistics: revision notes

1. Dispersion and range

Measures of central tendency (mean, median) do not show how spread out data are. The simplest measure of dispersion is the range \(=\) maximum \(-\) minimum; it uses only the two extreme values.

2. Mean deviation

About a central value \(a\) (the mean \(\bar x\) or the median \(M\)):

• Ungrouped: \(\text{MD}(a) = \dfrac{\sum |x_i - a|}{n}\).
• Discrete or grouped (use class marks for \(x_i\)): \(\text{MD}(a) = \dfrac{\sum f_i|x_i - a|}{N}\), \(N = \sum f_i\).
• For the median of a discrete frequency table, find the \(\left(\dfrac{N + 1}{2}\right)\)th value from the cumulative frequencies (\(N\) odd).

3. Variance and standard deviation

\(\sigma^2 = \dfrac{1}{N}\sum f_i(x_i - \bar x)^2 = \dfrac{1}{N}\sum f_ix_i^2 - \bar x^2\), and \(\sigma = \sqrt{\sigma^2}\). For raw data put every \(f_i = 1\) and \(N = n\).

• Adding a constant to every value leaves \(\sigma\) unchanged; multiplying every value by \(k\) multiplies \(\sigma\) by \(|k|\) and \(\sigma^2\) by \(k^2\).
• Step-deviation: with \(y_i = \dfrac{x_i - A}{h}\), \(\sigma_x = h\,\sigma_y\) (useful for large equal class widths).

4. Comparing spread

Of two sets with the same mean, the one with the smaller standard deviation (or variance) is less spread out, i.e. more consistent.

Worked example 1

Find the mean deviation about the mean of \(3, 5, 9, 11\).

\(\bar x = 7\); \(|x - 7| = 4, 2, 2, 4\): MD \(= \dfrac{12}{4} = 3\).

Worked example 2

Find the variance of \(2, 4, 6, 8\).

\(\bar x = 5\); squared deviations \(9, 1, 1, 9\): \(\sigma^2 = \dfrac{20}{4} = 5\).

Worked example 3

The SD of a set is \(3\). Find the SD after each value is doubled and then increased by \(7\).

\(2 \times 3 = 6\) (adding \(7\) changes nothing).

Common errors

• Dropping the modulus in the mean deviation (the deviations then add to \(0\)).
• Using the class limits instead of the class marks for grouped data.
• Dividing by \(n\) (number of classes) instead of \(N = \sum f_i\).
• Forgetting to take the square root for the standard deviation, or subtracting \(\bar x\) instead of \(\bar x^2\) in the short formula.

Exam tips

• Set out a table with columns \(x_i, f_i, f_ix_i, |x_i - \bar x|\) (or \((x_i - \bar x)^2\)) and totals: the method marks are in the table.
• Leave a standard deviation as a surd (for example \(4\sqrt2\)) unless a value is supplied.