Statistics Class 9: notes and important questions

Statistics: revision notes

1. Mean

The mean of \(n\) values is \(\bar x = \dfrac{x_1 + x_2 + \cdots + x_n}{n}\). For a frequency table, \(\bar x = \dfrac{\sum f_ix_i}{\sum f_i}\). Total \(=\) mean \(\times\) number of values: this is the key step in most "combine" and "correct the mean" questions.

2. Weighted mean

When values count unequally, give each a weight \(w_i\):

\[\bar x_w = \frac{w_1x_1 + w_2x_2 + \cdots + w_nx_n}{w_1 + w_2 + \cdots + w_n}\]

With percentage weights that add to \(100\%\), the weighted mean is simply \(\sum (\text{weight} \times \text{value})\). The ordinary mean is the weighted mean with equal weights.

3. Combining averages

Group \(A\) of \(n_1\) items with mean \(\bar x_1\) and group \(B\) of \(n_2\) items with mean \(\bar x_2\): combined mean \(= \dfrac{n_1\bar x_1 + n_2\bar x_2}{n_1 + n_2}\). This is a weighted mean with the group sizes as weights; it equals the plain average of \(\bar x_1\) and \(\bar x_2\) only when \(n_1 = n_2\). The combined mean always lies between the two group means, nearer the larger group.

4. Mixtures and concentration

Mixing quantities \(q_1, q_2\) with concentrations (or purities, or prices per kg) \(c_1, c_2\): the mixture has concentration \(\dfrac{q_1c_1 + q_2c_2}{q_1 + q_2}\). Average speed over two legs is total distance \(\div\) total time: a weighted mean of the speeds with the times as weights.

5. Stacked bar charts

A stacked bar chart shows each total as one bar split into its parts, so both totals and parts can be compared. A 100% stacked bar chart makes every bar the same height (100%) and shows each part as a percentage: use it to compare proportions (like a pie chart for each group), not totals.

6. Median and mode

The median is the middle value once the data are in order: for \(n\) values it is the \(\left(\dfrac{n + 1}{2}\right)\)th value when \(n\) is odd, and the mean of the \(\dfrac n2\)th and \(\left(\dfrac n2 + 1\right)\)th values when \(n\) is even. The mode is the value that occurs most often (a data set can have more than one mode). The mean uses every value, so one very large or very small value pulls it; the median does not move, which is why it is often the fairer “typical” value for incomes or prices.

Worked example 1

A report gives weight \(40\%\) to coursework and \(60\%\) to the exam. Find the result for coursework \(85\) and exam \(70\).

\(0.4 \times 85 + 0.6 \times 70 = 34 + 42 = 76\).

Worked example 2

Class A: \(30\) students, mean \(64\). Class B: \(20\) students, mean \(74\). Find the combined mean.

\(\dfrac{30 \times 64 + 20 \times 74}{50} = \dfrac{1920 + 1480}{50} = 68\).

Worked example 3

\(4\) litres of a \(10\%\) salt solution is mixed with \(6\) litres of a \(20\%\) solution. Find the concentration.

Salt \(= 0.4 + 1.2 = 1.6\) litres in \(10\) litres: \(16\%\).

Common errors

• Averaging two group means without weighting by the group sizes.
• Average speed as the mean of the two speeds when the times differ.
• Dividing a weighted total by the number of values instead of the sum of the weights.
• Comparing totals from a 100% stacked chart: it shows only proportions.
• Finding the median without first putting the values in order.

Exam tips

• Turn every mean into a total first (mean \(\times\) count), combine the totals, then divide.
• Check that a combined or weighted mean lies between the smallest and largest values.