Mode, median and mean for discrete and grouped data; estimating the mean of grouped data with midpoints; linear interpolation for the median; weighted and geometric mean; effect of changes and transformations on averages.

What this dot point is asking

Edexcel codes 2b.01 to 2b.03 require you to calculate the mode, median and arithmetic mean for discrete and grouped data, estimate the mean of grouped data using class midpoints, find the median of grouped data by linear interpolation, and (Higher tier) calculate the weighted mean and geometric mean. You must also justify which average suits a context and understand the effect on each average of adding or removing a value, or of simple transformations (scaling and translating the data).

Mode, median and mean

For a small ordered list of $n$ values, the median is at position $\frac{n+1}{2}$. For frequency data the mean is

The most common error is dividing by the number of categories instead of $\sum f$, the total frequency.

Grouped data: estimating the mean

For grouped data you do not know the individual values, so you assume each lies at its class midpoint $x$. The mean is then estimated as $\frac{\sum fx}{\sum f}$ using those midpoints. Because of this assumption the answer is an estimate, and Edexcel regularly asks you to state why. Use equal class widths at Foundation; Higher tier may use unequal widths.

Grouped data: median by interpolation

The median of grouped data is found by linear interpolation within the class that contains the middle value. Locate the $\frac{n}{2}$th value, find which class it falls in, then assume the values are evenly spread across that class:

where $L$ is the lower boundary of the median class, $F$ the cumulative frequency before it, $f$ the median class frequency and $w$ its width.

Weighted and geometric mean

The weighted mean gives different importance to different values:

It is used when components count unequally (for example a grade made of $70\%$ exam and $30\%$ coursework). The geometric mean of $n$ values is the $n$th root of their product, $\sqrt[n]{x_1 \times x_2 \times \cdots \times x_n}$, and suits averaging growth rates or ratios, where the arithmetic mean would mislead.

Effect of changes and transformations

Edexcel expects you to predict how averages respond to changes. Adding a value above the mean raises the mean; removing the largest value lowers the mean and may change the median. Under a transformation (Higher tier), if every value is multiplied by $k$ and then increased by $c$, the mean, mode and median are all transformed the same way: new average $= k \times (\text{old average}) + c$.