Averages, Spread and Charts
Averages and spread
An average is a single value that represents a typical value in a data set. There are three types, plus the range to measure spread.
The three averages
| Average | How to find it |
|---|---|
| Mean | Add all the values, then divide by how many there are |
| Median | Put values in order, take the middle one (if two middle values, average them) |
| Mode | The value that appears most often (there can be none or more than one) |
Range = highest value − lowest value (a measure of spread, not an average). A smaller range means the data is more consistent.
Example
Data: 4, 7, 7, 2, 10.
- Mean = (4 + 7 + 7 + 2 + 10) ÷ 5 = 30 ÷ 5 = 6.
- Median: in order 2, 4, 7, 7, 10 → middle = 7.
- Mode = 7 (appears twice).
- Range = 10 − 2 = 8.
Which average to use?
- Mean uses every value but is affected by extreme values (outliers).
- Median is better when there are outliers (it ignores extremes).
- Mode is the only average for non-numerical data (e.g. favourite colour).
Averages from a frequency table
- Mean = (Σ (value × frequency)) ÷ (Σ frequency) — total of all values ÷ total frequency.
- Modal value = the value with the highest frequency.
- Median position = the ((n + 1) ÷ 2)th value.
Charts and graphs
- Bar chart — compares categories (gaps between bars).
- Pie chart — shows proportions; each category's angle = (frequency ÷ total) × 360°.
- Scatter graph — shows correlation between two variables: positive (both increase), negative (one increases as the other decreases), or none. A line of best fit shows the trend and can be used to estimate values.
Worked example
Find the mean of 5, 8, 8, 11.
- Mean = (5 + 8 + 8 + 11) ÷ 4 = 32 ÷ 4 = 8. ✓
Common mistakes
- Forgetting to put data in order before finding the median.
- Confusing mode (most common) with median (middle).
- Forgetting the range is highest − lowest (spread), not an average.
Exam tips
- Learn the three averages and when each is most appropriate (mean vs outliers).
- For a pie chart, each angle = (frequency ÷ total) × 360°.
- Describe scatter-graph correlation as positive, negative or none, and use a line of best fit to estimate.
Key facts to remember
- Mean = total ÷ number; median = middle (in order); mode = most common; range = highest − lowest (spread).
- Mean is affected by outliers; median isn't; mode works for non-numerical data.
- Scatter graphs show correlation (positive/negative/none) via a line of best fit; pie-chart angle = (freq ÷ total) × 360°.