Kinds of data and the three averages
Quantitative data are numbers (a height, a mark); qualitative data are categories (a colour, a favourite sport). Statisticians also sort data into four scales. Nominal: names with no order (eye colour). Ordinal: an order, but the steps are not equal (first, second, third place). Interval: equal steps but no true zero (temperature in °C, or the clock time of day). Ratio: equal steps and a true zero, so ratios mean something (height, or how long a race lasted). Descriptive statistics summarise the data we have; inferential statistics use a sample to judge a whole group. Look at the marks of ten pupils in a quiz out of 10: 4, 6, 6, 7, 7, 7, 8, 9, 9, 10. A frequency table counts how often each mark occurs, and a histogram draws it with bars that touch. The mean is the sum divided by the count: 73 ÷ 10 = 7.3. The median is the middle value of the sorted list; with ten values it is the average of the 5th and 6th: (7 + 7) ÷ 2 = 7. The mode is the most frequent value: 7 occurs three times. A list can have two modes, as 3, 5, 5, 8, 8, 9 has 5 and 8, or none if every value occurs once.
| Scale | Order? | Equal steps? | True zero? | Example |
|---|---|---|---|---|
| nominal | no | – | – | eye colour |
| ordinal | yes | no | – | place in a race |
| interval | yes | yes | no | temperature in °C |
| ratio | yes | yes | yes | duration of a race |
| Measure | Calculation | Result |
|---|---|---|
| mean | 40 ÷ 5 | 8 |
| median | middle of 5, 7, 7, 9, 12 | 7 |
| mode | 7 occurs twice | 7 |
Collect (with permission) a list of ten numbers from your class, such as the number of siblings, using no names. Write the mean, median and mode, and say which scale of data it is. Which measure describes a typical pupil best, and why?