Descriptive statistics · Years 11–12
In Descriptive statistics (Years 11–12) you practise 8 skills: Frequency tables · Charts · The mean · The mode · The median · Quartiles and deciles · Measures of spread · Comparing distributions. A variable can be qualitative, like eye colour, or quantitative, like height. A frequency table shows how often each value appears, and the relative frequency is each value’s share of all the data. A bar chart suits comparing categories, a pie chart shows parts of a whole, and a histogram shows quantitative data in groups, with bars that touch.
The mean is the sum of the data divided by how many there are, and it is affected by extreme values. The median is the middle value when the data are in order, and extreme values do not affect it. The mode is the value that appears most often, and it is the only one that also works for a qualitative variable. In a weighted mean each value counts according to its weight. Measures of spread describe how spread out the data are: the range is the difference between the largest and the smallest, and the standard deviation measures the typical distance from the mean. Here the standard deviation divides by the number of data values. Adding a number to all the data does not change the spread, and multiplying by a number multiplies it. Quartiles and deciles are found by place: sort the data from smallest to largest, multiply the number of values by the number of the quartile and divide by four (for a decile, by ten), and round up. The value in that place is the quartile.
Tip: Before you find a median, put the data in order from smallest to largest. Most mistakes with the median come from data left out of order.
What you learn
Frequency tables
For example: The table shows how many books each pupil in a school year read in one month.
| Books read | Frequency |
|---|---|
| 0 | 3 |
| 1 | 1 |
| 2 | 6 |
| 3 | 4 |
| 4 | 3 |
| 5 | 3 |
Answer and solution
The answer: 5%
The number of values: 3 + 1 + 6 + 4 + 3 + 3 = 20 . The frequency of the value is 1 , so the relative frequency is 1 ÷ 20 × 100 = 5 percent.
Charts
For example: The pupils of a school year each chose one sport. The chart shows how many pupils chose each sport.
Answer and solution
The answer: 25
The bar for swimming reaches 25 .
The mean
For example: Find the mean of the data: 4, 1, 16, 6, 13
Answer and solution
The answer: 8
The sum of the values: 4 + 1 + 16 + 6 + 13 = 40 . The number of values: 5 . The mean is 40 ÷ 5 = 8
The mode
For example: Find the mode of the data: 12, 2, 6, 11, 9, 3, 3
Answer and solution
The answer: 3
The value 3 appears 2 times, more than any other value, so it is the mode.
The median
For example: Find the median of the data: 29, 5, 15, 27, 22
Answer and solution
The answer: 22
The data from smallest to largest: 5, 15, 22, 27, 29 . There are 5 values, so the median is the value in place 3: 22
Quartiles and deciles
For example: Find the lower quartile of the data: 8, 25, 45, 37, 10, 9, 2, 38, 14, 31, 18
Answer and solution
The answer: 9
The data from smallest to largest: 2, 8, 9, 10, 14, 18, 25, 31, 37, 38, 45 . The place: 1 × 11 ÷ 4 = 2.75 , rounded up the place is 3 . The value in that place is 9 .
Measures of spread
For example: Find the range of the data: 58, 9, 29, 54, 44, 12
Answer and solution
The answer: 49
The largest value is 58 and the smallest is 9 . The range: 58 − 9 = 49
Comparing distributions
For example: The data of the first group: 19, 19, 18, 17, 17 . The data of the second group: 14, 22, 18, 26, 10 . Both groups have the same mean. Which group is more spread out?
- The first group
- The second group
Answer and solution
The answer: The second group
The mean of both groups is 18 . The range of the first group is 2 and of the second 16 , so the second group is more spread out.
The exercises are made by rules and the answers are worked out in code, so there are new exercises every time. The topic follows the national curriculum in England: mathematics (key stages 3 and 4) and selected AS and A level mathematics practice in England (the GCE subject content).