Quartiles and deciles · Grade 11 · 3 units
In Quartiles and deciles (Grade 11 · 3 units) you practise 5 skills: What quartiles and deciles say · From a list · From a frequency table · From a bar chart or a pie chart · Who gets the benefit. Quartiles and deciles are places that split the values, from smallest to largest, into equal parts. The three quartiles split them into four quarters: a quarter of the values lie below the lower quartile, and three quarters below the upper quartile. The nine deciles split them into ten parts: a tenth of the values lie below the lower decile, and a tenth above the upper decile. The median is both the second quartile and the fifth decile.
To find a quartile, sort the values from smallest to largest and work out its place: the quartile’s number times one more than the number of values, divided by four. For a decile, divide by ten. For example, 1, 3, 4, 7, 8, 8, 10 are seven values, and the place of the lower quartile is eight divided by four, which is two. So the lower quartile is 3, the median 7 and the upper quartile 8. When the place is not whole, such as two and a quarter, take the mean of the two values on either side of it. In a frequency table, the cumulative column says how many values there are up to each value, including it. Find the first row whose cumulative count reaches the place: the value in that row is the value in that place. On a bar chart or a pie chart, add up the frequencies from the smallest value up in the same way. Note that other pages of this site may count places another way, so their answers can differ. Saying that something is in the top tenth means it is above the upper decile: among the largest ten percent. The middle half of the values lies between the lower and the upper quartile, and the gap between them shows how spread out it is. This is how to decide, for example, who gets a benefit that goes only to the top tenth.
Tip: Before working out a place, check that the values are sorted from smallest to largest, and count how many there are.
What you learn
What quartiles and deciles say
For example: A large survey found that 10 percent of the monthly salaries in a large company are below 5,400 shekels, 40 percent are between 5,400 shekels and 11,950 shekels, 40 percent between 11,950 shekels and 23,100 shekels, and 10 percent above 23,100 shekels. What is the upper decile?
Answer and solution
The answer: 23100 shekels
The upper decile is the value with 90 percent of the values below it. From the description, exactly 90 percent lie below 23,100 shekels. So the upper decile is 23,100 shekels.
From a list
For example: The marks in a test in one class, from lowest to highest: 41, 50, 51, 69, 84, 86, 93 What is the upper quartile?
Answer and solution
The answer: 86
There are 7 values, and the place of the upper quartile is: 3 × (7 + 1) ÷ 4 = 6 The place is whole, so the upper quartile is the value in place 6: 86.
From a frequency table
For example: The table shows the number of goals scored in each match of one season.
| Goals in a match | Frequency | Cumulative |
|---|---|---|
| 0 | 4 | 4 |
| 1 | 5 | 9 |
| 2 | 5 | 14 |
| 3 | 2 | 16 |
| 4 | 3 | 19 |
Answer and solution
The answer: 1
There are 19 values, and the place of the lower quartile is: (19 + 1) ÷ 4 = 5 Up to 0 there are 4 values, and up to 1 there are 9. So the value in place 5 is 1. That is, the lower quartile is 1.
From a bar chart or a pie chart
For example: The chart shows the number of flats in a building by how many rooms each has.
Answer and solution
The answer: 3
Add up the frequencies from the smallest value up: 3, 8, 15, 18, 19 There are 19 values, and the place of the second decile is: 2 × (19 + 1) ÷ 10 = 4 The place is whole, so the second decile is the value in place 4: 2. The values less than 2 are in the bar for 1, which holds 3. So 3 values are less than the second decile.
Who gets the benefit
For example: A benefit goes to large households: to every household whose number of people is at least the upper decile. The table shows the number of people in the households of one town.
| People in a household | Frequency | Cumulative |
|---|---|---|
| 1 | 15 | 15 |
| 2 | 33 | 48 |
| 3 | 24 | 72 |
| 4 | 15 | 87 |
| 5 | 12 | 99 |
- Yes
- No
Answer and solution
The answer: No
There are 99 values, and the place of the upper decile is: 9 × (99 + 1) ÷ 10 = 90 Up to 4 there are 87 values, and up to 5 there are 99. So the value in place 90 is 5. That is, the upper decile is 5. The value 1 is less than the upper decile, so the answer is no.
Not on the site yet
Parts of this topic that are not built yet: Building the cumulative table yourself · Quartiles in a spreadsheet.
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 Israeli Ministry of Education’s high school maths curriculum at 3 units (grades 10 and 11).