GCSE Maths: Averages from frequency tables (Higher)
Part of the GCSE Maths statistics chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Averages from frequency tables
When lots of values repeat, it is quicker to list them in a frequency table. The table shows each value and its frequency: how many times it occurs. For example, if 5 families have 2 children, the table shows the value 2 with frequency 5.
The same data can be shown in a vertical line chart (above). The height of each line is the frequency. Use it for discrete numerical data. The total frequency, found by adding the frequencies, is the number of values in the data set.
The mean from a frequency table. We cannot just add the values in the first column, because they occur different numbers of times. Instead add a third column, frequency × value. For the row with value 2 and frequency 6, the 6 values of 2 add to 12. Adding the new column gives the total of all the values, and the mean is total of (frequency × value) ÷ total frequency.
The mode is the value with the highest frequency. The median is the middle value of the data in order. To find it, work out the position of the middle, (n + 1) ÷ 2, and add a running total (cumulative frequency) down the table to find which value is in that position. The range is the biggest value minus the smallest value.
Key ideas
- Total frequency = number of values in the data set.
- Mean = total of (frequency × value) ÷ total frequency.
- Mode = the value with the highest frequency (not the frequency itself).
- Median: position (n + 1) ÷ 2; use a running total of the frequencies to find the value in that position.
- Range = biggest value − smallest value.
| Goals | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| Frequency | 4 | 7 | 6 | 2 | 1 |
Key words. A frequency table shows how many times each value occurs. The frequency (f) is the number of times. The total of the frequency column is the number of values in the data set.
Method. For a frequency table, multiply each value by its frequency, add these up, and divide by the total frequency.
- Add a column for frequency × value. Each value appears f times, so it contributes f × value to the total. This saves writing the value out f times.
Goals Frequency (f) f × goals 0 4 0 1 7 7 2 6 12 3 2 6 4 1 4 Total 20 29 - Add up the frequency column. This gives the number of values in the data set.4 + 7 + 6 + 2 + 1 = 20
- Add up the (f × value) column. This gives the total of all the values in the data set.0 + 7 + 12 + 6 + 4 = 29
- Divide the total by the number of values. mean = total ÷ number of values.29 ÷ 20 = 1.45
Check. The mean must be between the smallest and largest values (0 and 4). 1.45 is. Also 1.45 × 20 = 29. ✓
Watch out. Divide by the total frequency (the number of values), not by the number of rows in the table.
(a) Median.
Key words. A frequency table shows how many times each value occurs. The frequency (f) is the number of times. The total of the frequency column is the number of values in the data set.
Method. Use a running total of the frequencies to find which value is in the middle position.
- Add up the frequencies. This is the number of values, n.4 + 7 + 6 + 2 + 1 = 20
- Find the positions of the two middle values. With an even number of values we need the two middle ones, at n ÷ 2 and n ÷ 2 + 1.20 ÷ 2 = 10, so the 10th and 11th values.
- Add a running total of the frequencies. The running total tells us which values occupy which positions when the data is in order.
Goals Frequency Running total (cumulative) 0 4 4 1 7 11 2 6 17 3 2 19 4 1 20 - Find the 10th value. The running total first reaches 10 or more on the row for 1: that row holds positions 5 to 11.10th value = 1
- Find the 11th value. The running total first reaches 11 or more on the row for 1: that row holds positions 5 to 11.11th value = 1
Check. There are 20 values, so 10 lie below the middle (or middle pair) and the same number above. ✓
Watch out. The median is the value in the middle position, not the middle frequency or the middle row of the table.
(b) Mode.
Key words. A frequency table shows how many times each value occurs. The frequency (f) is the number of times. The total of the frequency column is the number of values in the data set.
Method. The mode is the value with the highest frequency.
- Find the highest frequency. The value that occurs most often has the biggest frequency.Frequencies: 4, 7, 6, 2, 1. The highest is 7.
- Read the value that goes with it. The question asks for the mode, which is the value, not the frequency.Mode = 1
Check. 1 occurs 7 times, more than any other value. ✓
Watch out. Do not give the frequency 7 as the answer. The mode is the value that has that frequency.
(c) Range.
Key words. A frequency table shows how many times each value occurs. The frequency (f) is the number of times. The total of the frequency column is the number of values in the data set.
Method. Range = biggest value − smallest value. Only look at values that actually occur (frequency more than 0).
- Find the smallest value that occurs. Ignore any row with frequency 0.Smallest = 0
- Find the biggest value that occurs. Biggest = 4
- Subtract. Biggest minus smallest.4 − 0 = 4
Check. 0 + 4 = 4. ✓
Watch out. Do not subtract the frequencies. The range uses the values.
| Books | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| Frequency | 2 | 5 | 8 | 6 | 3 | 1 |
Key words. A frequency table shows how many times each value occurs. The frequency (f) is the number of times. The total of the frequency column is the number of values in the data set.
Method. For a frequency table, multiply each value by its frequency, add these up, and divide by the total frequency.
- Add a column for frequency × value. Each value appears f times, so it contributes f × value to the total. This saves writing the value out f times.
Books Frequency (f) f × books 0 2 0 1 5 5 2 8 16 3 6 18 4 3 12 5 1 5 Total 25 56 - Add up the frequency column. This gives the number of values in the data set.2 + 5 + 8 + 6 + 3 + 1 = 25
- Add up the (f × value) column. This gives the total of all the values in the data set.0 + 5 + 16 + 18 + 12 + 5 = 56
- Divide the total by the number of values. mean = total ÷ number of values.56 ÷ 25 = 2.24
Check. The mean must be between the smallest and largest values (0 and 5). 2.24 is. Also 2.24 × 25 = 56. ✓
Watch out. Divide by the total frequency (the number of values), not by the number of rows in the table.
| Score | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Frequency | 5 | x | 8 | 3 |
Method. Write the total frequency and the total of (frequency × score) using x, then use the mean to make an equation.
- Write the total frequency. Add the frequencies, including x.5 + x + 8 + 3 = 16 + x
- Write the total of the scores. Multiply each score by its frequency.1 × 5 + 2 × x + 3 × 8 + 4 × 3 = 5 + 2x + 24 + 12 = 41 + 2x
- Use the mean to write an equation. mean = total of scores ÷ total frequency.(41 + 2x) ÷ (16 + x) = 2.5
- Multiply both sides by (16 + x). This removes the fraction.41 + 2x = 2.5 × (16 + x) = 40 + 2.5x
- Collect the x terms and the numbers. Take 2x and 40 from both sides.41 − 40 = 2.5x − 2x
1 = 0.5x - Solve. Divide both sides by 0.5.x = 1 ÷ 0.5 = 2
Check. With x = 2: total frequency = 18, total of scores = 41 + 4 = 45, and 45 ÷ 18 = 2.5. ✓
Watch out. The mean is the total of (frequency × score) divided by the total frequency, and both contain x.
Watch out
For the mean, divide by the total frequency (the number of values), not by the number of rows in the table. And the mode is the value with the highest frequency, not the frequency.Exercise 24D Averages from frequency tables
Non-calculator(a) How many households own 2 cars? (b) What is the mode? (c) How many households have more than 2 cars? (d) Work out the mean number of cars.
Method. Find 2 on the horizontal axis and read the height of its line.
- Find the line above 2. The horizontal axis shows the number of cars.Line above 2
- Read the height. The height of the line is the frequency.10 households
Check. The frequencies 3 + 7 + 10 + 6 + 3 + 1 add to 30, the number of households. ✓
Watch out. Read the height of the line, not its position on the horizontal axis.
Key words. A frequency table shows how many times each value occurs. The frequency (f) is the number of times. The total of the frequency column is the number of values in the data set.
Method. The mode is the value with the highest frequency.
- Find the highest frequency. The value that occurs most often has the biggest frequency.Frequencies: 3, 7, 10, 6, 3, 1. The highest is 10.
- Read the value that goes with it. The question asks for the mode, which is the value, not the frequency.Mode = 2
Check. 2 occurs 10 times, more than any other value. ✓
Watch out. Do not give the frequency 10 as the answer. The mode is the value that has that frequency.
Method. “More than 2” means 3, 4 or 5 cars, not 2.
- Read the frequencies for 3, 4 and 5. Do not include 2.3 cars: 6, 4 cars: 3, 5 cars: 1
- Add them. 6 + 3 + 1 = 10
Check. Households with 2 or fewer cars: 3 + 7 + 10 = 20, and 20 + 10 = 30. ✓
Watch out. “More than 2” does not include 2.
Key words. A frequency table shows how many times each value occurs. The frequency (f) is the number of times. The total of the frequency column is the number of values in the data set.
Method. For a frequency table, multiply each value by its frequency, add these up, and divide by the total frequency.
- Add a column for frequency × value. Each value appears f times, so it contributes f × value to the total. This saves writing the value out f times.
Cars Frequency (f) f × cars 0 3 0 1 7 7 2 10 20 3 6 18 4 3 12 5 1 5 Total 30 62 - Add up the frequency column. This gives the number of values in the data set.3 + 7 + 10 + 6 + 3 + 1 = 30
- Add up the (f × value) column. This gives the total of all the values in the data set.0 + 7 + 20 + 18 + 12 + 5 = 62
- Divide the total by the number of values. mean = total ÷ number of values.62 ÷ 30 = 2.0667… = 2.1 to 1 d.p.
Check. The mean must be between the smallest and largest values (0 and 5). 2.1 is. Also 2.1 × 30 ≈ 62. ✓
Watch out. Divide by the total frequency (the number of values), not by the number of rows in the table.
| Value | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Frequency | 3 | 6 | 8 | 3 |
Key words. A frequency table shows how many times each value occurs. The frequency (f) is the number of times. The total of the frequency column is the number of values in the data set.
Method. For a frequency table, multiply each value by its frequency, add these up, and divide by the total frequency.
- Add a column for frequency × value. Each value appears f times, so it contributes f × value to the total. This saves writing the value out f times.
Value Frequency (f) f × value 1 3 3 2 6 12 3 8 24 4 3 12 Total 20 51 - Add up the frequency column. This gives the number of values in the data set.3 + 6 + 8 + 3 = 20
- Add up the (f × value) column. This gives the total of all the values in the data set.3 + 12 + 24 + 12 = 51
- Divide the total by the number of values. mean = total ÷ number of values.51 ÷ 20 = 2.55
Check. The mean must be between the smallest and largest values (1 and 4). 2.55 is. Also 2.55 × 20 = 51. ✓
Watch out. Divide by the total frequency (the number of values), not by the number of rows in the table.
| Value | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| Frequency | 5 | 8 | 4 | 3 |
Key words. A frequency table shows how many times each value occurs. The frequency (f) is the number of times. The total of the frequency column is the number of values in the data set.
Method. For a frequency table, multiply each value by its frequency, add these up, and divide by the total frequency.
- Add a column for frequency × value. Each value appears f times, so it contributes f × value to the total. This saves writing the value out f times.
Value Frequency (f) f × value 0 5 0 1 8 8 2 4 8 3 3 9 Total 20 25 - Add up the frequency column. This gives the number of values in the data set.5 + 8 + 4 + 3 = 20
- Add up the (f × value) column. This gives the total of all the values in the data set.0 + 8 + 8 + 9 = 25
- Divide the total by the number of values. mean = total ÷ number of values.25 ÷ 20 = 1.25
Check. The mean must be between the smallest and largest values (0 and 3). 1.25 is. Also 1.25 × 20 = 25. ✓
Watch out. Divide by the total frequency (the number of values), not by the number of rows in the table.
| Value | 10 | 20 | 30 | 40 |
|---|---|---|---|---|
| Frequency | 2 | 4 | 3 | 1 |
Key words. A frequency table shows how many times each value occurs. The frequency (f) is the number of times. The total of the frequency column is the number of values in the data set.
Method. For a frequency table, multiply each value by its frequency, add these up, and divide by the total frequency.
- Add a column for frequency × value. Each value appears f times, so it contributes f × value to the total. This saves writing the value out f times.
Value Frequency (f) f × value 10 2 20 20 4 80 30 3 90 40 1 40 Total 10 230 - Add up the frequency column. This gives the number of values in the data set.2 + 4 + 3 + 1 = 10
- Add up the (f × value) column. This gives the total of all the values in the data set.20 + 80 + 90 + 40 = 230
- Divide the total by the number of values. mean = total ÷ number of values.230 ÷ 10 = 23
Check. The mean must be between the smallest and largest values (10 and 40). 23 is. Also 23 × 10 = 230. ✓
Watch out. Divide by the total frequency (the number of values), not by the number of rows in the table.
| Value | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Frequency | 3 | 6 | 8 | 3 |
Key words. A frequency table shows how many times each value occurs. The frequency (f) is the number of times. The total of the frequency column is the number of values in the data set.
Method. Use a running total of the frequencies to find which value is in the middle position.
- Add up the frequencies. This is the number of values, n.3 + 6 + 8 + 3 = 20
- Find the positions of the two middle values. With an even number of values we need the two middle ones, at n ÷ 2 and n ÷ 2 + 1.20 ÷ 2 = 10, so the 10th and 11th values.
- Add a running total of the frequencies. The running total tells us which values occupy which positions when the data is in order.
Value Frequency Running total (cumulative) 1 3 3 2 6 9 3 8 17 4 3 20 - Find the 10th value. The running total first reaches 10 or more on the row for 3: that row holds positions 10 to 17.10th value = 3
- Find the 11th value. The running total first reaches 11 or more on the row for 3: that row holds positions 10 to 17.11th value = 3
Check. There are 20 values, so 10 lie below the middle (or middle pair) and the same number above. ✓
Watch out. The median is the value in the middle position, not the middle frequency or the middle row of the table.
| Value | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| Frequency | 5 | 8 | 4 | 3 |
Key words. A frequency table shows how many times each value occurs. The frequency (f) is the number of times. The total of the frequency column is the number of values in the data set.
Method. Use a running total of the frequencies to find which value is in the middle position.
- Add up the frequencies. This is the number of values, n.5 + 8 + 4 + 3 = 20
- Find the positions of the two middle values. With an even number of values we need the two middle ones, at n ÷ 2 and n ÷ 2 + 1.20 ÷ 2 = 10, so the 10th and 11th values.
- Add a running total of the frequencies. The running total tells us which values occupy which positions when the data is in order.
Value Frequency Running total (cumulative) 0 5 5 1 8 13 2 4 17 3 3 20 - Find the 10th value. The running total first reaches 10 or more on the row for 1: that row holds positions 6 to 13.10th value = 1
- Find the 11th value. The running total first reaches 11 or more on the row for 1: that row holds positions 6 to 13.11th value = 1
Check. There are 20 values, so 10 lie below the middle (or middle pair) and the same number above. ✓
Watch out. The median is the value in the middle position, not the middle frequency or the middle row of the table.
| Value | 2 | 4 | 6 | 8 | 10 |
|---|---|---|---|---|---|
| Frequency | 1 | 4 | 7 | 5 | 3 |
Key words. A frequency table shows how many times each value occurs. The frequency (f) is the number of times. The total of the frequency column is the number of values in the data set.
Method. Use a running total of the frequencies to find which value is in the middle position.
- Add up the frequencies. This is the number of values, n.1 + 4 + 7 + 5 + 3 = 20
- Find the positions of the two middle values. With an even number of values we need the two middle ones, at n ÷ 2 and n ÷ 2 + 1.20 ÷ 2 = 10, so the 10th and 11th values.
- Add a running total of the frequencies. The running total tells us which values occupy which positions when the data is in order.
Value Frequency Running total (cumulative) 2 1 1 4 4 5 6 7 12 8 5 17 10 3 20 - Find the 10th value. The running total first reaches 10 or more on the row for 6: that row holds positions 6 to 12.10th value = 6
- Find the 11th value. The running total first reaches 11 or more on the row for 6: that row holds positions 6 to 12.11th value = 6
Check. There are 20 values, so 10 lie below the middle (or middle pair) and the same number above. ✓
Watch out. The median is the value in the middle position, not the middle frequency or the middle row of the table.
| Value | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Frequency | 3 | 6 | 8 | 3 |
Key words. A frequency table shows how many times each value occurs. The frequency (f) is the number of times. The total of the frequency column is the number of values in the data set.
Method. The mode is the value with the highest frequency.
- Find the highest frequency. The value that occurs most often has the biggest frequency.Frequencies: 3, 6, 8, 3. The highest is 8.
- Read the value that goes with it. The question asks for the mode, which is the value, not the frequency.Mode = 3
Check. 3 occurs 8 times, more than any other value. ✓
Watch out. Do not give the frequency 8 as the answer. The mode is the value that has that frequency.
| Value | 5 | 6 | 7 | 8 |
|---|---|---|---|---|
| Frequency | 9 | 4 | 2 | 5 |
Key words. A frequency table shows how many times each value occurs. The frequency (f) is the number of times. The total of the frequency column is the number of values in the data set.
Method. The mode is the value with the highest frequency.
- Find the highest frequency. The value that occurs most often has the biggest frequency.Frequencies: 9, 4, 2, 5. The highest is 9.
- Read the value that goes with it. The question asks for the mode, which is the value, not the frequency.Mode = 5
Check. 5 occurs 9 times, more than any other value. ✓
Watch out. Do not give the frequency 9 as the answer. The mode is the value that has that frequency.
| Value | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Frequency | 3 | 6 | 8 | 3 |
Key words. A frequency table shows how many times each value occurs. The frequency (f) is the number of times. The total of the frequency column is the number of values in the data set.
Method. Range = biggest value − smallest value. Only look at values that actually occur (frequency more than 0).
- Find the smallest value that occurs. Ignore any row with frequency 0.Smallest = 1
- Find the biggest value that occurs. Biggest = 4
- Subtract. Biggest minus smallest.4 − 1 = 3
Check. 1 + 3 = 4. ✓
Watch out. Do not subtract the frequencies. The range uses the values.
| Value | 5 | 6 | 7 | 8 |
|---|---|---|---|---|
| Frequency | 9 | 4 | 2 | 5 |
Key words. A frequency table shows how many times each value occurs. The frequency (f) is the number of times. The total of the frequency column is the number of values in the data set.
Method. Range = biggest value − smallest value. Only look at values that actually occur (frequency more than 0).
- Find the smallest value that occurs. Ignore any row with frequency 0.Smallest = 5
- Find the biggest value that occurs. Biggest = 8
- Subtract. Biggest minus smallest.8 − 5 = 3
Check. 5 + 3 = 8. ✓
Watch out. Do not subtract the frequencies. The range uses the values.
| Shoe size | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|
| Frequency | 2 | 5 | 9 | 7 | 4 | 3 |
Key words. A frequency table shows how many times each value occurs. The frequency (f) is the number of times. The total of the frequency column is the number of values in the data set.
Method. The mode is the value with the highest frequency.
- Find the highest frequency. The value that occurs most often has the biggest frequency.Frequencies: 2, 5, 9, 7, 4, 3. The highest is 9.
- Read the value that goes with it. The question asks for the mode, which is the value, not the frequency.Mode = 5
Check. 5 occurs 9 times, more than any other value. ✓
Watch out. Do not give the frequency 9 as the answer. The mode is the value that has that frequency.
Key words. A frequency table shows how many times each value occurs. The frequency (f) is the number of times. The total of the frequency column is the number of values in the data set.
Method. Use a running total of the frequencies to find which value is in the middle position.
- Add up the frequencies. This is the number of values, n.2 + 5 + 9 + 7 + 4 + 3 = 30
- Find the positions of the two middle values. With an even number of values we need the two middle ones, at n ÷ 2 and n ÷ 2 + 1.30 ÷ 2 = 15, so the 15th and 16th values.
- Add a running total of the frequencies. The running total tells us which values occupy which positions when the data is in order.
Shoe size Frequency Running total (cumulative) 3 2 2 4 5 7 5 9 16 6 7 23 7 4 27 8 3 30 - Find the 15th value. The running total first reaches 15 or more on the row for 5: that row holds positions 8 to 16.15th value = 5
- Find the 16th value. The running total first reaches 16 or more on the row for 5: that row holds positions 8 to 16.16th value = 5
Check. There are 30 values, so 15 lie below the middle (or middle pair) and the same number above. ✓
Watch out. The median is the value in the middle position, not the middle frequency or the middle row of the table.
Key words. A frequency table shows how many times each value occurs. The frequency (f) is the number of times. The total of the frequency column is the number of values in the data set.
Method. Range = biggest value − smallest value. Only look at values that actually occur (frequency more than 0).
- Find the smallest value that occurs. Ignore any row with frequency 0.Smallest = 3
- Find the biggest value that occurs. Biggest = 8
- Subtract. Biggest minus smallest.8 − 3 = 5
Check. 3 + 5 = 8. ✓
Watch out. Do not subtract the frequencies. The range uses the values.
Key words. A frequency table shows how many times each value occurs. The frequency (f) is the number of times. The total of the frequency column is the number of values in the data set.
Method. For a frequency table, multiply each value by its frequency, add these up, and divide by the total frequency.
- Add a column for frequency × value. Each value appears f times, so it contributes f × value to the total. This saves writing the value out f times.
Shoe size Frequency (f) f × shoe size 3 2 6 4 5 20 5 9 45 6 7 42 7 4 28 8 3 24 Total 30 165 - Add up the frequency column. This gives the number of values in the data set.2 + 5 + 9 + 7 + 4 + 3 = 30
- Add up the (f × value) column. This gives the total of all the values in the data set.6 + 20 + 45 + 42 + 28 + 24 = 165
- Divide the total by the number of values. mean = total ÷ number of values.165 ÷ 30 = 5.5
Check. The mean must be between the smallest and largest values (3 and 8). 5.5 is. Also 5.5 × 30 = 165. ✓
Watch out. Divide by the total frequency (the number of values), not by the number of rows in the table.
| Goals | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| Matches | 4 | 9 | 8 | 5 | 3 | 1 |
Key words. A frequency table shows how many times each value occurs. The frequency (f) is the number of times. The total of the frequency column is the number of values in the data set.
Method. For a frequency table, multiply each value by its frequency, add these up, and divide by the total frequency.
- Add a column for frequency × value. Each value appears f times, so it contributes f × value to the total. This saves writing the value out f times.
Goals Frequency (f) f × goals 0 4 0 1 9 9 2 8 16 3 5 15 4 3 12 5 1 5 Total 30 57 - Add up the frequency column. This gives the number of values in the data set.4 + 9 + 8 + 5 + 3 + 1 = 30
- Add up the (f × value) column. This gives the total of all the values in the data set.0 + 9 + 16 + 15 + 12 + 5 = 57
- Divide the total by the number of values. mean = total ÷ number of values.57 ÷ 30 = 1.90
Check. The mean must be between the smallest and largest values (0 and 5). 1.90 is. Also 1.90 × 30 = 57. ✓
Watch out. Divide by the total frequency (the number of values), not by the number of rows in the table.
| Passengers per car | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Cars | 9 | 14 | ? | 6 |
Method. The frequencies must add up to the total number of cars, 40.
- Add the known frequencies. 9 + 14 + 6 = 29
- Subtract from the total. What is left is the missing frequency.40 − 29 = 11
Check. 9 + 14 + 11 + 6 = 40. ✓
Watch out. Subtract from the total; do not add 40 to the other frequencies.
Method. Multiply each number of passengers by the number of cars, then add.
- Work out each row. Passengers × cars.1 × 9 = 9
2 × 14 = 28
3 × 11 = 33
4 × 6 = 24 - Add. 9 + 28 + 33 + 24 = 94
Check. 94 ÷ 40 = 2.35 passengers per car, a sensible value between 1 and 4. ✓
Watch out. Use the completed table with the missing frequency 11.
Class A
| Siblings | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| Frequency | 6 | 9 | 3 | 2 |
| Siblings | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| Frequency | 4 | 6 | 7 | 3 |
Method. Work out the mean for each class and compare them.
- Class A: find the totals. Multiply each number of siblings by its frequency and add; add the frequencies.Total siblings: 0×6 + 1×9 + 2×3 + 3×2 = 21
Students: 6 + 9 + 3 + 2 = 20 - Class A: divide. mean = total ÷ students.21 ÷ 20 = 1.05
- Class B: find the totals. Do the same for Class B.Total siblings: 0×4 + 1×6 + 2×7 + 3×3 = 29
Students: 4 + 6 + 7 + 3 = 20 - Class B: divide. 29 ÷ 20 = 1.45
- Compare. 1.45 is bigger than 1.05, so Class B has the higher mean.
Check. Class B has fewer students with no siblings and more with 2 or 3, so a higher mean is sensible. ✓
Watch out. Compare the means, not the totals: the classes are different sizes.
| Value | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Frequency | 3 | 8 | 6 | 3 |
Key words. A frequency table shows how many times each value occurs. The frequency (f) is the number of times. The total of the frequency column is the number of values in the data set.
Method. For a frequency table, multiply each value by its frequency, add these up, and divide by the total frequency.
- Find Jon’s mistake. He treated every value as though it occurred once. But 2 occurs 8 times, and 3 occurs 6 times.Jon ignored the frequencies.
- Add a column for frequency × value. Each value appears f times, so it contributes f × value to the total. This saves writing the value out f times.
Value Frequency (f) f × value 1 3 3 2 8 16 3 6 18 4 3 12 Total 20 49 - Add up the frequency column. This gives the number of values in the data set.3 + 8 + 6 + 3 = 20
- Add up the (f × value) column. This gives the total of all the values in the data set.3 + 16 + 18 + 12 = 49
- Divide the total by the number of values. mean = total ÷ number of values.49 ÷ 20 = 2.45
Check. The mean must be between the smallest and largest values (1 and 4). 2.45 is. Also 2.45 × 20 = 49. ✓
Watch out. Divide by the total frequency (the number of values), not by the number of rows in the table.
| Texts | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Frequency | 4 | 6 | 7 | 2 | 1 |
Key words. A frequency table shows how many times each value occurs. The frequency (f) is the number of times. The total of the frequency column is the number of values in the data set.
Method. For a frequency table, multiply each value by its frequency, add these up, and divide by the total frequency.
- Add a column for frequency × value. Each value appears f times, so it contributes f × value to the total. This saves writing the value out f times.
Texts Frequency (f) f × texts 1 4 4 2 6 12 3 7 21 4 2 8 5 1 5 Total 20 50 - Add up the frequency column. This gives the number of values in the data set.4 + 6 + 7 + 2 + 1 = 20
- Add up the (f × value) column. This gives the total of all the values in the data set.4 + 12 + 21 + 8 + 5 = 50
- Divide the total by the number of values. mean = total ÷ number of values.50 ÷ 20 = 2.5
Check. The mean must be between the smallest and largest values (1 and 5). 2.5 is. Also 2.5 × 20 = 50. ✓
Watch out. Divide by the total frequency (the number of values), not by the number of rows in the table.
Method. Find the mean from part (a), then add the frequencies of the values bigger than it.
- Recall the mean. From part (a).mean = 2.5
- Find values bigger than the mean. These are the values 3, 4 and 5 (2 is below the mean).3 texts: 7, 4 texts: 2, 5 texts: 1
- Add the frequencies. 7 + 2 + 1 = 10
Check. Students with 1 or 2 texts: 4 + 6 = 10, and 10 + 10 = 20. ✓
Watch out. Compare each value with the mean, then add the frequencies (not the values).
| Score | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Frequency | 4 | x | 6 | 3 |
Method. Write the total frequency and total score using x, then make an equation using the mean.
- Total frequency. Add the frequencies, including x.4 + x + 6 + 3 = 13 + x
- Total score. Multiply each score by its frequency and add.1 × 4 + 2x + 3 × 6 + 4 × 3 = 4 + 2x + 18 + 12 = 34 + 2x
- Write the equation. mean = total score ÷ total frequency.(34 + 2x) ÷ (13 + x) = 2.4
- Multiply both sides by (13 + x). This removes the fraction.34 + 2x = 2.4 × 13 + 2.4x = 31.2 + 2.4x
- Collect terms. Subtract 2x and 31.2 from both sides.34 − 31.2 = 2.4x − 2x
2.8 = 0.4x - Solve. Divide both sides by 0.4.x = 2.8 ÷ 0.4 = 7
Check. With x = 7: total frequency = 20, total score = 34 + 14 = 48, and 48 ÷ 20 = 2.4. ✓
Watch out. Remember the unknown x appears in both the total frequency and the total score.
Method. Make two equations, one from the total frequency and one from the mean, and solve them together.
- Use the total frequency. The frequencies add up to 20.3 + a + b + 4 + 2 = 20, so a + b = 11
- Find the total score from the mean. total = mean × 20.2.85 × 20 = 57
- Write the total score using a and b. Multiply each score by its frequency.1 × 3 + 2a + 3b + 4 × 4 + 5 × 2 = 29 + 2a + 3b
- Write the second equation. This must equal 57.29 + 2a + 3b = 57, so 2a + 3b = 28
- Eliminate a. Double the first equation (2a + 2b = 22) and subtract it from the second.(2a + 3b) − (2a + 2b) = 28 − 22
b = 6 - Find a. Use a + b = 11.a = 11 − 6 = 5
Check. Total score = 3 + 10 + 18 + 16 + 10 = 57 and 57 ÷ 20 = 2.85. ✓
Watch out. You need two equations for two unknowns.
Method. Use a very large value with a small frequency, so that it pulls the mean up but does not make it the mode.
- Choose the table. Make 3 the most frequent value, and include a big value such as 10.
Value 1 2 3 10 Frequency 1 2 5 2 - Check the mode. The highest frequency is 5, for the value 3.Mode = 3
- Work out the mean. Total of (frequency × value) ÷ total frequency.(1×1 + 2×2 + 3×5 + 10×2) ÷ 10 = 40 ÷ 10 = 4
Check. The mean 4 is bigger than the mode 3, because the two 10s pull it up. ✓
Watch out. The mean and the mode are different kinds of average and do not have to be equal.
| Rooms | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|
| Houses | 3 | 6 | 9 | 5 | 2 |
Key words. A frequency table shows how many times each value occurs. The frequency (f) is the number of times. The total of the frequency column is the number of values in the data set.
Method. For a frequency table, multiply each value by its frequency, add these up, and divide by the total frequency.
- Add a column for frequency × value. Each value appears f times, so it contributes f × value to the total. This saves writing the value out f times.
Rooms Frequency (f) f × rooms 2 3 6 3 6 18 4 9 36 5 5 25 6 2 12 Total 25 97 - Add up the frequency column. This gives the number of values in the data set.3 + 6 + 9 + 5 + 2 = 25
- Add up the (f × value) column. This gives the total of all the values in the data set.6 + 18 + 36 + 25 + 12 = 97
- Divide the total by the number of values. mean = total ÷ number of values.97 ÷ 25 = 3.88
Check. The mean must be between the smallest and largest values (2 and 6). 3.88 is. Also 3.88 × 25 = 97. ✓
Watch out. Divide by the total frequency (the number of values), not by the number of rows in the table.
Key words. A frequency table shows how many times each value occurs. The frequency (f) is the number of times. The total of the frequency column is the number of values in the data set.
Method. Use a running total of the frequencies to find which value is in the middle position.
- Add up the frequencies. This is the number of values, n.3 + 6 + 9 + 5 + 2 = 25
- Find the position of the median. The median is the middle value of the data in order: position (n + 1) ÷ 2.(25 + 1) ÷ 2 = 13, so the 13th value.
- Add a running total of the frequencies. The running total tells us which values occupy which positions when the data is in order.
Rooms Frequency Running total (cumulative) 2 3 3 3 6 9 4 9 18 5 5 23 6 2 25 - Find the 13th value. The running total first reaches 13 or more on the row for 4: that row holds positions 10 to 18.13th value = 4
Check. There are 25 values, so 12 lie below the middle (or middle pair) and the same number above. ✓
Watch out. The median is the value in the middle position, not the middle frequency or the middle row of the table.
Key words. A frequency table shows how many times each value occurs. The frequency (f) is the number of times. The total of the frequency column is the number of values in the data set.
Method. The mode is the value with the highest frequency.
- Find the highest frequency. The value that occurs most often has the biggest frequency.Frequencies: 3, 6, 9, 5, 2. The highest is 9.
- Read the value that goes with it. The question asks for the mode, which is the value, not the frequency.Mode = 4
Check. 4 occurs 9 times, more than any other value. ✓
Watch out. Do not give the frequency 9 as the answer. The mode is the value that has that frequency.
Key words. A frequency table shows how many times each value occurs. The frequency (f) is the number of times. The total of the frequency column is the number of values in the data set.
Method. Range = biggest value − smallest value. Only look at values that actually occur (frequency more than 0).
- Find the smallest value that occurs. Ignore any row with frequency 0.Smallest = 2
- Find the biggest value that occurs. Biggest = 6
- Subtract. Biggest minus smallest.6 − 2 = 4
Check. 2 + 4 = 6. ✓
Watch out. Do not subtract the frequencies. The range uses the values.
Key words. A frequency table shows how many times each value occurs. The frequency (f) is the number of times. The total of the frequency column is the number of values in the data set.
Method. Work out the mean and the range of each table, then compare them.
- Mean of table A. Total of (frequency × value) ÷ total frequency.(2 × 1 + 3 × 8 + 4 × 1) ÷ 10 = 30 ÷ 10 = 3
- Mean of table B. Do the same.(1 × 3 + 3 × 4 + 5 × 3) ÷ 10 = 30 ÷ 10 = 3
- Find the ranges. Largest value minus smallest value.Table A: 4 − 2 = 2
Table B: 5 − 1 = 4 - Conclude. The means are equal (3) but the ranges are different (2 and 4).
Check. Table A is bunched around 3 and table B is more spread out, which agrees with the ranges. ✓
Watch out. The same mean can come from data with very different spreads.