GCSE Maths: Quartiles and the interquartile range (Higher)
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Quartiles and the interquartile range
The median splits ordered data into two halves. The quartiles go further and split it into four equal parts. The lower quartile (Q1) has a quarter of the data below it. The median is Q2, with half the data below it. The upper quartile (Q3) has three quarters of the data below it. These three numbers cut the ordered data into four groups with the same number of values in each.
From a list. (1) Put the data in order. (2) Find the median. (3) Split the data into a lower half and an upper half (if there is an odd number of values, leave out the median). (4) Q1 is the median of the lower half and Q3 is the median of the upper half. With an odd number of values this gives the same answer as the rule “Q1 is the (n + 1) ÷ 4th value and Q3 is the 3(n + 1) ÷ 4th value”.
The interquartile range (IQR) is Q3 − Q1. It is the spread of the middle half of the data. Unlike the range, it is not affected by extreme values, so it is a more reliable measure of spread when there are outliers. A small IQR means the middle half of the data is close together (consistent).
From a frequency table. Add a running total (cumulative frequency) column. Work out the positions (n + 1) ÷ 4, (n + 1) ÷ 2 and 3(n + 1) ÷ 4. The quartile is the value in that position, found from the running total, exactly as for the median.
Key ideas
- Q1 = lower quartile (a quarter of the data is below it). Q2 = median. Q3 = upper quartile (three quarters below it).
- For a list: order the data, find the median, then the median of each half (leave out the median when n is odd).
- IQR = Q3 − Q1: the spread of the middle 50% of the data.
- The IQR is not affected by outliers, but the range is.
- For a frequency table, use positions (n + 1) ÷ 4 and 3(n + 1) ÷ 4 with a running total.
Key words. The quartiles split ordered data into four equal parts. Q1 is the lower quartile, Q2 is the median and Q3 is the upper quartile. The interquartile range (IQR) is Q3 minus Q1: the spread of the middle half.
Method. Quartiles split ordered data into four equal parts. Q1 is the median of the lower half, Q3 is the median of the upper half, and IQR = Q3 − Q1.
- Write the values in order. Quartiles are found from ordered data.3, 5, 8, 12, 15, 18, 21, 25, 30
- Find the median (Q2). The median splits the data into a lower half and an upper half.n = 9, median = 15 (the middle value)
- Split into two halves, leaving out the median. With an odd number of values the median belongs to neither half.Lower half: 3, 5, 8, 12
Upper half: 18, 21, 25, 30 - Q1 is the median of the lower half. It is the value a quarter of the way through the data.Q1 = 6.5
- Q3 is the median of the upper half. It is the value three quarters of the way through the data.Q3 = 23
- Subtract to find the interquartile range. IQR = Q3 − Q1: the spread of the middle half of the data.23 − 6.5 = 16.5
Check. About a quarter of the values are below Q1 and a quarter are above Q3. Here 2 are below 6.5 and 2 are above 23. ✓
Watch out. Do not forget to put the data in order first, and leave out the median when you split an odd number of values.
Key words. The quartiles split ordered data into four equal parts. Q1 is the lower quartile, Q2 is the median and Q3 is the upper quartile. The interquartile range (IQR) is Q3 minus Q1: the spread of the middle half.
Method. Quartiles split ordered data into four equal parts. Q1 is the median of the lower half, Q3 is the median of the upper half, and IQR = Q3 − Q1.
- Write the values in order. Quartiles are found from ordered data.8, 9, 11, 12, 14, 16, 17, 19, 21, 25
- Find the median (Q2). The median splits the data into a lower half and an upper half.n = 10, median = 15 (halfway between the two middle values)
- Split into two equal halves. With an even number of values, the lower half is the first half of the list.Lower half: 8, 9, 11, 12, 14
Upper half: 16, 17, 19, 21, 25 - Q1 is the median of the lower half. It is the value a quarter of the way through the data.Q1 = 11
- Q3 is the median of the upper half. It is the value three quarters of the way through the data.Q3 = 19
- Subtract to find the interquartile range. IQR = Q3 − Q1: the spread of the middle half of the data.19 − 11 = 8
Check. Half of the 10 values lie between Q1 and Q3. ✓
Watch out. Do not forget to put the data in order first, and leave out the median when you split an odd number of values.
| Calls | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Hours | 2 | 6 | 9 | 8 | 6 |
Key words. The quartiles split ordered data into four equal parts. Q1 is the lower quartile, Q2 is the median and Q3 is the upper quartile. The interquartile range (IQR) is Q3 minus Q1: the spread of the middle half.
Method. For a frequency table, find the position of each quartile, then use a running total of the frequencies to find the value in that position.
- Add up the frequencies. This is the number of values, n.2 + 6 + 9 + 8 + 6 = 31
- Work out the positions of the quartiles. Q1 is at (n + 1) ÷ 4, the median at (n + 1) ÷ 2, and Q3 at 3(n + 1) ÷ 4.(31 + 1) ÷ 4 = 8
(31 + 1) ÷ 2 = 16
3 × (31 + 1) ÷ 4 = 24 - Add a running total column. The running total shows which positions each value occupies.
Calls Frequency Running total 1 2 2 2 6 8 3 9 17 4 8 25 5 6 31 - Find the value at each position. Look for the first row where the running total reaches the position.Position 8: row for 2 (positions 3 to 8), so Q1 = 2
Position 16: row for 3, so median = 3
Position 24: row for 4 (positions 18 to 25), so Q3 = 4 - Subtract to find the interquartile range. IQR = Q3 − Q1.4 − 2 = 2
Check. The running total before the row for Q1 is less than 8, and in the row for Q1 it is at least 8. ✓
Watch out. Use (n + 1) ÷ 4 for a frequency table of single values. Give the value, not the position.
Key words. The quartiles split ordered data into four equal parts. Q1 is the lower quartile, Q2 is the median and Q3 is the upper quartile. The interquartile range (IQR) is Q3 minus Q1: the spread of the middle half.
Method. Work out the range and IQR before and after adding the new value, then compare the changes.
- Before: range. Biggest minus smallest.22 − 12 = 10
- Before: IQR. The 7 values give Q1 = 14 and Q3 = 20.20 − 14 = 6
- After: range. The biggest value is now 60.60 − 12 = 48
- After: IQR. The 8 values give lower half 12, 14, 15, 17 (Q1 = 14.5) and upper half 18, 20, 22, 60 (Q3 = 21).21 − 14.5 = 6.5
- Compare. The range jumped from 10 to 48. The IQR changed from 6 to 6.5.The range is badly affected by the outlier. The IQR is hardly affected.
Check. The new value is far above the others, so it is at the very end of the data and only nudges the quartiles. ✓
Watch out. Use the IQR instead of the range when the data has extreme values.
Watch out
Put the data in order first. And with an odd number of values, leave out the median when you split the data into two halves.Exercise 24L Quartiles and the interquartile range
Non-calculatorKey words. The quartiles split ordered data into four equal parts. Q1 is the lower quartile, Q2 is the median and Q3 is the upper quartile. The interquartile range (IQR) is Q3 minus Q1: the spread of the middle half.
Method. Quartiles split ordered data into four equal parts. Q1 is the median of the lower half, Q3 is the median of the upper half, and IQR = Q3 − Q1.
- Write the values in order. Quartiles are found from ordered data.2, 4, 7, 9, 12, 15, 20
- Find the median (Q2). The median splits the data into a lower half and an upper half.n = 7, median = 9 (the middle value)
- Split into two halves, leaving out the median. With an odd number of values the median belongs to neither half.Lower half: 2, 4, 7
Upper half: 12, 15, 20 - Q1 is the median of the lower half. It is the value a quarter of the way through the data.Q1 = 4
- Q3 is the median of the upper half. It is the value three quarters of the way through the data.Q3 = 15
- Subtract to find the interquartile range. IQR = Q3 − Q1: the spread of the middle half of the data.15 − 4 = 11
Check. About a quarter of the values are below Q1 and a quarter are above Q3. Here 1 are below 4 and 1 are above 15. ✓
Watch out. Do not forget to put the data in order first, and leave out the median when you split an odd number of values.
Key words. The quartiles split ordered data into four equal parts. Q1 is the lower quartile, Q2 is the median and Q3 is the upper quartile. The interquartile range (IQR) is Q3 minus Q1: the spread of the middle half.
Method. Quartiles split ordered data into four equal parts. Q1 is the median of the lower half, Q3 is the median of the upper half, and IQR = Q3 − Q1.
- Write the values in order. Quartiles are found from ordered data.3, 5, 7, 8, 10, 12, 15, 17, 20
- Find the median (Q2). The median splits the data into a lower half and an upper half.n = 9, median = 10 (the middle value)
- Split into two halves, leaving out the median. With an odd number of values the median belongs to neither half.Lower half: 3, 5, 7, 8
Upper half: 12, 15, 17, 20 - Q1 is the median of the lower half. It is the value a quarter of the way through the data.Q1 = 6
- Q3 is the median of the upper half. It is the value three quarters of the way through the data.Q3 = 16
- Subtract to find the interquartile range. IQR = Q3 − Q1: the spread of the middle half of the data.16 − 6 = 10
Check. About a quarter of the values are below Q1 and a quarter are above Q3. Here 2 are below 6 and 2 are above 16. ✓
Watch out. Do not forget to put the data in order first, and leave out the median when you split an odd number of values.
Key words. The quartiles split ordered data into four equal parts. Q1 is the lower quartile, Q2 is the median and Q3 is the upper quartile. The interquartile range (IQR) is Q3 minus Q1: the spread of the middle half.
Method. Quartiles split ordered data into four equal parts. Q1 is the median of the lower half, Q3 is the median of the upper half, and IQR = Q3 − Q1.
- Write the values in order. Quartiles are found from ordered data.12, 15, 16, 18, 20, 22, 24, 25, 26, 28, 30
- Find the median (Q2). The median splits the data into a lower half and an upper half.n = 11, median = 22 (the middle value)
- Split into two halves, leaving out the median. With an odd number of values the median belongs to neither half.Lower half: 12, 15, 16, 18, 20
Upper half: 24, 25, 26, 28, 30 - Q1 is the median of the lower half. It is the value a quarter of the way through the data.Q1 = 16
- Q3 is the median of the upper half. It is the value three quarters of the way through the data.Q3 = 26
- Subtract to find the interquartile range. IQR = Q3 − Q1: the spread of the middle half of the data.26 − 16 = 10
Check. About a quarter of the values are below Q1 and a quarter are above Q3. Here 2 are below 16 and 2 are above 26. ✓
Watch out. Do not forget to put the data in order first, and leave out the median when you split an odd number of values.
Key words. The quartiles split ordered data into four equal parts. Q1 is the lower quartile, Q2 is the median and Q3 is the upper quartile. The interquartile range (IQR) is Q3 minus Q1: the spread of the middle half.
Method. Quartiles split ordered data into four equal parts. Q1 is the median of the lower half, Q3 is the median of the upper half, and IQR = Q3 − Q1.
- Write the values in order. Quartiles are found from ordered data.0.8, 1.5, 1.9, 2.1, 2.4, 2.7, 3.2
- Find the median (Q2). The median splits the data into a lower half and an upper half.n = 7, median = 2.1 (the middle value)
- Split into two halves, leaving out the median. With an odd number of values the median belongs to neither half.Lower half: 0.8, 1.5, 1.9
Upper half: 2.4, 2.7, 3.2 - Q1 is the median of the lower half. It is the value a quarter of the way through the data.Q1 = 1.5
- Q3 is the median of the upper half. It is the value three quarters of the way through the data.Q3 = 2.7
- Subtract to find the interquartile range. IQR = Q3 − Q1: the spread of the middle half of the data.2.7 − 1.5 = 1.2
Check. About a quarter of the values are below Q1 and a quarter are above Q3. Here 1 are below 1.5 and 1 are above 2.7. ✓
Watch out. Do not forget to put the data in order first, and leave out the median when you split an odd number of values.
Key words. The quartiles split ordered data into four equal parts. Q1 is the lower quartile, Q2 is the median and Q3 is the upper quartile. The interquartile range (IQR) is Q3 minus Q1: the spread of the middle half.
Method. Quartiles split ordered data into four equal parts. Q1 is the median of the lower half, Q3 is the median of the upper half, and IQR = Q3 − Q1.
- Write the values in order. Quartiles are found from ordered data.4, 7, 9, 12, 15, 18
- Find the median (Q2). The median splits the data into a lower half and an upper half.n = 6, median = 10.5 (halfway between the two middle values)
- Split into two equal halves. With an even number of values, the lower half is the first half of the list.Lower half: 4, 7, 9
Upper half: 12, 15, 18 - Q1 is the median of the lower half. It is the value a quarter of the way through the data.Q1 = 7
- Q3 is the median of the upper half. It is the value three quarters of the way through the data.Q3 = 15
- Subtract to find the interquartile range. IQR = Q3 − Q1: the spread of the middle half of the data.15 − 7 = 8
Check. Half of the 6 values lie between Q1 and Q3. ✓
Watch out. Do not forget to put the data in order first, and leave out the median when you split an odd number of values.
Key words. The quartiles split ordered data into four equal parts. Q1 is the lower quartile, Q2 is the median and Q3 is the upper quartile. The interquartile range (IQR) is Q3 minus Q1: the spread of the middle half.
Method. Quartiles split ordered data into four equal parts. Q1 is the median of the lower half, Q3 is the median of the upper half, and IQR = Q3 − Q1.
- Write the values in order. Quartiles are found from ordered data.22, 25, 28, 30, 31, 35, 38, 40
- Find the median (Q2). The median splits the data into a lower half and an upper half.n = 8, median = 30.5 (halfway between the two middle values)
- Split into two equal halves. With an even number of values, the lower half is the first half of the list.Lower half: 22, 25, 28, 30
Upper half: 31, 35, 38, 40 - Q1 is the median of the lower half. It is the value a quarter of the way through the data.Q1 = 26.5
- Q3 is the median of the upper half. It is the value three quarters of the way through the data.Q3 = 36.5
- Subtract to find the interquartile range. IQR = Q3 − Q1: the spread of the middle half of the data.36.5 − 26.5 = 10
Check. Half of the 8 values lie between Q1 and Q3. ✓
Watch out. Do not forget to put the data in order first, and leave out the median when you split an odd number of values.
| Value | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Frequency | 2 | 6 | 9 | 8 | 6 |
Key words. The quartiles split ordered data into four equal parts. Q1 is the lower quartile, Q2 is the median and Q3 is the upper quartile. The interquartile range (IQR) is Q3 minus Q1: the spread of the middle half.
Method. For a frequency table, find the position of each quartile, then use a running total of the frequencies to find the value in that position.
- Add up the frequencies. This is the number of values, n.2 + 6 + 9 + 8 + 6 = 31
- Work out the positions of the quartiles. Q1 is at (n + 1) ÷ 4, the median at (n + 1) ÷ 2, and Q3 at 3(n + 1) ÷ 4.(31 + 1) ÷ 4 = 8
(31 + 1) ÷ 2 = 16
3 × (31 + 1) ÷ 4 = 24 - Add a running total column. The running total shows which positions each value occupies.
Value Frequency Running total 1 2 2 2 6 8 3 9 17 4 8 25 5 6 31 - Find the value at each position. Look for the first row where the running total reaches the position.Position 8: row for 2 (positions 3 to 8), so Q1 = 2
Position 16: row for 3, so median = 3
Position 24: row for 4 (positions 18 to 25), so Q3 = 4 - Subtract to find the interquartile range. IQR = Q3 − Q1.4 − 2 = 2
Check. The running total before the row for Q1 is less than 8, and in the row for Q1 it is at least 8. ✓
Watch out. Use (n + 1) ÷ 4 for a frequency table of single values. Give the value, not the position.
| Value | 10 | 20 | 30 | 40 |
|---|---|---|---|---|
| Frequency | 4 | 7 | 9 | 3 |
Key words. The quartiles split ordered data into four equal parts. Q1 is the lower quartile, Q2 is the median and Q3 is the upper quartile. The interquartile range (IQR) is Q3 minus Q1: the spread of the middle half.
Method. For a frequency table, find the position of each quartile, then use a running total of the frequencies to find the value in that position.
- Add up the frequencies. This is the number of values, n.4 + 7 + 9 + 3 = 23
- Work out the positions of the quartiles. Q1 is at (n + 1) ÷ 4, the median at (n + 1) ÷ 2, and Q3 at 3(n + 1) ÷ 4.(23 + 1) ÷ 4 = 6
(23 + 1) ÷ 2 = 12
3 × (23 + 1) ÷ 4 = 18 - Add a running total column. The running total shows which positions each value occupies.
Value Frequency Running total 10 4 4 20 7 11 30 9 20 40 3 23 - Find the value at each position. Look for the first row where the running total reaches the position.Position 6: row for 20 (positions 5 to 11), so Q1 = 20
Position 12: row for 30, so median = 30
Position 18: row for 30 (positions 12 to 20), so Q3 = 30 - Subtract to find the interquartile range. IQR = Q3 − Q1.30 − 20 = 10
Check. The running total before the row for Q1 is less than 6, and in the row for Q1 it is at least 6. ✓
Watch out. Use (n + 1) ÷ 4 for a frequency table of single values. Give the value, not the position.
Key words. The quartiles split ordered data into four equal parts. Q1 is the lower quartile, Q2 is the median and Q3 is the upper quartile. The interquartile range (IQR) is Q3 minus Q1: the spread of the middle half.
Method. Quartiles split ordered data into four equal parts. Q1 is the median of the lower half, Q3 is the median of the upper half, and IQR = Q3 − Q1.
- Write the values in order. Quartiles are found from ordered data.19, 22, 23, 27, 29, 31, 33, 35, 38, 42, 45
- Find the median (Q2). The median splits the data into a lower half and an upper half.n = 11, median = 31 (the middle value)
- Split into two halves, leaving out the median. With an odd number of values the median belongs to neither half.Lower half: 19, 22, 23, 27, 29
Upper half: 33, 35, 38, 42, 45 - Q1 is the median of the lower half. It is the value a quarter of the way through the data.Q1 = 23
- Q3 is the median of the upper half. It is the value three quarters of the way through the data.Q3 = 38
- Subtract to find the interquartile range. IQR = Q3 − Q1: the spread of the middle half of the data.38 − 23 = 15
Check. About a quarter of the values are below Q1 and a quarter are above Q3. Here 2 are below 23 and 2 are above 38. ✓
Watch out. Do not forget to put the data in order first, and leave out the median when you split an odd number of values.
Key words. The quartiles split ordered data into four equal parts. Q1 is the lower quartile, Q2 is the median and Q3 is the upper quartile. The interquartile range (IQR) is Q3 minus Q1: the spread of the middle half.
Method. The data must be in order before the quartiles can be found.
- Find the mistake. Hannah took the first and last numbers of the unordered list as the quartiles. The quartiles must be found from the ordered data.She did not put the data in order.
- Order the data. Smallest to largest.3, 4, 6, 7, 9, 12, 15
- Find the median. 7 values, so the 4th.Median = 7
- Find Q1 and Q3. Lower half 3, 4, 6; upper half 9, 12, 15. The medians of the halves are the middle values.Q1 = 4, Q3 = 12
- Subtract. 12 − 4 = 8
Check. The middle half of the data runs from 4 to 12, a width of 8. ✓
Watch out. Always order first.
Key words. The quartiles split ordered data into four equal parts. Q1 is the lower quartile, Q2 is the median and Q3 is the upper quartile. The interquartile range (IQR) is Q3 minus Q1: the spread of the middle half.
Method. IQR = Q3 − Q1, so Q3 = IQR + Q1.
- Write the rule. IQR = Q3 − Q1
- Rearrange. Add Q1 to both sides.Q3 = IQR + Q1
- Substitute. 12 + 9 = 21
Check. 21 − 9 = 12, which is the IQR. ✓
Watch out. The median is not needed for this question.
Key words. The quartiles split ordered data into four equal parts. Q1 is the lower quartile, Q2 is the median and Q3 is the upper quartile. The interquartile range (IQR) is Q3 minus Q1: the spread of the middle half.
Method. Quartiles split the data into four equal groups. The values between Q1 and Q3 are the middle two groups.
- Find the number in each quarter. Four equal parts of 80.80 ÷ 4 = 20
- Find how many are between Q1 and Q3. That is the middle two quarters.2 × 20 = 40
Check. The IQR contains 50% of the data and 50% of 80 is 40. ✓
Watch out. The IQR covers the middle half of the data.
Key words. An average is one number that stands for a whole set of data. The mean is the total divided by how many values there are. The median is the middle value when the data is in order. The mode is the value that appears most often. The range is the biggest value minus the smallest value.
Method. Range = biggest value − smallest value.
- Find the biggest value. The range measures how spread out the data is, from one end to the other.Biggest = 60
- Find the smallest value. Smallest = 12
- Subtract: biggest minus smallest. The range is the gap between the two ends.60 − 12 = 48
Check. Count on from 12 by 48 and you reach 60. ✓
Watch out. The range is one number (the difference), not the pair “from 12 to 60”.
Key words. The quartiles split ordered data into four equal parts. Q1 is the lower quartile, Q2 is the median and Q3 is the upper quartile. The interquartile range (IQR) is Q3 minus Q1: the spread of the middle half.
Method. Quartiles split ordered data into four equal parts. Q1 is the median of the lower half, Q3 is the median of the upper half, and IQR = Q3 − Q1.
- Write the values in order. Quartiles are found from ordered data.12, 14, 15, 17, 18, 20, 22, 60
- Find the median (Q2). The median splits the data into a lower half and an upper half.n = 8, median = 17.5 (halfway between the two middle values)
- Split into two equal halves. With an even number of values, the lower half is the first half of the list.Lower half: 12, 14, 15, 17
Upper half: 18, 20, 22, 60 - Q1 is the median of the lower half. It is the value a quarter of the way through the data.Q1 = 14.5
- Q3 is the median of the upper half. It is the value three quarters of the way through the data.Q3 = 21
- Subtract to find the interquartile range. IQR = Q3 − Q1: the spread of the middle half of the data.21 − 14.5 = 6.5
Check. Half of the 8 values lie between Q1 and Q3. ✓
Watch out. Do not forget to put the data in order first, and leave out the median when you split an odd number of values.
Key words. The quartiles split ordered data into four equal parts. Q1 is the lower quartile, Q2 is the median and Q3 is the upper quartile. The interquartile range (IQR) is Q3 minus Q1: the spread of the middle half.
Method. Compare what each measure uses.
- The range. It uses only the biggest and the smallest values, which could be outliers.Range = biggest − smallest
- The IQR. It uses the quartiles, which are in the middle of the data, so extreme values are ignored.IQR = Q3 − Q1
- Conclude. One very large or small value can change the range a lot, but hardly changes the IQR.Zara is right.
Check. Adding 60 to 12 to 22 changed the range from 10 to 48 but the IQR only from 6 to 6.5. ✓
Watch out. Mention outliers in the explanation.
Key words. The quartiles split ordered data into four equal parts. Q1 is the lower quartile, Q2 is the median and Q3 is the upper quartile. The interquartile range (IQR) is Q3 minus Q1: the spread of the middle half.
Method. Work out the median and IQR for both lists and compare.
- Original list. Q1, median and Q3 from the 7 values.Q1 = 5, median = 10, Q3 = 16, IQR = 11
- New list. The list is now 2, 5, 7, 10, 10, 13, 16, 20. The median is halfway between the 4th and 5th values. Lower half: 2, 5, 7, 10. Upper half: 10, 13, 16, 20.Median = (10 + 10) ÷ 2 = 10
Q1 = (5 + 7) ÷ 2 = 6
Q3 = (13 + 16) ÷ 2 = 14.5
IQR = 14.5 − 6 = 8.5 - Compare. The median is still 10, and the IQR went from 11 to 8.5, which is smaller.
Check. Adding a value at the median moves it into the middle, so both quartiles move towards the median. ✓
Watch out. Work out both lists carefully, remembering to leave out the median only when n is odd.
Club A: 12, 18, 20, 22, 25, 27, 30
Club B: 15, 16, 18, 19, 21, 22, 24
Key words. An average is one number that stands for a whole set of data. The mean is the total divided by how many values there are. The median is the middle value when the data is in order. The mode is the value that appears most often. The range is the biggest value minus the smallest value.
Method. Compare an average (the median) and a measure of spread (the interquartile range) for each group, then write sentences in context.
- Work out the median for Club A. An average tells us what is typical.In order: 12, 18, 20, 22, 25, 27, 30; median = 22
- Work out the median for Club B. In order: 15, 16, 18, 19, 21, 22, 24; median = 19
- Work out the interquartile range for Club A. A spread tells us how consistent the data is.Q1 = 18, Q3 = 27, IQR = 27 − 18 = 9
- Work out the interquartile range for Club B. Q1 = 16, Q3 = 22, IQR = 22 − 16 = 6
- Write the comparison. Say something about the average and something about the spread, and use the context (the words of the question).On average, Club A has the higher age (median 22 compared with 19).
Club B is more consistent (smaller interquartile range: 6 compared with 9), so Club A is more spread out.
Check. Both sentences use numbers from the working. ✓
Watch out. A comparison needs both an average and a spread, in the context of the question. A smaller range (or IQR) means more consistent.
Key words. The quartiles split ordered data into four equal parts. Q1 is the lower quartile, Q2 is the median and Q3 is the upper quartile. The interquartile range (IQR) is Q3 minus Q1: the spread of the middle half.
Method. The IQR is Q3 − Q1 and the range is largest − smallest. With seven numbers, Q1 is the 2nd value, the median is the 4th value and Q3 is the 6th value.
- Find the IQR. Upper quartile minus lower quartile.12 − 5 = 7
- Find the range. Largest minus smallest.20 − 2 = 18
- Place the known values. In order the numbers are a, b, c, d, e, f, g with a = 2, b = 5, d = 8, f = 12 and g = 20.2, 5, c, 8, e, 12, 20
- Choose c and e. c must be between 5 and 8, and e between 8 and 12.For example c = 6 and e = 10: 2, 5, 6, 8, 10, 12, 20
Check. For 2, 5, 6, 8, 10, 12, 20: median 8, Q1 = 5, Q3 = 12, range 18. ✓
Watch out. Seven numbers means Q1 and Q3 are actual values in the list.