Median Quartiles and IQR
The median is the middle value of a data set when the values are placed in order, and the quartiles split the ordered data into four equal parts. The interquartile range (IQR) measures the spread of the middle 50% of the data, calculated as the upper quartile minus the lower quartile, and unlike the range it is not distorted by extreme outliers.
Method
- Put all the data values in order from smallest to largest.
- Find the median: for n values it is at position (n+1)/2; if this falls between two values, take their mean.
- Find the lower quartile (Q1): the median of the lower half of the data, the values below the overall median.
- Find the upper quartile (Q3): the median of the upper half of the data, the values above the overall median.
- Calculate the interquartile range: IQR = Q3 - Q1.
- Use the IQR to compare the spread of two data sets, since it ignores outliers that would distort the range.
Worked example
Find the median, quartiles and interquartile range of this ordered set of 11 test scores: 12, 15, 18, 19, 21, 23, 25, 27, 29, 31, 34.
- The data is already in order and there are n = 11 values.
- The median is the (11+1)/2 = 6th value = 23.
- The lower half is 12, 15, 18, 19, 21 (5 values); its median, the 3rd value, is Q1 = 18.
- The upper half is 25, 27, 29, 31, 34 (5 values); its median, the 3rd value, is Q3 = 29.
- IQR = Q3 - Q1 = 29 - 18 = 11.
Practice questions
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Q1Find the median of: 4, 7, 9, 12, 15.Show answer
Answer: 9
Q2Find the median of: 3, 5, 8, 10.Show answer
Answer: 6.5 (mean of the two middle values, 5 and 8)
Q3For the ordered data 2, 4, 5, 7, 8, 9, 11, 13, 14, find the lower quartile (Q1) and upper quartile (Q3).Show answer
Answer: Q1 = 4.5, Q3 = 12
Q4For the ordered data 6, 9, 11, 14, 16, 18, 20, 23, 25, 28, find the interquartile range.Show answer
Answer: IQR = 12 (Q1 = 11, Q3 = 23)
Q5A data set has Q1 = 22 and Q3 = 35. State the interquartile range and explain what it measures.Show answer
Answer: IQR = 13; it measures the spread of the middle 50% of the data, ignoring outliers.
Q6Two classes have the same median test score. Class A has IQR = 6 and Class B has IQR = 15. Which class's scores are more consistent, and why?Show answer
Answer: Class A; a smaller IQR means the middle 50% of its scores are more closely clustered around the median.
Exam-style questions
Written in the style of a GCSE Statistics exam paper, with a full mark scheme.
A data set of 8 values has lower quartile 12 and upper quartile 27. Calculate the interquartile range and state what a large IQR tells you about the spread of the data.
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The ordered ages, in years, of 13 people at a club are: 14, 16, 17, 19, 20, 22, 24, 25, 27, 29, 31, 33, 36. Find the median and the interquartile range.
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The ordered heights, in cm, of 15 plants are: 20, 22, 23, 25, 26, 28, 29, 30, 32, 34, 35, 37, 39, 41, 44. (a) Find the median height. (b) Find the interquartile range. (c) A gardener claims 'more than half the plants are taller than 30 cm'. Use your median to comment on this claim.
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Free printable worksheet
Want more practice on paper? Download the median quartiles and iqr worksheet pack - 16 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
This topic is chapter 8 of GCSE Statistics Foundation Workbook 2 and chapter 12 of GCSE Statistics Higher Workbook 2, the whole course as one free printable PDF.
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