Cumulative frequency and box plots
By the end of this chapter you will be able to
- Build a cumulative frequency table, and plot the points at the upper class boundaries
- Read the median, quartiles and a percentile from a cumulative frequency curve
- Find the five-number summary from raw data, and draw a box plot
- Compare two box plots using medians, quartiles and interquartile ranges
- Comment on the shape and spread of a data set in context
You should already know
- Averages and the range from lists and frequency tables (Year 8, Chapter 13)
- Grouped frequency tables and class boundaries (Year 8, Chapter 13)
- Drawing and reading graphs with axes and scales (Year 8 and Year 9, Chapter 6)
Where this leads
- Histograms and comparing distributions (Year 9, Chapter 13)
- Statistical sampling and bias in GCSE
- Scatter graphs and correlation in GCSE
National Curriculum for England, Key Stage 3 mathematics, Year 9. Content: Construct a cumulative frequency table and plot a cumulative frequency curve or polygon at the upper class boundaries, Estimate the median, quartiles, interquartile range and a percentile from a cumulative frequency graph, Draw a box plot from a five-number summary and from raw data, Compare two distributions using box plots and cumulative frequency graphs, Interpret and comment on the shape and spread of a distribution, using averages and the interquartile range.
Exercise P Do you remember?
Non-calculatorMethod. Put the values in order, then take the middle one.
- Order the values. They are already in order: 3, 7, 9, 12, 15.3, 7, 9, 12, 15
- Find the middle. There are 5 values, so the 3rd is the middle.9
Check. There are two values below 9 and two above it.
Method. Add the values, then divide by how many there are.
- Add. 4 + 6 + 8 + 10 = 28.28
- Divide. 28 ÷ 4 = 7.7
Check. 7 × 4 = 28.
Method. Range = highest − lowest.
- Find the highest and lowest. Highest 14, lowest 2.14 − 2
- Subtract. 12
Check. The range measures the spread of all the data.
Method. The median of an even number of values is the mean of the middle two.
- Find the middle. With 50 values, the middle two are the 25th and 26th.25th and 26th
- Write the answer. The 25th and 26th values
Check. The median is the mean of the 25th and 26th values.
Method. Read the frequency column.
- Read the column. Each frequency is a count.frequency = count
- Write the answer. How many times the value or group occurs
Check. The frequencies add up to the total number of values.
Cumulative frequency tables and curves
The cumulative frequency is the running total of the frequencies. Plot each cumulative frequency at the upper class boundary, then join the points with straight lines to make a cumulative frequency polygon, or a smooth curve. The first point is at the lower boundary of the first class, with cumulative frequency 0.
Key ideas
- Cumulative frequency = running total of the frequencies.
- Plot each cumulative frequency at the upper class boundary.
- Start the curve at (lower boundary, 0).
- The final cumulative frequency equals the total number of values.
Method. Add the frequencies in order, one class at a time.
- First class. The first cumulative frequency is just the first frequency: 3.3
- Add each frequency. Add the next frequency to the running total.3 + 8 = 11, 11 + 15 = 26, 26 + 10 = 36, 36 + 4 = 40
- Check. The last cumulative frequency should equal the total, 40.40 = 40
Check. The total of the frequencies is 40.
Method. Use the upper boundary and the cumulative frequency at that boundary.
- Upper boundary. The class from 20 to 30 has upper boundary 30.x = 30
- Cumulative frequency. The cumulative frequency up to 30 is 26.y = 26
- Write the answer. (30, 26)
Check. 26 plants are 30 cm or shorter.
Method. Use the upper boundary and its cumulative frequency.
- Upper boundary. The class from 30 to 40 has upper boundary 40.x = 40
- Cumulative frequency. The cumulative frequency up to 40 is 36.y = 36
- Write the answer. (40, 36)
Check. 36 plants are 40 cm or shorter.
Watch out
Watch out: plot the cumulative frequency at the upper boundary, not the middle of the class. A point at the middle would be wrong.
Exam tip
Exam tip. Label the axes with the units, and join the points with a smooth curve or straight lines as asked.
Exercise 12A Cumulative frequency tables and curves
Non-calculator| Class | 0 < x ≤ 10 | 10 < x ≤ 20 | 20 < x ≤ 30 | 30 < x ≤ 40 | 40 < x ≤ 50 |
|---|---|---|---|---|---|
| Frequency | 3 | 8 | 15 | 10 | 4 |
| Cumulative frequency | 3 | 11 | 26 | 36 | 40 |
Method. Add the frequencies one after another.
- Running total. Add each frequency to the total so far.3, 3 + 8 = 11, 11 + 15 = 26, 26 + 10 = 36, 36 + 4 = 40
- Write the answer. 3, 11, 26, 36, 40
Check. The last total is 40, the number of plants.
Method. Use the upper boundaries and the cumulative frequencies.
- Start point. The first point is (0, 0), at the lower boundary of the first class.(0, 0)
- Other points. Use each upper boundary with its cumulative frequency.(10, 3), (20, 11), (30, 26), (40, 36), (50, 40)
- Write the answer. (0, 0), (10, 3), (20, 11), (30, 26), (40, 36), (50, 40)
Check. The last point is at (50, 40).
Method. Read the cumulative frequency at 20 cm.
- Find 20 cm. The cumulative frequency at the upper boundary 20 is 11.11
- Write the answer. 11
Check. 11 of the 40 plants are 20 cm or shorter.
Method. Subtract the cumulative frequency at 30 from the total.
- Cumulative frequency at 30. There are 26 plants at most 30 cm.26
- Subtract from the total. 40 − 26 = 14.14
Check. 26 + 14 = 40.
Method. Think about what cumulative frequency adds up to.
- Total. The last cumulative frequency is the sum of all the frequencies.40
- Write the answer. So the curve ends at the total number of plants.40, the total number of plants
Check. All the plants are counted by 50 cm.
Method. Add the frequencies one after another.
- Running total. Add each frequency to the total so far.5, 15, 27, 36, 40
- Write the answer. 5, 15, 27, 36, 40
Check. The last total is 40.
Method. Use the upper boundary and its cumulative frequency.
- Upper boundary. The upper boundary is 30.x = 30
- Cumulative frequency. The cumulative frequency at 30 is 27.y = 27
- Write the answer. (30, 27)
Check. 27 plants are at most 30 cm tall.
Method. Read the y-value at x = 20.
- Read. The point (20, 11) means 11 plants are at most 20 cm.11
- Write the answer. 11
Check. This is the same as the table value.
Method. The total is the final cumulative frequency.
- Final point. The last point is at (50, 40).40
- Write the answer. 40
Check. The total is the last value on the curve.
Method. Subtract the earlier cumulative frequency.
- Cumulative frequencies. 4 up to 10, and 13 up to 20.13 − 4
- Subtract. The frequency of the middle class is 13 − 4 = 9.9
Check. 4 + 9 = 13.
Method. Think about what the running total means.
- Look at the running total. The cumulative frequency counts all values up to the upper boundary.up to the upper boundary
- Write the answer. It counts all values up to that boundary
Check. Plotting at the lower boundary would miss values in the class.
Reading the median, quartiles and percentiles
For n values, the median is at cumulative frequency n/2, the lower quartile at n/4, and the upper quartile at 3n/4. Read across to the curve and then down to the axis. The interquartile range (IQR) is the upper quartile minus the lower quartile.
Key ideas
- Median: cumulative frequency n/2. Lower quartile: n/4. Upper quartile: 3n/4.
- Read across from the cumulative frequency, then down to the x-axis.
- IQR = upper quartile − lower quartile.
- A percentile is read at the same fraction of n: the 90th percentile is at 0.9n.
Method. Read the value for the median from the cumulative frequency curve.
- Find the cumulative frequency. For the median, the cumulative frequency is 20 out of 40.k = 20
- Find the point on the curve. Find the two points on the cumulative frequency polygon either side of this height.
- Interpolate. The polygon is a straight line between the points, so the value is found in the same proportion.x = 26.0000
- Write the answer. 26.0
Check. Check: the value sits between the points on either side of cumulative frequency 20.
Watch out. Use the upper class boundary for each point on the curve, and plot the curve from the lower boundary of the first class.
Method. Read the value for the lower quartile from the cumulative frequency curve.
- Find the cumulative frequency. For the lower quartile, the cumulative frequency is 10 out of 40.k = 10
- Find the point on the curve. Find the two points on the cumulative frequency polygon either side of this height.
- Interpolate. The polygon is a straight line between the points, so the value is found in the same proportion.x = 18.7500
- Write the answer. 18.8
Check. Check: the value sits between the points on either side of cumulative frequency 10.
Watch out. Use the upper class boundary for each point on the curve, and plot the curve from the lower boundary of the first class.
Method. Read the value for the 90th percentile from the cumulative frequency curve.
- Find the cumulative frequency. For the 90th percentile, the cumulative frequency is 36 out of 40.k = 36
- Find the point on the curve. Find the two points on the cumulative frequency polygon either side of this height.
- Interpolate. The polygon is a straight line between the points, so the value is found in the same proportion.x = 40.0000
- Write the answer. 40.0
Check. Check: the value sits between the points on either side of cumulative frequency 36.
Watch out. Use the upper class boundary for each point on the curve, and plot the curve from the lower boundary of the first class.
Watch out
Watch out: the lower quartile is at n/4, not at n/2 minus something. For 40 plants it is at cumulative frequency 10.
Exam tip
Exam tip. Draw the lines on the graph as you read them, and write the reading next to each line.
Exercise 12B Median, quartiles and percentiles from a curve
Non-calculatorMethod. Read the value for the median from the cumulative frequency curve.
- Find the cumulative frequency. For the median, the cumulative frequency is 20 out of 40.k = 20
- Find the point on the curve. Find the two points on the cumulative frequency polygon either side of this height.
- Interpolate. The polygon is a straight line between the points, so the value is found in the same proportion.x = 26.0000
- Write the answer. 26.0
Check. Check: the value sits between the points on either side of cumulative frequency 20.
Watch out. Use the upper class boundary for each point on the curve, and plot the curve from the lower boundary of the first class.
Method. Read the value for the lower quartile from the cumulative frequency curve.
- Find the cumulative frequency. For the lower quartile, the cumulative frequency is 10 out of 40.k = 10
- Find the point on the curve. Find the two points on the cumulative frequency polygon either side of this height.
- Interpolate. The polygon is a straight line between the points, so the value is found in the same proportion.x = 18.7500
- Write the answer. 18.8
Check. Check: the value sits between the points on either side of cumulative frequency 10.
Watch out. Use the upper class boundary for each point on the curve, and plot the curve from the lower boundary of the first class.
Method. Read the value for the upper quartile from the cumulative frequency curve.
- Find the cumulative frequency. For the upper quartile, the cumulative frequency is 30 out of 40.k = 30
- Find the point on the curve. Find the two points on the cumulative frequency polygon either side of this height.
- Interpolate. The polygon is a straight line between the points, so the value is found in the same proportion.x = 34.0000
- Write the answer. 34.0
Check. Check: the value sits between the points on either side of cumulative frequency 30.
Watch out. Use the upper class boundary for each point on the curve, and plot the curve from the lower boundary of the first class.
Method. Read the value for the 90th percentile from the cumulative frequency curve.
- Find the cumulative frequency. For the 90th percentile, the cumulative frequency is 36 out of 40.k = 36
- Find the point on the curve. Find the two points on the cumulative frequency polygon either side of this height.
- Interpolate. The polygon is a straight line between the points, so the value is found in the same proportion.x = 40.0000
- Write the answer. 40.0
Check. Check: the value sits between the points on either side of cumulative frequency 36.
Watch out. Use the upper class boundary for each point on the curve, and plot the curve from the lower boundary of the first class.
Method. Read the value for the median from the cumulative frequency curve.
- Find the cumulative frequency. For the median, the cumulative frequency is 20 out of 40.k = 20
- Find the point on the curve. Find the two points on the cumulative frequency polygon either side of this height.
- Interpolate. The polygon is a straight line between the points, so the value is found in the same proportion.x = 24.1667
- Write the answer. 24.2
Check. Check: the value sits between the points on either side of cumulative frequency 20.
Watch out. Use the upper class boundary for each point on the curve, and plot the curve from the lower boundary of the first class.
Method. Read the value for the lower quartile from the cumulative frequency curve.
- Find the cumulative frequency. For the lower quartile, the cumulative frequency is 10 out of 40.k = 10
- Find the point on the curve. Find the two points on the cumulative frequency polygon either side of this height.
- Interpolate. The polygon is a straight line between the points, so the value is found in the same proportion.x = 15.0000
- Write the answer. 15.0
Check. Check: the value sits between the points on either side of cumulative frequency 10.
Watch out. Use the upper class boundary for each point on the curve, and plot the curve from the lower boundary of the first class.
Method. Read the value for the upper quartile from the cumulative frequency curve.
- Find the cumulative frequency. For the upper quartile, the cumulative frequency is 30 out of 40.k = 30
- Find the point on the curve. Find the two points on the cumulative frequency polygon either side of this height.
- Interpolate. The polygon is a straight line between the points, so the value is found in the same proportion.x = 33.3333
- Write the answer. 33.3
Check. Check: the value sits between the points on either side of cumulative frequency 30.
Watch out. Use the upper class boundary for each point on the curve, and plot the curve from the lower boundary of the first class.
Method. IQR = upper quartile − lower quartile.
- Upper quartile. The upper quartile is 34 (at cumulative frequency 30).34
- Lower quartile. The lower quartile is 18.75 (at cumulative frequency 10).18.75
- Subtract. 34 − 18.75 = 15.25.15.25
- Write the answer. Rounded to 1 d.p., the IQR is 15.3.15.3
Check. The IQR is the width of the middle half of the data.
Method. IQR = upper quartile − lower quartile.
- Upper quartile. Read at cumulative frequency 30: 33.3.33.3
- Lower quartile. Read at cumulative frequency 10: 15.0.15.0
- Subtract. 33.3 − 15.0 = 18.3.18.3
Check. The IQR is 18.3.
Method. Find half of the total.
- Half of 40. The median splits the data into two equal halves.40 ÷ 2 = 20
- Write the answer. 20
Check. There are 20 values below the median and 20 above.
Method. Half of the plants are taller than the median.
- Half of 40. The median splits the plants into two halves of 20.20
- Write the answer. 20
Check. The median is the middle value, so half the plants are above it.
Method. Think about the data.
- Grouped data. The values are only known to be in a class, not exactly.grouped
- Write the answer. So the graph can only give an estimate of the median.An estimate, because the data are grouped
Check. Grouped data loses the exact values.
Method. Read the value for the upper quartile from the cumulative frequency curve.
- Find the cumulative frequency. For the upper quartile, the cumulative frequency is 30 out of 40.k = 30
- Find the point on the curve. Find the two points on the cumulative frequency polygon either side of this height.
- Interpolate. The polygon is a straight line between the points, so the value is found in the same proportion.x = 34.0000
- Write the answer. 34.0
Check. Check: the value sits between the points on either side of cumulative frequency 30.
Watch out. Use the upper class boundary for each point on the curve, and plot the curve from the lower boundary of the first class.
Box plots from raw data and a summary
The five-number summary is the lowest value, the lower quartile, the median, the upper quartile and the highest value. The box runs from the lower to the upper quartile, and the whiskers go out to the lowest and highest values.
Key ideas
- Five-number summary: lowest, Q1, median, Q3, highest.
- The box goes from Q1 to Q3. The IQR is the width of the box.
- For an odd number of values, leave out the median when you find the quartiles.
- Draw the box plot on a scale that covers all the data.
Method. Order the data, find the median, then the medians of each half.
- Put the data in order. Write the values from smallest to largest.12, 15, 17, 19, 20, 22, 25, 28, 30
- Find the median. There are 9 values, so the median is in the middle.median = 20
- Find the quartiles. The lower quartile is the median of the lower half, and the upper quartile is the median of the upper half.Q1 = 16, Q3 = 26.5
- Write the summary. Include the lowest and highest values as well.lowest 12, lower quartile 16, median 20, upper quartile 26.5, highest 30
Check. The interquartile range is 10.5.
Watch out. If there are an odd number of values, leave out the median when you find the quartiles.
Method. Order the data, find the median, then the medians of each half.
- Put the data in order. Write the values from smallest to largest.5, 7, 8, 10, 12, 14, 15, 18
- Find the median. There are 8 values, so the median is in the middle.median = 11
- Find the quartiles. The lower quartile is the median of the lower half, and the upper quartile is the median of the upper half.Q1 = 7.5, Q3 = 14.5
- Write the summary. Include the lowest and highest values as well.lowest 5, lower quartile 7.5, median 11, upper quartile 14.5, highest 18
Check. The interquartile range is 7.
Watch out. If there are an odd number of values, leave out the median when you find the quartiles.
Method. The interquartile range (IQR) measures the spread of the middle half of the data.
- Write the quartiles. The lower quartile is Q1 and the upper quartile is Q3.Q1 = 16, Q3 = 26.5
- Subtract. The interquartile range is the upper quartile minus the lower quartile.26.5 − 16 = 10.5
- Write the answer. 10.5
Check. The IQR is the width of the box in a box plot.
Watch out. The range uses the highest and lowest values, the IQR uses the quartiles.
Watch out
Watch out: the quartiles are medians of the halves, not the quarter points of the list.
Exam tip
Exam tip. Draw the box plot with a ruled scale, and label the five values on the diagram.
Exercise 12C Box plots
Non-calculatorMethod. Order the data, find the median, then the medians of each half.
- Put the data in order. Write the values from smallest to largest.12, 15, 17, 19, 20, 22, 25, 28, 30
- Find the median. There are 9 values, so the median is in the middle.median = 20
- Find the quartiles. The lower quartile is the median of the lower half, and the upper quartile is the median of the upper half.Q1 = 16, Q3 = 26.5
- Write the summary. Include the lowest and highest values as well.lowest 12, lower quartile 16, median 20, upper quartile 26.5, highest 30
Check. The interquartile range is 10.5.
Watch out. If there are an odd number of values, leave out the median when you find the quartiles.
Method. Order the data, find the median, then the medians of each half.
- Put the data in order. Write the values from smallest to largest.5, 7, 8, 10, 12, 14, 15, 18
- Find the median. There are 8 values, so the median is in the middle.median = 11
- Find the quartiles. The lower quartile is the median of the lower half, and the upper quartile is the median of the upper half.Q1 = 7.5, Q3 = 14.5
- Write the summary. Include the lowest and highest values as well.lowest 5, lower quartile 7.5, median 11, upper quartile 14.5, highest 18
Check. The interquartile range is 7.
Watch out. If there are an odd number of values, leave out the median when you find the quartiles.
Method. Order the data, find the median, then the medians of each half.
- Put the data in order. Write the values from smallest to largest.3, 4, 6, 6, 7, 9, 10, 12, 15, 21, 30
- Find the median. There are 11 values, so the median is in the middle.median = 9
- Find the quartiles. The lower quartile is the median of the lower half, and the upper quartile is the median of the upper half.Q1 = 6, Q3 = 15
- Write the summary. Include the lowest and highest values as well.lowest 3, lower quartile 6, median 9, upper quartile 15, highest 30
Check. The interquartile range is 9.
Watch out. If there are an odd number of values, leave out the median when you find the quartiles.
Method. Order the data, find the median, then the medians of each half.
- Put the data in order. Write the values from smallest to largest.40, 45, 52, 55, 58, 60, 61, 67, 70, 75
- Find the median. There are 10 values, so the median is in the middle.median = 59
- Find the quartiles. The lower quartile is the median of the lower half, and the upper quartile is the median of the upper half.Q1 = 52, Q3 = 67
- Write the summary. Include the lowest and highest values as well.lowest 40, lower quartile 52, median 59, upper quartile 67, highest 75
Check. The interquartile range is 15.
Watch out. If there are an odd number of values, leave out the median when you find the quartiles.
Method. The interquartile range (IQR) measures the spread of the middle half of the data.
- Write the quartiles. The lower quartile is Q1 and the upper quartile is Q3.Q1 = 52, Q3 = 67
- Subtract. The interquartile range is the upper quartile minus the lower quartile.67 − 52 = 15
- Write the answer. 15
Check. The IQR is the width of the box in a box plot.
Watch out. The range uses the highest and lowest values, the IQR uses the quartiles.
Method. The interquartile range (IQR) measures the spread of the middle half of the data.
- Write the quartiles. The lower quartile is Q1 and the upper quartile is Q3.Q1 = 6, Q3 = 15
- Subtract. The interquartile range is the upper quartile minus the lower quartile.15 − 6 = 9
- Write the answer. 9
Check. The IQR is the width of the box in a box plot.
Watch out. The range uses the highest and lowest values, the IQR uses the quartiles.
Method. Range = highest − lowest. IQR = upper quartile − lower quartile.
- Range. 30 − 4 = 26.26
- IQR. 22 − 10 = 12.12
- Write the answer. Range 26, IQR 12
Check. The IQR is always smaller than or equal to the range.
Method. Use the five values as the box and whisker positions.
- Scale. Use a scale from 0 to 35.0 to 35
- Box. The box runs from 10 to 22, with a line at the median 15.box 10 to 22
- Whiskers. Whiskers go to 4 and to 30.whiskers 4 and 30
- Write the answer. Box from 10 to 22, median 15, whiskers 4 and 30
Check. The picture matches the five values.
Method. Look at the halves.
- Odd number. With an odd number of values, the median is one value.median = middle value
- Halves. The halves are the values below and above the median.two equal halves
- Write the answer. The median is not in either half
Check. This gives equal halves for the quartiles.
Method. Find the median and the medians of the halves.
- Median. The middle two values are 17 and 19, so the median is 18.18
- Quartiles. The lower half is 9, 11, 13, 15, 17, with median 13. The upper half is 19, 21, 23, 25, 27, with median 23.Q1 = 13, Q3 = 23
- Write the answer. Median 18, Q1 13, Q3 23
Check. Check the halves carefully.
Comparing two distributions
To compare, always say something about the average (the median) and the spread (the IQR or range), in the context of the data.
Key ideas
- Compare medians for the typical value.
- Compare IQRs for the spread of the middle half.
- Say what this means in context, not just the numbers.
- Use the same scale for both box plots.
Method. Compare the medians, then the IQRs.
- Medians. Class A has the higher median, 20, compared with 11.20 > 11
- IQR. Class A has the larger IQR, 10.5, compared with 7, so its scores are more spread out.10.5 > 7
- Write the comparison. Class A scored higher on average and was more spread out
Check. Both comparisons use the same measures, so they are fair.
Method. The medians are the same, so compare the IQRs.
- Medians. The medians are equal, so the typical scores are the same.15 = 15
- IQR. Class D has a larger IQR, so its middle half is more spread out.12 > 4
- Write the comparison. Same average, but Class D is more spread out
Check. A larger IQR means more variation in the middle half.
Method. Compare the averages, then the spreads.
- Averages. The medians are both 30, so the typical scores are the same.30 = 30
- Spread. Class Y has the larger IQR, 15, compared with 8 for Class X.15 > 8
- Write the comparison. Same average; Class Y is more spread out
Check. The IQR shows the spread of the middle half.
Watch out
Watch out: a comparison needs both an average and a spread. Do not say one class is better without a reason.
Exam tip
Exam tip. Use the phrase "on average" for the median, and "more spread out" or "more consistent" for the IQR.
Exercise 12D Comparing distributions
Non-calculatorMethod. Compare the medians.
- Medians. 20 is bigger than 11.20 > 11
- Write the answer. Class A
Check. The median is the average.
Method. Compare the IQRs. The smaller IQR is more consistent.
- IQR. 7 is smaller than 10.5.7 < 10.5
- Write the answer. Class B
Check. A smaller IQR means the middle half is closer together.
Method. Compare the averages and the spreads.
- Average. The medians are equal.15 = 15
- Spread. The IQRs are 4 and 12, so the second is more spread out.12 > 4
- Write the answer. Same average; the second has a larger spread
Check. The IQR shows the middle half spread.
Method. Think about what the range measures.
- Range. The range uses only the two extreme values.two values
- IQR. The IQR uses the middle half, which is a better measure of consistency.middle half
- Write the answer. The IQR shows the spread of the middle half
Check. An outlier can change the range but not the IQR much.
Method. Compare the medians.
- Medians. 26 is bigger than 14.2.26 > 14.2
- Write the answer. The first garden has taller plants on average
Check. Medians are the typical values.
Method. The length of the box is the IQR.
- Length. The length of the box is the IQR.box length = IQR
- Write the answer. Box A has a larger IQR, so more spread
Check. Box A has more variation in its middle half.
Method. The smaller IQR is more consistent.
- Compare. 6 is smaller than 15.6 < 15
- Write the answer. Class P
Check. Class P is more consistent.
Shape and spread of a distribution
A distribution is symmetrical if the median is in the middle of the box and the whiskers are about equal. It is positively skewed if the tail is on the right (the median is nearer the lower quartile). It is negatively skewed if the tail is on the left.
Key ideas
- Symmetrical: median in the middle of the box, equal whiskers.
- Positive skew: long tail to the right, median nearer Q1.
- Negative skew: long tail to the left, median nearer Q3.
- Comment in context, using the median and the IQR.
Method. Look at where the median sits, and the length of each whisker.
- Median in the box. The median 12 is near the lower end of the box (10 to 20).12 is near Q1
- Whiskers. The upper whisker (20 to 40) is much longer than the lower one (8 to 10).upper whisker is longer
- Write the answer. Positively skewed
Check. The long tail is on the right.
Method. Compare the IQRs.
- Compare. The second IQR is 5 times the first.15 ÷ 3 = 5
- Check the medians. The medians are equal, so the comparison is only about spread.15 = 15
- Comment. The second set is more spread out in the middle half.The second set is more spread out
Check. A larger IQR means a wider spread of the middle values.
Method. Look at the whiskers and the position of the median.
- Median in the box. The median 5 is in the middle of the box from 3 to 7.5 is central
- Whiskers. The lower whisker (1 to 3) is shorter than the upper whisker (7 to 25).upper whisker is longer
- Write the answer. Positively skewed
Check. The long tail is on the right.
Watch out
Watch out: skew is about where the tail is, not where the median is on the axis. Look at the whiskers and the median position in the box.
Exam tip
Exam tip. Always link your comment to the context, such as "waiting times are more spread out", not just "the IQR is bigger".
Exercise 12E Shape and spread
Non-calculatorMethod. The long whisker is on the right.
- Median. The median is near the lower quartile.near Q1
- Whisker. The upper whisker is much longer.long right tail
- Write the answer. Positively skewed
Check. A long right tail means positive skew.
Method. Look at the median and the whiskers.
- Median. The median 50 is nearer the upper quartile 60.near Q3
- Whisker. The lower whisker from 20 to 30 is longer than the upper one.long left tail
- Write the answer. Negatively skewed
Check. The long left tail means negative skew.
Method. The median is in the middle and the whiskers are equal.
- Median. The median 20 is exactly in the middle of the box.20 is the middle
- Whiskers. Both whiskers are 5 long.5 and 5
- Write the answer. Symmetrical
Check. Equal whiskers and a central median mean symmetry.
Method. Compare the IQRs.
- Compare. 12 is bigger than 4.12 > 4
- Write the answer. The second data set is more spread out
Check. The IQR measures the middle half.
Method. Think about the tail.
- Tail. A long tail to the right means some very large values.large values on the right
- Write the answer. Most values are at the lower end, with a few large ones.Most waits are short, with a few long ones
Check. The median is usually closer to the lower end when the tail is on the right.
Method. Think about the large values.
- Effect. A few large values pull the mean up.mean is pulled up
- Write the answer. So the median can be a better typical value.The median is better
Check. Compare the median and mean to see the skew.
Method. The interquartile range (IQR) measures the spread of the middle half of the data.
- Write the quartiles. The lower quartile is Q1 and the upper quartile is Q3.Q1 = 28, Q3 = 45
- Subtract. The interquartile range is the upper quartile minus the lower quartile.45 − 28 = 17
- Write the answer. 17
Check. The IQR is the width of the box in a box plot.
Watch out. The range uses the highest and lowest values, the IQR uses the quartiles.
Chapter summary
| Skill | Method |
|---|---|
| Cumulative frequency | Running total. Plot at the upper class boundary. |
| Median, quartiles | At cumulative frequency n/2, n/4 and 3n/4 on the curve. |
| Five-number summary | Lowest, lower quartile, median, upper quartile, highest. |
| IQR | Upper quartile minus lower quartile: the width of the box. |
| Shape | Symmetrical, positive skew (tail right), negative skew (tail left). |
| Comparing | Compare the medians (average) and the IQRs (spread), in context. |
Can you…?
- I can build a cumulative frequency table and plot it at the upper boundaries
- I can read the median and quartiles from a cumulative frequency curve
- I can find a five-number summary and draw a box plot
- I can compare two distributions with medians and IQRs
- I can describe the shape and spread of a distribution in context
Key terms
- cumulative frequency
- the running total of the frequencies up to a given value
- quartile
- a value that splits the ordered data into four equal parts
- interquartile range
- the upper quartile minus the lower quartile
- five-number summary
- the lowest value, lower quartile, median, upper quartile and highest value
- skew
- the way a distribution has a longer tail on one side
Exercise 12Z Check-up
Non-calculatorMethod. Order the data, find the median, then the medians of each half.
- Put the data in order. Write the values from smallest to largest.2, 5, 6, 9, 11, 14, 17
- Find the median. There are 7 values, so the median is in the middle.median = 9
- Find the quartiles. The lower quartile is the median of the lower half, and the upper quartile is the median of the upper half.Q1 = 5, Q3 = 14
- Write the summary. Include the lowest and highest values as well.lowest 2, lower quartile 5, median 9, upper quartile 14, highest 17
Check. The interquartile range is 9.
Watch out. If there are an odd number of values, leave out the median when you find the quartiles.
Method. Read the value for the median from the cumulative frequency curve.
- Find the cumulative frequency. For the median, the cumulative frequency is 20 out of 40.k = 20
- Find the point on the curve. Find the two points on the cumulative frequency polygon either side of this height.
- Interpolate. The polygon is a straight line between the points, so the value is found in the same proportion.x = 26.0000
- Write the answer. 26.0
Check. Check: the value sits between the points on either side of cumulative frequency 20.
Watch out. Use the upper class boundary for each point on the curve, and plot the curve from the lower boundary of the first class.
Method. Read the value for the upper quartile from the cumulative frequency curve.
- Find the cumulative frequency. For the upper quartile, the cumulative frequency is 30 out of 40.k = 30
- Find the point on the curve. Find the two points on the cumulative frequency polygon either side of this height.
- Interpolate. The polygon is a straight line between the points, so the value is found in the same proportion.x = 34.0000
- Write the answer. 34.0
Check. Check: the value sits between the points on either side of cumulative frequency 30.
Watch out. Use the upper class boundary for each point on the curve, and plot the curve from the lower boundary of the first class.
Method. Read the value for the lower quartile from the cumulative frequency curve.
- Find the cumulative frequency. For the lower quartile, the cumulative frequency is 10 out of 40.k = 10
- Find the point on the curve. Find the two points on the cumulative frequency polygon either side of this height.
- Interpolate. The polygon is a straight line between the points, so the value is found in the same proportion.x = 15.0000
- Write the answer. 15.0
Check. Check: the value sits between the points on either side of cumulative frequency 10.
Watch out. Use the upper class boundary for each point on the curve, and plot the curve from the lower boundary of the first class.
Method. The interquartile range (IQR) measures the spread of the middle half of the data.
- Write the quartiles. The lower quartile is Q1 and the upper quartile is Q3.Q1 = 16, Q3 = 26.5
- Subtract. The interquartile range is the upper quartile minus the lower quartile.26.5 − 16 = 10.5
- Write the answer. 10.5
Check. The IQR is the width of the box in a box plot.
Watch out. The range uses the highest and lowest values, the IQR uses the quartiles.
Method. Range and IQR.
- Range. 31 − 4 = 27.27
- IQR. 20 − 9 = 11.11
- Write the answer. Range 27, IQR 11
Check. 27 and 11.
Method. Compare the averages and spreads.
- Average. The medians are both 30.30 = 30
- Spread. Class B has the larger IQR.14 > 6
- Write the answer. Same median; Class B is more spread out
Check. The IQR shows the middle half.
Method. Add the frequencies one after another.
- Running total. 4, 4 + 6 = 10, 10 + 9 = 19, 19 + 3 = 22.4, 10, 19, 22
- Write the answer. 4, 10, 19, 22
Check. The total is 22.
Method. Look at the whiskers.
- Whiskers. The lower whisker from 4 to 12 is longer than the upper one from 24 to 26.long left tail
- Write the answer. Negatively skewed
Check. The long tail is on the left.
Method. Order the data and find the middle value.
- Order. 3, 6, 8, 9, 12, 15, 20.3, 6, 8, 9, 12, 15, 20
- Middle. The 4th value is 9.9
Check. There are 3 values on each side.
Total 23 marks. Allow about 23 minutes.
Exercise 12X Practice test questions
Non-calculatorMethod. Read the value for the median from the cumulative frequency curve.
- Find the cumulative frequency. For the median, the cumulative frequency is 20 out of 40.k = 20
- Find the point on the curve. Find the two points on the cumulative frequency polygon either side of this height.
- Interpolate. The polygon is a straight line between the points, so the value is found in the same proportion.x = 26.0000
- Write the answer. 26.0
Check. Check: the value sits between the points on either side of cumulative frequency 20.
Watch out. Use the upper class boundary for each point on the curve, and plot the curve from the lower boundary of the first class.
Method. Order the data, find the median, then the medians of each half.
- Put the data in order. Write the values from smallest to largest.3, 7, 8, 10, 12, 15, 18, 21
- Find the median. There are 8 values, so the median is in the middle.median = 11
- Find the quartiles. The lower quartile is the median of the lower half, and the upper quartile is the median of the upper half.Q1 = 7.5, Q3 = 16.5
- Write the summary. Include the lowest and highest values as well.lowest 3, lower quartile 7.5, median 11, upper quartile 16.5, highest 21
Check. The interquartile range is 9.
Watch out. If there are an odd number of values, leave out the median when you find the quartiles.
Method. The interquartile range (IQR) measures the spread of the middle half of the data.
- Write the quartiles. The lower quartile is Q1 and the upper quartile is Q3.Q1 = 52, Q3 = 67
- Subtract. The interquartile range is the upper quartile minus the lower quartile.67 − 52 = 15
- Write the answer. 15
Check. The IQR is the width of the box in a box plot.
Watch out. The range uses the highest and lowest values, the IQR uses the quartiles.
Method. Read the value for the upper quartile from the cumulative frequency curve.
- Find the cumulative frequency. For the upper quartile, the cumulative frequency is 30 out of 40.k = 30
- Find the point on the curve. Find the two points on the cumulative frequency polygon either side of this height.
- Interpolate. The polygon is a straight line between the points, so the value is found in the same proportion.x = 33.3333
- Write the answer. 33.3
Check. Check: the value sits between the points on either side of cumulative frequency 30.
Watch out. Use the upper class boundary for each point on the curve, and plot the curve from the lower boundary of the first class.
Method. Compare the medians, then the IQRs.
- Medians. 60 is bigger than 52, so Class X has the higher average.60 > 52
- IQR. 8 is smaller than 20, so Class X is more consistent.8 < 20
- Write the answer. Class X: higher average and more consistent
Check. Both comparisons use the averages and spreads.
Method. Look at the whiskers.
- Whiskers. The upper whisker from 36 to 60 is longer than the lower whisker from 10 to 25.long right tail
- Write the answer. Positively skewed
Check. The long tail is on the right.
Method. Find the median and the medians of the halves.
- Median. The middle value of 9 is the 5th, which is 10.10
- Quartiles. Lower half 2, 4, 6, 8: Q1 = 5. Upper half 12, 14, 16, 18: Q3 = 15.Q1 = 5, Q3 = 15
- Write the answer. Median 10, Q1 5, Q3 15
Check. The IQR is 10.
Method. Think about grouped data.
- Grouped. The data are in classes, so exact values are lost.classes
- Write the answer. The data are grouped, so the answer is an estimate
Check. Estimates from grouped data are approximate.
Method. IQR = upper quartile − lower quartile.
- Read. Upper quartile 34, lower quartile 18.75.34 and 18.75
- Subtract. 34 − 18.75 = 15.25, which is 15.3 to 1 d.p.15.25
- Write the answer. 15.3
Check. The IQR is about 15.3.
Method. The tail is on the left, so the skew is negative.
- Name the skew. A tail on the left is negative skew.negative
- Write the answer. Negatively skewed
Check. The median is usually nearer the upper quartile.