GCSE Maths: Cumulative frequency graphs (Higher)
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Cumulative frequency graphs
Cumulative frequency is a running total of the frequencies. It tells us how many values are up to and including the end of each class. A cumulative frequency graph plots each cumulative frequency against the upper boundary of its class and joins the points with a smooth S-shaped curve. It rises from 0 at the start and ends at the total number of values.
Reading the graph. To find the median, go to half of the total (n ÷ 2) on the vertical axis, across to the curve and down to the horizontal axis. For the lower quartile use n ÷ 4 and for the upper quartile use 3n ÷ 4. The interquartile range is the upper quartile minus the lower quartile. A percentile works in the same way: the 90th percentile is at 90% of n. These readings are estimates. Notice that on a cumulative frequency graph we use n ÷ 2, n ÷ 4 and 3n ÷ 4, not (n + 1) ÷ 2 and so on.
To find how many values are less than a given value, go up from that value to the curve and across to the cumulative frequency axis. To find how many are more than a value, subtract that reading from the total.
Key ideas
- Cumulative frequency = running total of the frequencies.
- Plot (upper class boundary, cumulative frequency), starting with (lowest boundary, 0), and join with a smooth curve.
- Median at n ÷ 2, lower quartile at n ÷ 4, upper quartile at 3n ÷ 4. IQR = Q3 − Q1.
- Percentiles: the p-th percentile is at p% of n.
- To find the number above a value, read the cumulative frequency and subtract it from the total.
- Readings from a graph are estimates.
| Time | Frequency |
|---|---|
| 0 < t ≤ 10 min | 4 |
| 10 < t ≤ 20 min | 10 |
| 20 < t ≤ 30 min | 22 |
| 30 < t ≤ 40 min | 25 |
| 40 < t ≤ 50 min | 13 |
| 50 < t ≤ 60 min | 6 |
Key words. Cumulative frequency is a running total of the frequencies. A cumulative frequency graph plots each running total against the upper end of its class.
Method. Cumulative frequency is a running total: add each frequency to the total of all the earlier ones.
- Start the running total with the first frequency. Cumulative frequency means “how many values so far”.First class: 4
- Add the next frequency to the total so far (10 < t ≤ 20 min). Each cumulative frequency is the previous one plus this class’s frequency.4 + 10 = 14
- Add the next frequency to the total so far (20 < t ≤ 30 min). Each cumulative frequency is the previous one plus this class’s frequency.14 + 22 = 36
- Add the next frequency to the total so far (30 < t ≤ 40 min). Each cumulative frequency is the previous one plus this class’s frequency.36 + 25 = 61
- Add the next frequency to the total so far (40 < t ≤ 50 min). Each cumulative frequency is the previous one plus this class’s frequency.61 + 13 = 74
- Add the next frequency to the total so far (50 < t ≤ 60 min). Each cumulative frequency is the previous one plus this class’s frequency.74 + 6 = 80
- Check the last number. The last cumulative frequency must equal the total number of values.Total of frequencies = 80 = last cumulative frequency 80
Check. The last running total (80) equals the sum of all the frequencies. ✓
Watch out. The cumulative frequency never goes down, and the last one is the total number of data values.
Key words. Cumulative frequency is a running total of the frequencies. A cumulative frequency graph plots each running total against the upper end of its class.
Method. Make a cumulative frequency table, then plot each cumulative frequency against the upper end of its class, and join the points with a smooth curve.
- Work out the cumulative frequencies. Add each frequency to the total so far.
Time Frequency Cumulative frequency 0 < t ≤ 10 min 4 4 10 < t ≤ 20 min 10 14 20 < t ≤ 30 min 22 36 30 < t ≤ 40 min 25 61 40 < t ≤ 50 min 13 74 50 < t ≤ 60 min 6 80 - Write the points to plot. Plot each cumulative frequency at the upper class boundary, because that is the point by which that many values have been counted. Start at (0, 0): nothing is below the first lower boundary.(0, 0), (10, 4), (20, 14), (30, 36), (40, 61), (50, 74), (60, 80)
- Choose scales and plot the points. The horizontal axis goes from the lowest to the highest class boundary. The vertical axis goes from 0 to the total frequency.Horizontal axis: Time taken (minutes). Vertical axis: cumulative frequency up to 80.
- Join the points with a smooth curve. The curve is S-shaped and never goes down.
Check. The curve starts at 0 and ends at the total frequency, 80. ✓
Watch out. Plot at the upper boundary of each class, not the midpoint and not the lower boundary.
Key words. Cumulative frequency is a running total of the frequencies. A cumulative frequency graph plots each running total against the upper end of its class.
Method. On a cumulative frequency graph, the median is at height n ÷ 2, the lower quartile at n ÷ 4 and the upper quartile at 3n ÷ 4. Read across to the curve and down to the axis.
- Find the total number of values. The last cumulative frequency is the total, n.n = 80
- Work out the heights to read at. Lower quartile at n ÷ 4, median at n ÷ 2, upper quartile at 3n ÷ 4.n ÷ 4 = 20
n ÷ 2 = 40
3n ÷ 4 = 60 - Draw a horizontal line across from each height to the curve, then straight down to the axis. The value on the horizontal axis is the quartile.
- Read the three values. Read the scale carefully. The answers are estimates.Q1 ≈ 23.5 min
Median ≈ 31.5 min
Q3 ≈ 39.5 min - Subtract to find the interquartile range. IQR = Q3 − Q1.39.5 − 23.5 = 16 min
Check. Going up from each reading to the curve and across should give the heights again. ✓
Watch out. Read heights from the cumulative frequency axis and answers from the data axis. Do not use n + 1 here.
Key words. Cumulative frequency is a running total of the frequencies. A cumulative frequency graph plots each running total against the upper end of its class.
Method. Read the cumulative frequency at the value. This is the number of values up to that value; subtract from the total to get the number above it.
- Find 45 minutes on the horizontal axis and go up to the curve. The curve shows how many values are at or below each value.
- Read across to the cumulative frequency axis. This gives the number of values up to 45.Cumulative frequency = 69
- Subtract from the total to find how many are more than that. The last cumulative frequency is the total.80 − 69 = 11
Check. 69 below plus 11 above makes 80, the total. ✓
Watch out. The reading gives the number below the value. For “more than” you must subtract from the total.
Key words. Cumulative frequency is a running total of the frequencies. A cumulative frequency graph plots each running total against the upper end of its class.
Method. Cumulative frequency graphs are read by going across from the cumulative frequency to the curve, then down to the axis.
- Find the position on the cumulative frequency axis. The total cumulative frequency is n = 80. The 90th percentile is 90% of the way through the data: 90 ÷ 100 × 80 = 72.Position = 72
- Draw a horizontal line across from that height to the curve. We want the data value that has this many values at or below it.
- Draw a vertical line down to the horizontal axis and read the value. Read the scale carefully. The result is an estimate.90th percentile ≈ 48 min
Check. Going up from your answer to the curve and across should give the same height again. ✓
Watch out. Read across from the cumulative frequency axis. The frequency axis is the vertical one, the data axis is the horizontal one.
Watch out
Plot at the upper class boundary (not the midpoint), and start the curve at (lowest boundary, 0). When reading, use n ÷ 2, n ÷ 4 and 3n ÷ 4: do not add 1.Exercise 24M Cumulative frequency graphs
Non-calculator| Height | Frequency |
|---|---|
| 10 < h ≤ 20 cm | 5 |
| 20 < h ≤ 30 cm | 12 |
| 30 < h ≤ 40 cm | 24 |
| 40 < h ≤ 50 cm | 14 |
| 50 < h ≤ 60 cm | 5 |
Key words. Cumulative frequency is a running total of the frequencies. A cumulative frequency graph plots each running total against the upper end of its class.
Method. Cumulative frequency is a running total: add each frequency to the total of all the earlier ones.
- Start the running total with the first frequency. Cumulative frequency means “how many values so far”.First class: 5
- Add the next frequency to the total so far (20 < h ≤ 30 cm). Each cumulative frequency is the previous one plus this class’s frequency.5 + 12 = 17
- Add the next frequency to the total so far (30 < h ≤ 40 cm). Each cumulative frequency is the previous one plus this class’s frequency.17 + 24 = 41
- Add the next frequency to the total so far (40 < h ≤ 50 cm). Each cumulative frequency is the previous one plus this class’s frequency.41 + 14 = 55
- Add the next frequency to the total so far (50 < h ≤ 60 cm). Each cumulative frequency is the previous one plus this class’s frequency.55 + 5 = 60
- Check the last number. The last cumulative frequency must equal the total number of values.Total of frequencies = 60 = last cumulative frequency 60
Check. The last running total (60) equals the sum of all the frequencies. ✓
Watch out. The cumulative frequency never goes down, and the last one is the total number of data values.
Key words. Cumulative frequency is a running total of the frequencies. A cumulative frequency graph plots each running total against the upper end of its class.
Method. Make a cumulative frequency table, then plot each cumulative frequency against the upper end of its class, and join the points with a smooth curve.
- Work out the cumulative frequencies. Add each frequency to the total so far.
Height Frequency Cumulative frequency 10 < h ≤ 20 cm 5 5 20 < h ≤ 30 cm 12 17 30 < h ≤ 40 cm 24 41 40 < h ≤ 50 cm 14 55 50 < h ≤ 60 cm 5 60 - Write the points to plot. Plot each cumulative frequency at the upper class boundary, because that is the point by which that many values have been counted. Start at (10, 0): nothing is below the first lower boundary.(10, 0), (20, 5), (30, 17), (40, 41), (50, 55), (60, 60)
- Choose scales and plot the points. The horizontal axis goes from the lowest to the highest class boundary. The vertical axis goes from 0 to the total frequency.Horizontal axis: Height (cm). Vertical axis: cumulative frequency up to 60.
- Join the points with a smooth curve. The curve is S-shaped and never goes down.
Check. The curve starts at 0 and ends at the total frequency, 60. ✓
Watch out. Plot at the upper boundary of each class, not the midpoint and not the lower boundary.
Key words. Cumulative frequency is a running total of the frequencies. A cumulative frequency graph plots each running total against the upper end of its class.
Method. On a cumulative frequency graph, the median is at height n ÷ 2, the lower quartile at n ÷ 4 and the upper quartile at 3n ÷ 4. Read across to the curve and down to the axis.
- Find the total number of values. The last cumulative frequency is the total, n.n = 80
- Work out the heights to read at. Lower quartile at n ÷ 4, median at n ÷ 2, upper quartile at 3n ÷ 4.n ÷ 4 = 20
n ÷ 2 = 40
3n ÷ 4 = 60 - Draw a horizontal line across from each height to the curve, then straight down to the axis. The value on the horizontal axis is the quartile.
- Read the three values. Read the scale carefully. The answers are estimates.Q1 ≈ 23.5 min
Median ≈ 31.5 min
Q3 ≈ 39.5 min
Check. Going up from each reading to the curve and across should give the heights again. ✓
Watch out. Read heights from the cumulative frequency axis and answers from the data axis. Do not use n + 1 here.
Key words. Cumulative frequency is a running total of the frequencies. A cumulative frequency graph plots each running total against the upper end of its class.
Method. On a cumulative frequency graph, the median is at height n ÷ 2, the lower quartile at n ÷ 4 and the upper quartile at 3n ÷ 4. Read across to the curve and down to the axis.
- Find the total number of values. The last cumulative frequency is the total, n.n = 80
- Work out the heights to read at. Lower quartile at n ÷ 4, median at n ÷ 2, upper quartile at 3n ÷ 4.n ÷ 4 = 20
n ÷ 2 = 40
3n ÷ 4 = 60 - Draw a horizontal line across from each height to the curve, then straight down to the axis. The value on the horizontal axis is the quartile.
- Read the three values. Read the scale carefully. The answers are estimates.Q1 ≈ 23.5 min
Median ≈ 31.5 min
Q3 ≈ 39.5 min
Check. Going up from each reading to the curve and across should give the heights again. ✓
Watch out. Read heights from the cumulative frequency axis and answers from the data axis. Do not use n + 1 here.
Key words. Cumulative frequency is a running total of the frequencies. A cumulative frequency graph plots each running total against the upper end of its class.
Method. On a cumulative frequency graph, the median is at height n ÷ 2, the lower quartile at n ÷ 4 and the upper quartile at 3n ÷ 4. Read across to the curve and down to the axis.
- Find the total number of values. The last cumulative frequency is the total, n.n = 80
- Work out the heights to read at. Lower quartile at n ÷ 4, median at n ÷ 2, upper quartile at 3n ÷ 4.n ÷ 4 = 20
n ÷ 2 = 40
3n ÷ 4 = 60 - Draw a horizontal line across from each height to the curve, then straight down to the axis. The value on the horizontal axis is the quartile.
- Read the three values. Read the scale carefully. The answers are estimates.Q1 ≈ 23.5 min
Median ≈ 31.5 min
Q3 ≈ 39.5 min
Check. Going up from each reading to the curve and across should give the heights again. ✓
Watch out. Read heights from the cumulative frequency axis and answers from the data axis. Do not use n + 1 here.
Key words. Cumulative frequency is a running total of the frequencies. A cumulative frequency graph plots each running total against the upper end of its class.
Method. On a cumulative frequency graph, the median is at height n ÷ 2, the lower quartile at n ÷ 4 and the upper quartile at 3n ÷ 4. Read across to the curve and down to the axis.
- Find the total number of values. The last cumulative frequency is the total, n.n = 80
- Work out the heights to read at. Lower quartile at n ÷ 4, median at n ÷ 2, upper quartile at 3n ÷ 4.n ÷ 4 = 20
n ÷ 2 = 40
3n ÷ 4 = 60 - Draw a horizontal line across from each height to the curve, then straight down to the axis. The value on the horizontal axis is the quartile.
- Read the three values. Read the scale carefully. The answers are estimates.Q1 ≈ 23.5 min
Median ≈ 31.5 min
Q3 ≈ 39.5 min - Subtract to find the interquartile range. IQR = Q3 − Q1.39.5 − 23.5 = 16 min
Check. Going up from each reading to the curve and across should give the heights again. ✓
Watch out. Read heights from the cumulative frequency axis and answers from the data axis. Do not use n + 1 here.
Key words. Cumulative frequency is a running total of the frequencies. A cumulative frequency graph plots each running total against the upper end of its class.
Method. Read the cumulative frequency at the value. This is the number of values up to that value; subtract from the total to get the number above it.
- Find 45 minutes on the horizontal axis and go up to the curve. The curve shows how many values are at or below each value.
- Read across to the cumulative frequency axis. This gives the number of values up to 45.Cumulative frequency = 69
- Subtract from the total to find how many are more than that. The last cumulative frequency is the total.80 − 69 = 11
Check. 69 below plus 11 above makes 80, the total. ✓
Watch out. The reading gives the number below the value. For “more than” you must subtract from the total.
Key words. Cumulative frequency is a running total of the frequencies. A cumulative frequency graph plots each running total against the upper end of its class.
Method. Read the cumulative frequency at the value. This is the number of values up to that value; subtract from the total to get the number above it.
- Find 25 minutes on the horizontal axis and go up to the curve. The curve shows how many values are at or below each value.
- Read across to the cumulative frequency axis. This gives the number of values up to 25.Cumulative frequency = 24
Check. 24 below plus 56 above makes 80, the total. ✓
Watch out. The reading gives the number below the value. For “more than” you must subtract from the total.
Key words. Cumulative frequency is a running total of the frequencies. A cumulative frequency graph plots each running total against the upper end of its class.
Method. Read the number of workers who took less than 20 minutes, then write it as a percentage of the total.
- Read the graph at 20 minutes. The point on the curve at 20 minutes is the cumulative frequency, the number of workers who took 20 minutes or less.14 workers
- Write it as a fraction of 80. The total is 80.1480
- Change to a percentage. Divide by 80 and multiply by 100.14 ÷ 80 × 100 = 17.5%
Check. 17.5% of 80 = 14. ✓
Watch out. The whole is 80, the total number of workers.
Key words. Cumulative frequency is a running total of the frequencies. A cumulative frequency graph plots each running total against the upper end of its class.
Method. Cumulative frequency graphs are read by going across from the cumulative frequency to the curve, then down to the axis.
- Find the position on the cumulative frequency axis. The total cumulative frequency is n = 80. The 90th percentile is 90% of the way through the data: 90 ÷ 100 × 80 = 72.Position = 72
- Draw a horizontal line across from that height to the curve. We want the data value that has this many values at or below it.
- Draw a vertical line down to the horizontal axis and read the value. Read the scale carefully. The result is an estimate.90th percentile ≈ 48 min
Check. Going up from your answer to the curve and across should give the same height again. ✓
Watch out. Read across from the cumulative frequency axis. The frequency axis is the vertical one, the data axis is the horizontal one.
| Mark | Frequency |
|---|---|
| 0 < m ≤ 20 | 4 |
| 20 < m ≤ 30 | 10 |
| 30 < m ≤ 40 | 18 |
| 40 < m ≤ 50 | 24 |
| 50 < m ≤ 60 | 22 |
| 60 < m ≤ 80 | 16 |
| 80 < m ≤ 100 | 6 |
Key words. Cumulative frequency is a running total of the frequencies. A cumulative frequency graph plots each running total against the upper end of its class.
Method. Make a cumulative frequency table, then plot each cumulative frequency against the upper end of its class, and join the points with a smooth curve.
- Work out the cumulative frequencies. Add each frequency to the total so far.
Mark Frequency Cumulative frequency 0 < m ≤ 20 4 4 20 < m ≤ 30 10 14 30 < m ≤ 40 18 32 40 < m ≤ 50 24 56 50 < m ≤ 60 22 78 60 < m ≤ 80 16 94 80 < m ≤ 100 6 100 - Write the points to plot. Plot each cumulative frequency at the upper class boundary, because that is the point by which that many values have been counted. Start at (0, 0): nothing is below the first lower boundary.(0, 0), (20, 4), (30, 14), (40, 32), (50, 56), (60, 78), (80, 94), (100, 100)
- Choose scales and plot the points. The horizontal axis goes from the lowest to the highest class boundary. The vertical axis goes from 0 to the total frequency.Horizontal axis: Mark. Vertical axis: cumulative frequency up to 100.
- Join the points with a smooth curve. The curve is S-shaped and never goes down.
Check. The curve starts at 0 and ends at the total frequency, 100. ✓
Watch out. Plot at the upper boundary of each class, not the midpoint and not the lower boundary.
Key words. Cumulative frequency is a running total of the frequencies. A cumulative frequency graph plots each running total against the upper end of its class.
Method. Read at n ÷ 2 = 20 on the vertical axis for each curve.
- Class A (blue). Go across at 20 to the blue curve, then down.Median ≈ 50
- Class B (orange). Go across at 20 to the orange curve, then down.Median ≈ 64
Check. The orange curve is to the right of the blue curve at height 20. ✓
Watch out. Read the right curve.
Key words. Cumulative frequency is a running total of the frequencies. A cumulative frequency graph plots each running total against the upper end of its class.
Method. Read at n ÷ 4 = 10 and 3n ÷ 4 = 30 for each curve.
- Class A. Q1 ≈ 36 and Q3 ≈ 64.64 − 36 = 28
- Class B. Q1 ≈ 49 and Q3 ≈ 75.75 − 49 = 26
Check. Both readings are estimates, so small differences are fine. ✓
Watch out. IQR = Q3 − Q1.
Key words. Cumulative frequency is a running total of the frequencies. A cumulative frequency graph plots each running total against the upper end of its class.
Method. Compare an average (median) and a spread (IQR).
- Compare medians. Class B 64 is bigger than class A 50
- Compare IQRs. A: 28, B: 26
- Write in context. Class B scored higher on average, and the spread of the marks is compared above.
Check. Each sentence uses numbers read from the graphs. ✓
Watch out. Compare medians with medians and IQRs with IQRs.
Key words. Cumulative frequency is a running total of the frequencies. A cumulative frequency graph plots each running total against the upper end of its class.
Method. Cumulative frequency graphs are read by going across from the cumulative frequency to the curve, then down to the axis.
- Find the position on the cumulative frequency axis. The total cumulative frequency is n = 80. The 15th percentile is 15% of the way through the data: 15 ÷ 100 × 80 = 12.Position = 12
- Draw a horizontal line across from that height to the curve. We want the data value that has this many values at or below it.
- Draw a vertical line down to the horizontal axis and read the value. Read the scale carefully. The result is an estimate.15th percentile ≈ 18.5 min
Check. Going up from your answer to the curve and across should give the same height again. ✓
Watch out. Read across from the cumulative frequency axis. The frequency axis is the vertical one, the data axis is the horizontal one.
Key words. Cumulative frequency is a running total of the frequencies. A cumulative frequency graph plots each running total against the upper end of its class.
Method. Read the cumulative frequency at 45 minutes and at 25 minutes, then subtract: that leaves the workers between the two times.
- Read the graph at 45 minutes. Go up from 45 to the curve and across.69 workers took up to 45 minutes
- Read the graph at 25 minutes. Go up from 25 to the curve and across.24 workers took up to 25 minutes
- Subtract. The difference is the number between the two times.69 − 24 = 45
Check. 24 + 45 = 69. ✓
Watch out. Subtract the two cumulative frequencies; do not add them.
Key words. Cumulative frequency is a running total of the frequencies. A cumulative frequency graph plots each running total against the upper end of its class.
Method. Think about what the cumulative frequency value means.
- What does the running total mean? The cumulative frequency for a class counts every value in that class and all the classes before it.For the class 20 < t ≤ 30, the running total counts everybody up to 30 minutes.
- Where do those values end? Only at the upper boundary can we say that all of them are at or below it.Upper boundary = 30
- Conclude. We plot at the upper boundary.
Check. At the midpoint (25) we would not have counted all the values in that class. ✓
Watch out. Plot (upper boundary, cumulative frequency).
Key words. Cumulative frequency is a running total of the frequencies. A cumulative frequency graph plots each running total against the upper end of its class.
Method. The frequency in a class is the difference between consecutive cumulative frequencies.
- Read the two cumulative frequencies. At 30 and at 20.At 30: 70. At 20: 40.
- Subtract. 70 − 40 = 30
Check. 40 + 30 = 70. ✓
Watch out. Subtract consecutive cumulative frequencies.
Key words. Cumulative frequency is a running total of the frequencies. A cumulative frequency graph plots each running total against the upper end of its class.
Method. The cumulative frequency at 20 is 40, and the total is 80.
- Find the fraction. Part over whole.4080 = 12
- Change to a percentage. 50%
Check. Half of 80 is 40. ✓
Watch out. The total is the last cumulative frequency (80).
| Time, t (minutes) | Frequency | Cumulative frequency |
|---|---|---|
| 0 < t ≤ 10 | 6 | 6 |
| 10 < t ≤ 20 | 14 | ? |
| 20 < t ≤ 30 | ? | 45 |
| 30 < t ≤ 40 | 9 | ? |
| 40 < t ≤ 50 | ? | 60 |
Key words. Cumulative frequency is a running total of the frequencies. A cumulative frequency graph plots each running total against the upper end of its class.
Method. Each cumulative frequency is the previous one plus this class’s frequency. Use this in both directions to find the missing numbers.
- Class 10 < t ≤ 20. Cumulative frequency = 6 + 14.6 + 14 = 20
- Class 20 < t ≤ 30. The cumulative frequency went from 20 to 45, so the frequency is the difference.45 − 20 = 25
- Class 30 < t ≤ 40. Add 9 to the previous cumulative frequency.45 + 9 = 54
- Class 40 < t ≤ 50. The total goes from 54 to 60.60 − 54 = 6
Check. 6 + 14 + 25 + 9 + 6 = 60, the last cumulative frequency. ✓
Watch out. Frequency = difference between consecutive cumulative frequencies.
Key words. Cumulative frequency is a running total of the frequencies. A cumulative frequency graph plots each running total against the upper end of its class.
Method. The lower quartile has a quarter of the values below it, and the upper quartile has three quarters below. So the middle half lies between them.
- Find the number below Q1. A quarter of 120.120 ÷ 4 = 30
- Find the number below Q3. Three quarters of 120.3 × 30 = 90
- Subtract. The values between the quartiles.90 − 30 = 60
Check. The IQR holds 50% of 120 = 60. ✓
Watch out. Between the quartiles is always half of the data.