Cumulative frequency
Probability and statistics
- Higher
- Non-calculator
- 8 questions
- 29 marks
These are original practice questions, not a past paper. This site is independent and not affiliated with AQA or Pearson.
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The frequency table shows times in seconds. Complete the cumulative frequencies.
Times taken by 50 people Time, t (s) Frequency Cumulative frequency 0 < t ≤ 10 5 10 < t ≤ 20 10 20 < t ≤ 30 20 30 < t ≤ 40 10 40 < t ≤ 50 5 2 marks
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How many people were timed?
2 marks
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The cumulative frequency graph plots the points $(10, 5)$, $(20, 15)$, $(30, 35)$, $(40, 45)$ and $(50, 50)$, joined by straight lines. Estimate the median.
Cumulative frequency graph of the times 3 marks
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Estimate the lower quartile from the graph.
3 marks
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Estimate the upper quartile and the interquartile range.
4 marks
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Estimate how many people took more than $40$ seconds, and how many took $30$ seconds or less.
4 marks
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Estimate how many people had a time between $15$ and $35$ seconds.
5 marks
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A second group has median $30$ seconds and interquartile range $8$ seconds. Compare the two groups. Estimate how many people in the first group took $20$ seconds or less.
6 marks
Mark scheme
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Answer. $5, 15, 35, 45, 50$
- Add the frequencies in order.
- The final cumulative frequency is $50$.
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Answer. $50$
- The frequencies sum to $50$.
- $50$
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Answer. $25$ seconds
- The median is at the $25$th value.
- From $(20, 15)$ to $(30, 35)$, $10$ more people take $10$ seconds.
- The median is $20 + 5 = 25$ seconds.
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Answer. $17.5$ seconds
- The lower quartile is at position $12.5$.
- Between $10$ and $20$ seconds, the cumulative frequency rises from $5$ to $15$.
- $10 + 7.5 = 17.5$ seconds.
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Answer. Upper quartile $32.5$ seconds, IQR $15$ seconds
- The upper quartile is at position $37.5$.
- Between $30$ and $40$, it is $2.5$ seconds past $30$, so $32.5$.
- IQR $= 32.5 - 17.5 = 15$ seconds.
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Answer. $5$ people took more than $40$ seconds, and $35$ took $30$ seconds or less.
- The cumulative frequency at $40$ is $45$, so $50 - 45 = 5$.
- The cumulative frequency at $30$ is $35$.
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Answer. $30$
- At $15$ seconds the cumulative frequency is $10$.
- At $35$ seconds the cumulative frequency is $40$.
- $40 - 10 = 30$ people.
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Answer. The second group is typically slower but more consistent. $15$ people took $20$ seconds or less.
- The second median, $30$, is higher than $25$, so those times are typically longer.
- The second IQR, $8$, is smaller than $15$, so those times are more consistent.
- The cumulative frequency at $20$ seconds is $15$.