Hard A-Level Data presentation Questions

Challenging, exam-style A-Level Data presentation questions with worked solutions. Stretch yourself on the hardest outliers, IQR, box plot, cumulative frequency problems.

outliersIQRbox plotcumulative frequencypercentileinterpolation
A-Level34 questionsStep-by-step solutions
Question 1
8 markschallenging
Test scores for groups A and B. A: min 4040, Q1=55Q_1=55, median 6060, Q3=65Q_3=65, max 8080. B: min 3030, Q1=50Q_1=50, median 6262, Q3=74Q_3=74, max 9595. A teacher says B is 'higher on average but less consistent'. Is this a fair summary?
Show worked solution

Worked solution

  1. Write down A's five-number summary

    40, 55, 60, 65, 8040,\ 55,\ 60,\ 65,\ 80

    Min, Q1, median, Q3, max for A.

  2. Write down B's five-number summary

    30, 50, 62, 74, 9530,\ 50,\ 62,\ 74,\ 95

    Min, Q1, median, Q3, max for B.

  3. Compute A's interquartile range

    IQRA=65−55=10\text{IQR}_A = 65 - 55 = 10

    Spread of A's middle half.

  4. Compute B's interquartile range

    IQRB=74−50=24\text{IQR}_B = 74 - 50 = 24

    Spread of B's middle half.

  5. Compute A's range

    rangeA=80−40=40\text{range}_A = 80 - 40 = 40

    Total spread of A.

  6. Compute B's range

    rangeB=95−30=65\text{range}_B = 95 - 30 = 65

    Total spread of B.

  7. Compare the medians (location)

    60<6260 < 62

    The median compares typical values.

  8. Compare the interquartile ranges (spread)

    10<2410 < 24

    The IQR compares consistency.

  9. Compare the ranges

    40<6540 < 65

    The range compares total spread.

  10. Check A's skew from the box

    A: Q3−Q2=5, Q2−Q1=5A:\ Q_3-Q_2=5,\ Q_2-Q_1=5

    Compare the two halves of A's box.

  11. Check B's skew from the box

    B: Q3−Q2=12, Q2−Q1=12B:\ Q_3-Q_2=12,\ Q_2-Q_1=12

    Compare the two halves of B's box.

  12. Summarise the location difference

    medianA<medianB\text{median}_A < \text{median}_B

    Which group is typically larger.

  13. Summarise the spread difference

    IQRA<IQRB\text{IQR}_A < \text{IQR}_B

    Which group is more consistent.

  14. Interpret both in context

    location and spread compared\text{location and spread compared}

    B tends to score a little higher but with more variable results.

  15. State the overall comparison

    conclusion drawn\text{conclusion drawn}

    B's median is slightly higher and its IQR is larger, so the summary is fair.

Answer
Yes: B has a slightly higher median and a larger IQR
Question 2
8 markschallenging
Two data sets. Set A: min 00, Q1=30Q_1=30, median 3434, Q3=38Q_3=38, max 8080. Set B: min 1010, Q1=32Q_1=32, median 3636, Q3=40Q_3=40, max 4545. Which statement about skew is correct?
Show worked solution

Worked solution

  1. Write down A's five-number summary

    0, 30, 34, 38, 800,\ 30,\ 34,\ 38,\ 80

    Min, Q1, median, Q3, max for A.

  2. Write down B's five-number summary

    10, 32, 36, 40, 4510,\ 32,\ 36,\ 40,\ 45

    Min, Q1, median, Q3, max for B.

  3. Compute A's interquartile range

    IQRA=38−30=8\text{IQR}_A = 38 - 30 = 8

    Spread of A's middle half.

  4. Compute B's interquartile range

    IQRB=40−32=8\text{IQR}_B = 40 - 32 = 8

    Spread of B's middle half.

  5. Compute A's range

    rangeA=80−0=80\text{range}_A = 80 - 0 = 80

    Total spread of A.

  6. Compute B's range

    rangeB=45−10=35\text{range}_B = 45 - 10 = 35

    Total spread of B.

  7. Compare the medians (location)

    34<3634 < 36

    The median compares typical values.

  8. Compare the interquartile ranges (spread)

    8=88 = 8

    The IQR compares consistency.

  9. Compare the ranges

    80>3580 > 35

    The range compares total spread.

  10. Check A's skew from the box

    A: Q3−Q2=4, Q2−Q1=4A:\ Q_3-Q_2=4,\ Q_2-Q_1=4

    Compare the two halves of A's box.

  11. Check B's skew from the box

    B: Q3−Q2=4, Q2−Q1=4B:\ Q_3-Q_2=4,\ Q_2-Q_1=4

    Compare the two halves of B's box.

  12. Summarise the location difference

    medianA<medianB\text{median}_A < \text{median}_B

    Which group is typically larger.

  13. Summarise the spread difference

    IQRA=IQRB\text{IQR}_A = \text{IQR}_B

    Which group is more consistent.

  14. Interpret both in context

    location and spread compared\text{location and spread compared}

    A is pulled to the right by high values; B is fairly balanced.

  15. State the overall comparison

    conclusion drawn\text{conclusion drawn}

    A has a long upper whisker (positive skew); B is roughly symmetric.

Answer
A is positively skewed, B is roughly symmetric
Question 3
8 markschallenging
Waiting times (minutes) at clinics A and B. A: min 1010, Q1=25Q_1=25, median 3030, Q3=35Q_3=35, max 5050. B: min 55, Q1=20Q_1=20, median 3030, Q3=48Q_3=48, max 7070. Which statement best compares the waiting times?
Show worked solution

Worked solution

  1. Write down A's five-number summary

    10, 25, 30, 35, 5010,\ 25,\ 30,\ 35,\ 50

    Min, Q1, median, Q3, max for A.

  2. Write down B's five-number summary

    5, 20, 30, 48, 705,\ 20,\ 30,\ 48,\ 70

    Min, Q1, median, Q3, max for B.

  3. Compute A's interquartile range

    IQRA=35−25=10\text{IQR}_A = 35 - 25 = 10

    Spread of A's middle half.

  4. Compute B's interquartile range

    IQRB=48−20=28\text{IQR}_B = 48 - 20 = 28

    Spread of B's middle half.

  5. Compute A's range

    rangeA=50−10=40\text{range}_A = 50 - 10 = 40

    Total spread of A.

  6. Compute B's range

    rangeB=70−5=65\text{range}_B = 70 - 5 = 65

    Total spread of B.

  7. Compare the medians (location)

    30=3030 = 30

    The median compares typical values.

  8. Compare the interquartile ranges (spread)

    10<2810 < 28

    The IQR compares consistency.

  9. Compare the ranges

    40<6540 < 65

    The range compares total spread.

  10. Check A's skew from the box

    A: Q3−Q2=5, Q2−Q1=5A:\ Q_3-Q_2=5,\ Q_2-Q_1=5

    Compare the two halves of A's box.

  11. Check B's skew from the box

    B: Q3−Q2=18, Q2−Q1=10B:\ Q_3-Q_2=18,\ Q_2-Q_1=10

    Compare the two halves of B's box.

  12. Summarise the location difference

    medianA>medianB\text{median}_A > \text{median}_B

    Which group is typically larger.

  13. Summarise the spread difference

    IQRA<IQRB\text{IQR}_A < \text{IQR}_B

    Which group is more consistent.

  14. Interpret both in context

    location and spread compared\text{location and spread compared}

    Clinics have the same typical wait, but B's waits vary much more.

  15. State the overall comparison

    conclusion drawn\text{conclusion drawn}

    Equal medians but B has a much larger IQR and range, so B is less consistent.

Answer
Same median, but B is more spread out
Question 4
8 markschallenging
Two classes sat the same test (out of 100). Class A box plot: min 2020, Q1=40Q_1=40, median 5050, Q3=60Q_3=60, max 8080. Class B box plot: min 3030, Q1=45Q_1=45, median 6565, Q3=75Q_3=75, max 9090. Which statement best compares the classes?
Show worked solution

Worked solution

  1. Write down A's five-number summary

    20, 40, 50, 60, 8020,\ 40,\ 50,\ 60,\ 80

    Min, Q1, median, Q3, max for A.

  2. Write down B's five-number summary

    30, 45, 65, 75, 9030,\ 45,\ 65,\ 75,\ 90

    Min, Q1, median, Q3, max for B.

  3. Compute A's interquartile range

    IQRA=60−40=20\text{IQR}_A = 60 - 40 = 20

    Spread of A's middle half.

  4. Compute B's interquartile range

    IQRB=75−45=30\text{IQR}_B = 75 - 45 = 30

    Spread of B's middle half.

  5. Compute A's range

    rangeA=80−20=60\text{range}_A = 80 - 20 = 60

    Total spread of A.

  6. Compute B's range

    rangeB=90−30=60\text{range}_B = 90 - 30 = 60

    Total spread of B.

  7. Compare the medians (location)

    50<6550 < 65

    The median compares typical values.

  8. Compare the interquartile ranges (spread)

    20<3020 < 30

    The IQR compares consistency.

  9. Compare the ranges

    60=6060 = 60

    The range compares total spread.

  10. Check A's skew from the box

    A: Q3−Q2=10, Q2−Q1=10A:\ Q_3-Q_2=10,\ Q_2-Q_1=10

    Compare the two halves of A's box.

  11. Check B's skew from the box

    B: Q3−Q2=10, Q2−Q1=20B:\ Q_3-Q_2=10,\ Q_2-Q_1=20

    Compare the two halves of B's box.

  12. Summarise the location difference

    medianA<medianB\text{median}_A < \text{median}_B

    Which group is typically larger.

  13. Summarise the spread difference

    IQRA<IQRB\text{IQR}_A < \text{IQR}_B

    Which group is more consistent.

  14. Interpret both in context

    location and spread compared\text{location and spread compared}

    Class B scored higher typically, with comparable variability.

  15. State the overall comparison

    conclusion drawn\text{conclusion drawn}

    B has a higher median with similar spread, so B performed better on average.

Answer
Class B has a higher median with similar IQR
Question 5
8 markschallenging
A histogram of the distance (km) has total frequency 8888 with classes: 0≤d<50\le d<5 frequency 99; 5≤d<105\le d<10 frequency 2121; 10≤d<1510\le d<15 frequency 2727; 15≤d<2515\le d<25 frequency UNKNOWN; 25≤d<3025\le d<30 frequency 1111. One class frequency is unknown. Reconstruct the frequencies and find the missing frequency.
Show worked solution

Worked solution

  1. Recall that each bar's frequency is density times width

    frequency=density×width\text{frequency} = \text{density}\times\text{width}

    The area of each bar equals its frequency.

  2. List the class widths

    5, 5, 5, 10, 55,\ 5,\ 5,\ 10,\ 5

    Differences of the class boundaries.

  3. Frequency of the first known class

    f1=9f_1 = 9

    Read from its bar.

  4. Frequency of the second known class

    f2=21f_2 = 21

    Read from its bar.

  5. Frequency of the remaining known classes

    27+11=3827+11 = 38

    The other bars with known heights.

  6. Add all known frequencies

    9+21+27+11=689+21+27+11 = 68

    Total of the classes with known bars.

  7. Find the missing frequency

    88−68=2088 - 68 = 20

    Subtract from the overall total.

  8. Find the density of the missing bar

    2010=2\frac{20}{10} = 2

    Its frequency divided by its width.

  9. Write the class midpoints

    2.5, 7.5, 12.5, 20, 27.52.5,\ 7.5,\ 12.5,\ 20,\ 27.5

    Midpoints estimate the values in each class.

  10. Multiply the first two midpoints by their frequencies

    2.5×9+7.5×21=1802.5\times9 + 7.5\times21 = 180

    Contribution of the first two classes.

  11. Multiply the next classes by their frequencies

    12.5×27+20×20=73812.5\times27 + 20\times20 = 738

    Contribution of the middle classes.

  12. Multiply the last class by its frequency

    27.5×11=30227.5\times11 = 302

    Contribution of the final class.

  13. Add all the products

    ∑fx=1220\sum fx = 1220

    The estimated total of all the data values.

  14. Divide by the total frequency

    xˉ=122088\bar{x} = \frac{1220}{88}

    Mean equals total of values divided by count.

  15. State the estimated mean

    xˉ≈13.9\bar{x} \approx 13.9

    The estimated mean of the data.

Answer
missing frequency=20\text{missing frequency} = 20

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