A-Level Data presentation Practice Questions

Free A-Level Data presentation practice questions with full step-by-step worked solutions. Covers histogram, frequency density, frequency from area, box plot. Practise exam-style problems and check your method.

histogramfrequency densityfrequency from areabox plotIQRrange
A-Level70 questionsStep-by-step solutions
Question 1
2 markseasy
In a histogram of the mass, in grams, the class 10≤m<2010\le m<20 contains 1515 items. Find the frequency density of this class.
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Worked solution

  1. Recall the frequency density formula

    frequency density=frequencyclass width\text{frequency density} = \frac{\text{frequency}}{\text{class width}}

    On a histogram the bar height is the frequency density, not the frequency.

  2. Substitute the frequency and the class width

    =1520−10=1510= \frac{15}{20 - 10} = \frac{15}{10}

    The class width is the difference of the upper and lower boundaries.

  3. Evaluate the frequency density

    =1.5= 1.5

    This is the height of the bar for this class.

Answer
frequency density=1.5\text{frequency density} = 1.5
Question 2
2 markseasy
A box plot has Q1=20Q_1 = 20, median =24= 24 and Q3=40Q_3 = 40, so the median is much closer to Q1Q_1 than to Q3Q_3. What does this suggest about the skew?
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Worked solution

  1. Compare the two halves of the box

    Q2−Q1=4,Q3−Q2=16Q_2 - Q_1 = 4,\quad Q_3 - Q_2 = 16

    The upper part of the box is much longer than the lower part.

  2. Interpret the longer upper tail

    16>416 > 4

    A longer stretch above the median indicates a tail to the right.

  3. State the type of skew

    positive skew\text{positive skew}

    When the median is closer to Q1, the data is positively skewed.

Answer
Positive skew
Question 3
3 marksintermediate
A cumulative frequency graph is drawn for a set of test scores. To estimate the median, at what cumulative frequency should you read across to the curve?
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Worked solution

  1. Let the total frequency be n

    n=total frequencyn = \text{total frequency}

    The total is read from the top of the cumulative frequency curve.

  2. Recall the median splits the data in half

    median position=n2\text{median position} = \tfrac{n}{2}

    Half the values lie below the median.

  3. Read across at this cumulative frequency

    read across at n2\text{read across at } \tfrac{n}{2}

    Go horizontally from n/2 to the curve.

  4. Then read down to the value axis

    read down to x\text{read down to } x

    The x-value at that point estimates the median.

  5. Contrast with using n+1 (grouped data)

    use n2 for a curve\text{use } \tfrac{n}{2} \text{ for a curve}

    For a continuous cumulative frequency curve we use n/2.

  6. State the reading

    n2\tfrac{n}{2}

    This gives the estimated median.

Answer
At a cumulative frequency of n/2
Question 4
5 markshard
Cumulative frequency curves for towns P and Q show that P's curve lies entirely to the left of Q's for house prices. What does this tell you about the two towns?
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Worked solution

  1. Recall what a cumulative frequency curve shows

    y=number of values≤xy = \text{number of values} \le x

    Higher up the curve means more data below that value.

  2. Interpret P's curve being to the left

    P reaches high totals at lower x\text{P reaches high totals at lower } x

    For any price, more of P's houses are below it.

  3. Interpret Q's curve being to the right

    Q needs higher x for same total\text{Q needs higher } x \text{ for same total}

    Q's prices reach the same cumulative totals only at higher values.

  4. Compare at the median level (n/2)

    medianP<medianQ\text{median}_P < \text{median}_Q

    Read across at half the total for each town.

  5. Compare at the lower quartile level

    Q1,P<Q1,QQ_{1,P} < Q_{1,Q}

    P's lower quartile is also to the left.

  6. Compare at the upper quartile level

    Q3,P<Q3,QQ_{3,P} < Q_{3,Q}

    Every quartile of P is below that of Q.

  7. Deduce the effect on location

    all quantiles lower for P\text{all quantiles lower for P}

    P's prices are systematically lower.

  8. Translate to context

    P cheaper than Q\text{P cheaper than Q}

    House prices in P are generally lower.

  9. State the comparison of typical values

    medianP<medianQ\text{median}_P < \text{median}_Q

    P has a lower typical (median) price.

  10. Conclude

    P generally cheaper\text{P generally cheaper}

    House prices in town P are generally lower than in town Q.

Answer
Town P generally has lower house prices
Question 5
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?
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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

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