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AQA GCSE · Question 18.2 · Statistics
For type Q, the median was 126 days and the interquartile range was 57 days.
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Compare the number of days that types P and Q lasted.
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Make one statement about the average and one statement about the spread.
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Use statistical measures to support your statements.
For type Q, the median was 126 days and the interquartile range was 57 days.
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Compare the number of days that types P and Q lasted.
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Make one statement about the average and one statement about the spread.
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Use statistical measures to support your statements.
How to approach this question
1. Use the cumulative frequency graph from part (a) to find the median and interquartile range (IQR) for battery type P.
- Median is at the 50th value (since there are 100 batteries).
- Lower Quartile (LQ) is at the 25th value.
- Upper Quartile (UQ) is at the 75th value.
- IQR = UQ - LQ.
2. You are given the median and IQR for type Q.
3. For the "average", compare the medians of P and Q. A higher median means it lasted longer on average.
4. For the "spread", compare the IQRs of P and Q. A smaller IQR means the data is more consistent or less spread out.
5. Write two clear sentences, one for average and one for spread, quoting the statistical values you have found or been given.
Full Answer
First, we need to find the median and interquartile range (IQR) for battery type P from the cumulative frequency graph drawn in part (a). There are 100 batteries.
- **Median (Q2):** 50% of 100 = 50th value. Reading from the graph at a cumulative frequency of 50, the median for P is approximately 132 days.
- **Lower Quartile (Q1):** 25% of 100 = 25th value. Reading from the graph at a cumulative frequency of 25, the lower quartile is approximately 108 days.
- **Upper Quartile (Q3):** 75% of 100 = 75th value. Reading from the graph at a cumulative frequency of 75, the upper quartile is approximately 154 days.
- **Interquartile Range (IQR) for P:** Q3 - Q1 = 154 - 108 = 46 days.
Now we compare these values with the given values for type Q (Median = 126, IQR = 57).
**Comparison of Average:**
The median for P (132 days) is higher than the median for Q (126 days). This means that on average, battery type P lasted longer than battery type Q.
**Comparison of Spread:**
The IQR for P (46 days) is smaller than the IQR for Q (57 days). This means that the lifespan of battery type P was more consistent (less spread out) than that of battery type Q.
**Summary:**
**Average:** On average, type P batteries lasted longer than type Q batteries (Median 132 > 126).
**Spread:** The lifespan of type P batteries was more consistent than type Q batteries (IQR 46 < 57).
To compare the two types of batteries, we need to calculate the median and interquartile range (IQR) for type P from our cumulative frequency graph and compare them to the given values for type Q.
**For Type P (from the graph):**
Total batteries = 100.
- **Median:** Find the 50th value (100 / 2). Go to 50 on the cumulative frequency axis, read across to the curve, and then down to the days axis. This gives a median of approximately **132 days**.
- **Lower Quartile (Q1):** Find the 25th value (100 / 4). This gives Q1 ≈ **108 days**.
- **Upper Quartile (Q3):** Find the 75th value (3 * 100 / 4). This gives Q3 ≈ **154 days**.
- **Interquartile Range (IQR):** IQR = Q3 - Q1 ≈ 154 - 108 = **46 days**.
**For Type Q (given):**
- Median = **126 days**
- IQR = **57 days**
**Comparison:**
- **Average:** The median for P (132 days) is higher than the median for Q (126 days). This suggests that, on average, type P batteries last longer.
- **Spread:** The IQR for P (46 days) is smaller than the IQR for Q (57 days). This suggests that the lifespan of type P batteries is more consistent and less varied than type Q.
Common mistakes
✗ Making comments without quoting the statistical values to support them.
✗ Confusing median and IQR. For example, saying "The median of P is 46, which is smaller than Q's 57".
✗ Misinterpreting the meaning of the IQR. A smaller IQR means more consistency, not less.
✗ Reading the values from the graph inaccurately.
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