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The cumulative frequency graph summarises the annual salary, p (£ thousands), of the 60 workers in a factory - OCR - GCSE Maths - Question 12 - 2019 - Paper 4
Question 12
The cumulative frequency graph summarises the annual salary, p (£ thousands), of the 60 workers in a factory. (a) Use the graph to estimate the median annual salary... show full transcript
Worked Solution & Example Answer:The cumulative frequency graph summarises the annual salary, p (£ thousands), of the 60 workers in a factory - OCR - GCSE Maths - Question 12 - 2019 - Paper 4
Step 1
Use the graph to estimate the median annual salary.
Answer
To find the median annual salary, we first note that the median is the value that separates the higher half from the lower half of the data. Given that there are 60 workers, the median will be the value at the 30th worker.
From the cumulative frequency graph, we locate the 30 mark on the cumulative frequency axis. Drawing a horizontal line from this point to intersect the curve, and then dropping a vertical line down to the annual salary axis, we estimate the median annual salary to be approximately £24,000.
Step 2
Step 3
Use the information in the cumulative frequency table to calculate an estimate of the mean annual salary.
Answer
To calculate the mean annual salary, we need to find the midpoints of each salary range and multiply these by their respective frequencies:
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Calculate midpoints:
- p < 10: midpoint = 5
- p < 20: midpoint = 15
- p < 30: midpoint = 25
- p < 50: midpoint = 40
- p < 80: midpoint = 65
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Frequency of ranges:
- p < 10: 14
- p < 20: 12 (26 - 14)
- p < 30: 14 (40 - 26)
- p < 50: 16 (56 - 40)
- p < 80: 4 (60 - 56)
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Calculate the estimated mean:
ext{Mean} = rac{(5 imes 14) + (15 imes 12) + (25 imes 14) + (40 imes 16) + (65 imes 4)}{60}
= rac{70 + 180 + 350 + 640 + 260}{60}
= rac{1500}{60} = 25
Thus, the estimated mean annual salary is approximately £28,500.
Step 4
Explain why your estimate of the median is more reliable than your estimate of the mean.
Answer
The estimate of the median is often more reliable than the mean because the median is less sensitive to extreme values or outliers in the data. In this case, if there are a few workers earning significantly higher salaries, it can skew the mean upward, making it less representative of the typical worker's salary. In contrast, the median represents the middle salary value, providing a better indication of the central tendency for skewed distributions.