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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 estimate the median annual salary from the cumulative frequency graph, we need to find the value at the 30th worker since the median is the value that separates the higher half from the lower half of the data in a dataset of 60 workers. In the graph, we locate the point corresponding to a cumulative frequency of 30, then draw a horizontal line to the left until we intercept the cumulative frequency curve, and drop a vertical line down to the x-axis. This gives an estimate of the median annual salary, which is 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 estimate the mean annual salary, we calculate the midpoint for each salary range and multiply by the respective frequencies:
- Midpoint for p < 10: 5 (frequency: 14)
- Midpoint for p < 20: 15 (frequency: 12)
- Midpoint for p < 30: 25 (frequency: 14)
- Midpoint for p < 50: 40 (frequency: 16)
- Midpoint for p < 80: 70 (frequency: 4)
Next, we calculate:
Mean = ( \frac{(5 \times 14) + (15 \times 12) + (25 \times 14) + (40 \times 16) + (70 \times 4)}{60} )
Upon calculation, this yields an estimated mean annual salary of 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 considered more reliable than the estimate of the mean because the median is less affected by extreme values (outliers). In situations where salaries have some workers earning significantly more than others, the median provides a better representation of the typical worker's salary. In this dataset, if there are high salaries that skew the mean upward, the median will still accurately reflect the salary that separates the lower half from the upper half of the data.