The table shows the distribution of ages of 22 students in a school. Age (years) 12-14 15-17 18-20 21-23 24-26 Frequency 6 10 3 2 1 Using an assumed mean of...
Question 1 Report
The table shows the distribution of ages of 22 students in a school.
| Age (years) | 12-14 | 15-17 | 18-20 | 21-23 | 24-26 |
| Frequency | 6 | 10 | 3 | 2 | 1 |
Using an assumed mean of 19, calculate, correct to three significant figures, the :
(a) mean age ; (b) standard deviation ; of the distribution.
Answer Details
Class width \(=3\); mid-values \(13,16,19,22,25\). Code with \(u=\dfrac{x-19}{3}\), \(A=19\).
| Age | Mid \(x\) | \(f\) | \(u\) | \(fu\) | \(fu^2\) |
|---|---|---|---|---|---|
| 12-14 | 13 | 6 | -2 | -12 | 24 |
| 15-17 | 16 | 10 | -1 | -10 | 10 |
| 18-20 | 19 | 3 | 0 | 0 | 0 |
| 21-23 | 22 | 2 | 1 | 2 | 2 |
| 24-26 | 25 | 1 | 2 | 2 | 4 |
| Total | 22 | -18 | 40 |
(a) Mean age.\[\bar{x}=19+\frac{\sum fu}{\sum f}\times 3=19+\frac{-18}{22}\times 3=19-2.4545=16.5\text{ years}\]
(b) Standard deviation.\[\sigma=c\sqrt{\frac{\sum fu^2}{\sum f}-\left(\frac{\sum fu}{\sum f}\right)^2}=3\sqrt{\frac{40}{22}-\left(\frac{-18}{22}\right)^2}\]\[=3\sqrt{1.8182-0.6694}=3\sqrt{1.1488}=3(1.0718)=3.22\]
To three significant figures: mean age = 16.5 years and standard deviation = 3.22 years.
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