June 2013 p61 q1
2525
A summary of 30 values of x gave the following information:
\(\Sigma(x-c) = 234\), \(\Sigma(x-c)^2 = 1957.5\),
where c is a constant.
- Find the standard deviation of these values of x.
- Given that the mean of these values is 86, find the value of c.
Solution
(i) To find the standard deviation, use the formula:
\(\text{sd}^2 = \frac{\Sigma(x-c)^2}{n} - \left(\frac{\Sigma(x-c)}{n}\right)^2\)
Substitute the given values:
\(\text{sd}^2 = \frac{1957.5}{30} - \left(\frac{234}{30}\right)^2\)
\(\text{sd}^2 = 65.25 - 6.084\)
\(\text{sd}^2 = 59.166\)
\(\text{sd} = \sqrt{59.166} \approx 2.1\)
(ii) Given the mean is 86, use the formula for the mean:
\(\bar{x} = \frac{\Sigma x}{n} = 86\)
\(86 = \frac{234}{30} + c\)
\(86 = 7.8 + c\)
\(c = 86 - 7.8\)
\(c = 78.2\)
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