Question

find (a) the range and (b) the standard deviation of the set of data. 27, 9, 33, 24, 27, 12, 15 (a) the range is. (simplify your answer.)

Explanation:

Step1: Find the maximum and minimum values

The data set is \(27,9,33,24,27,12,15\). The maximum value \(M = 33\) and the minimum value \(m=9\).

Step2: Calculate the range

The formula for the range \(R\) of a data - set is \(R = M - m\). Substituting the values, we get \(R=33 - 9=24\).

Step3: Calculate the mean

The mean \(\bar{x}=\frac{27 + 9+33+24+27+12+15}{7}=\frac{147}{7}=21\).

Step4: Calculate the squared differences

\((27 - 21)^2=36\), \((9 - 21)^2 = 144\), \((33 - 21)^2=144\), \((24 - 21)^2 = 9\), \((27 - 21)^2=36\), \((12 - 21)^2 = 81\), \((15 - 21)^2=36\).

Step5: Calculate the variance

The variance \(s^{2}=\frac{36+144+144 + 9+36+81+36}{7}=\frac{486}{7}\approx69.43\).

Step6: Calculate the standard deviation

The standard deviation \(s=\sqrt{\frac{486}{7}}\approx8.33\).

Answer:

(a) 24
(b) \(\sqrt{\frac{486}{7}}\approx8.33\)