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Explanation:

Step1: Calculate the mean

First, find the mid - points of each class. Let (x) be the mid - point and (f) be the frequency. The mid - points are (x_1 = 7), (x_2=8), (x_3 = 9) with frequencies (f_1 = 4), (f_2=7), (f_3 = 4) respectively. The total frequency (n=f_1 + f_2+f_3=4 + 7+4=15). The mean (\bar{x}=\frac{\sum_{i = 1}^{3}f_ix_i}{n}=\frac{4\times7 + 7\times8+4\times9}{15}=\frac{28 + 56+36}{15}=\frac{120}{15}=8).

Step2: Calculate the squared differences

Calculate ((x_i-\bar{x})^2) for each (i). For (x_1 = 7), ((7 - 8)^2=1); for (x_2 = 8), ((8 - 8)^2=0); for (x_3 = 9), ((9 - 8)^2=1).

Step3: Calculate the weighted squared differences

Calculate (f_i(x_i-\bar{x})^2) for each (i). (f_1(x_1-\bar{x})^2=4\times1 = 4), (f_2(x_2-\bar{x})^2=7\times0 = 0), (f_3(x_3-\bar{x})^2=4\times1 = 4).

Step4: Calculate the variance

The variance (s^2=\frac{\sum_{i = 1}^{3}f_i(x_i-\bar{x})^2}{n - 1}=\frac{4+0 + 4}{15 - 1}=\frac{8}{14}\approx0.5714).

Step5: Calculate the standard deviation

The standard deviation (s=\sqrt{s^2}=\sqrt{\frac{8}{14}}\approx0.76).

Answer:

0.76