A survey was conducted to find out the number of books read by students in a class. The frequency distribution of this data is given as follows:\n\nNumber of books: \([0, 1, 2, 3, 4, 5]\)\nFrequency: \([5, 10, 7, 5, 2, 1]\)\n\nFind the variance of the frequency table.
Answer
Next, we calculate the variance (\(\sigma^2\)), which is the sum of the product of the square of the difference between each value and the mean, times its frequency, divided by the total frequency.\n\n\[\sigma^2 = \frac{\sum (x-\mu)^2*f}{\sum f}\]\n\nSubstituting the values:\n\n\[\sigma^2 = \frac{(0-1.67)^2*5+(1-1.67)^2*10+(2-1.67)^2*7+(3-1.67)^2*5+(4-1.67)^2*2+(5-1.67)^2*1}{30} = \frac{38.89}{30} = 1.30\]
Step 1 ：First, we calculate the mean (\(\mu\)) of the frequency distribution, which is the sum of the products of each value and its frequency, divided by the total frequency.\n\n\[\mu = \frac{\sum x*f}{\sum f}\]\n\nSubstituting the given values:\n\n\[\mu = \frac{(0*5)+(1*10)+(2*7)+(3*5)+(4*2)+(5*1)}{5+10+7+5+2+1} = \frac{50}{30} = 1.67\]
Step 2 ：Next, we calculate the variance (\(\sigma^2\)), which is the sum of the product of the square of the difference between each value and the mean, times its frequency, divided by the total frequency.\n\n\[\sigma^2 = \frac{\sum (x-\mu)^2*f}{\sum f}\]\n\nSubstituting the values:\n\n\[\sigma^2 = \frac{(0-1.67)^2*5+(1-1.67)^2*10+(2-1.67)^2*7+(3-1.67)^2*5+(4-1.67)^2*2+(5-1.67)^2*1}{30} = \frac{38.89}{30} = 1.30\]
