Find the standard deviation of 2, 4, 4, 4, 5, 5, 7, 9

Problem: Find the standard deviation of the dataset $2, 4, 4, 4, 5, 5, 7, 9$.

Steps

1. Calculate the mean
   $$\mu = \frac{2 + 4 + 4 + 4 + 5 + 5 + 7 + 9}{8} = \frac{40}{8} = 5$$
2. Subtract the mean from each data point and square the differences
   $$(2-5)^2 = (-3)^2 = 9$$
   $$(4-5)^2 = (-1)^2 = 1$$
   $$(4-5)^2 = (-1)^2 = 1$$
   $$(4-5)^2 = (-1)^2 = 1$$
   $$(5-5)^2 = 0^2 = 0$$
   $$(5-5)^2 = 0^2 = 0$$
   $$(7-5)^2 = 2^2 = 4$$
   $$(9-5)^2 = 4^2 = 16$$
3. Sum the squared differences
   $$\sum (x_i - \mu)^2 = 9 + 1 + 1 + 1 + 0 + 0 + 4 + 16 = 32$$
4. Divide by the number of data points to find the variance (population standard deviation)
   $$\sigma^2 = \frac{32}{8} = 4$$
5. Take the square root of the variance
   $$\sigma = \sqrt{4} = 2$$