To calculate the standard deviation, we first need to find the mean of the data set.
Mean = (4*1 + 5*3 + 6*1) / (1+3+1) = (4 + 15 + 6) / 5 = 25 / 5 = 5
Next, calculate the squared differences between each value and the mean, and then multiply each squared difference by the frequency of that value and sum them up:
(4-5)^2 * 1 + (5-5)^2 * 3 + (6-5)^2 * 1
= 1 * 1 + 0 * 3 + 1 * 1
= 1 + 0 + 1
= 2
Now, divide the sum by the total number of observations minus 1, and then take the square root of that result to get the standard deviation:
Standard deviation = sqrt(2 / (5-1))
= sqrt(2 / 4)
= sqrt(0.5)
≈ 0.71
Therefore, the standard deviation of the data set is approximately 0.71.