Given the following frequency distribution, find the mean, variance, and standard deviation. Please show all of your work.

Errors Frequency
51-53 9
54-56 19
57-59 16
60-62 24
63-65 25

For grouped data, the mean can be calculated using the following formula:

mean = (Efx)/n
...where n represents the total frequencies (add up the number of frequencies), x represents the midpoint of each class interval (which is found by adding the lowest and the highest number in the interval together, then dividing by 2), and E represents the sum of each midpoint times each frequency f. It might be helpful to set up a table of midpoints for each interval, then multiply each midpoint by its respective frequency, then add those numbers to get a total. Then divide the total by n.

Standard deviation is the square root of the variance. For grouped data, the variance can be calculated using the following formula:
variance = Ef(x - mean)^2/(n-1)
...where E represents sum of frequency f times (x-mean)^2, and n = total frequencies. (x represents the midpoint of each class interval.)
Again it might be helpful to set up tables with the following headings:

x - mean (x - mean)^2 f(x - mean)^2

Add up the last column for a total, then divide by n - 1. When you are finished calculating the variance, take the square root of the variance and you will have your standard deviation for this group of data.

Hope this will help get you started on this problems.