What is the standard deviation for the data given:
(12, 13, 29, 18, 61, 35, 21}
a) 293.67
a
c) 17.14
b) 15.87
d) 41.98

Step-by-step explanation:

Let N be the number of samples (In this case N=7), The long comments and questions in parentheses are not necessary for doing the calculation, but are meant to enhance your understanding.

1.) What is the mean value of those numbers? Write it down.

2. )What are the differences between the numbers and their mean? List them.

3.) What are the values of the differences squared (multiplying each number by itself)? Write them down. (Not necessary but something to think about: What happens when you take the square of the difference when the number is less than the mean and you square the difference?) List them.

4.) What is the sum of all those squared differences? Write it down.

5.) Divide by the number of samples. (Compare the procedure of taking the sum of the squares and dividing by N with the procedure for calculating the mean of N samples.) Write it down.

6.) How do you correct for the fact that all the numbers in the sum are squares, not the deviations themselves? Calculate the square root of the sum of squares divided by N. Write it down.

That’s the standard deviation.

That may seem like a lot of steps, but it’s easy when you actually do it.

This kind of problem lends itself to calculating with a spreadsheet, especially if you have multiple problems like this one. If you have a spreadsheet program (eg. Excel) and you know how to use it, try using it to calculate the std. deviation. (Hint: to check your spreadsheet calculation, take a sample of perhaps only 4 of the samples you were given and compare the spreadsheet value with your hand calculation. A smaller sample is used to make the hand calculation easier.)

There are different kinds of standard deviations, depending on their intended use. If you simply want the standard deviation of a list of numbers without reference to some distribution from which they were drawn, then you should divide by N-1, not N.

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