The Sample Standard Deviation

ExampleLet’s jump right into an example to think about some ideas.

Let’s think about the number of Big Mac sandwiches sold during the noon hour this next Friday at 5 McDonald’s restaurants in Northeast Nebraska (you know the Wayne Store, West Point and so on). Say the numbers are 12, 20, 16, 17, and 23. (Of course I made the numbers up and I could be way off – but let’s not get lost in the example.)

You probably already noticed that all stores did not have the same number of units sold. Let’s not worry now about why each store does not have the same amount of units sold.

The Sample MeanIf I asked you to calculate the average of the values you would likely do the following:12 + 20 + 16 + 17 + 23 = 88 and then 88/5 = 17.6. So, the average number of units sold is 17.6.Well, in statistics we call the average the mean. When you added up the 5 numbers you performed an operation we call summation in statistics. The symbol Σ is the Greek capital letter sigma and when we use it we mean take the summation of a bunch of numbers. (page 543 shows the whole Greek alphabet.)On the next slide I will put the same numbers in a table and we will add columns as we introduce some additional ideas. Let’s say that when I have a column with the heading Xi that Xi stands for the variable we are working with, here the number of units sold at each store in our example. When we have sample size n the i stands for each value up to n. In our example n = 5 so I goes from 1 to 5.

The Sample MeanXi1220161723ΣXi = 88.

So here to sum the column I put ΣXi.

The sample mean is denoted as X = (ΣXi)/n = 88/5 = 17.6In the notes I will refer to the sample mean as xbar because that is easier than putting a line over the X. DO you understand?

To get the sample varianceXi Xi – xbar12 12 – 17.6 = -5.620 20 – 17.6 = 2.416 16 – 17.6 = -1.617 17 – 17.6 = -.623 23 - 17.6 = 5.4ΣXi = 88.xbar = 17.6You will notice that to get the sample variance we have to take each data point and subtract off the sample mean. We get some positive values and some negative values. All the values add up to zero. The adding up to zero is not useful, but some smart folks thought that about squaring each Xi – xbar. By the way Xi – xbar is called a difference or deviation.

To get the sample varianceXi Xi – xbar (Xi – xbar)2

12 12 – 17.6 = -5.6 31.3620 20 – 17.6 = 2.4 5.7616 16 – 17.6 = -1.6 2.5617 17 – 17.6 = -.6 .3623 23 - 17.6 = 5.4 29.16ΣXi = 88. Σ(Xi – xbar) = 69.2xbar = 17.6

In this 3rd column I took the result in the 2nd column and squared the value. At the bottom of the 3rd column I took the sum of the squared differences! The sample variance is S2 = Σ(Xi – xbar) / (n – 1) = 69.2/4 = 17.3(please check all my work here!)

The sample standard deviationThe sample standard deviation is the square root of the sample variance.So, to get the sample standard deviation you take the sum of the squared differences of each value from the mean and you divide by the sample size minus one and take the square root of the whole result. In our example we get

S = √S2 = √17.3 = 4.16 when you round to 2 decimals.

What does 4.16 mean? I have no idea! Really, though, 4.16 is just a measure of variation. The bigger the number the more the variation. Remember we were looking at how many Big Mac’s were sold at various store. S = 4.16 is just a measure of variability across the stores.