## Tuesday, July 23, 2013

### M&M Activity: Part 4

For this, we will again return to the data from the first table:
This time we are interested in the variation of the total number of M&Ms in each bag.  This is given in the row along the bottom. In order to find the average, we could simply take the total 177 and divide by 10 to get 17.7, but we will need the standard deviation too, and we will be doing something more complicated later. Therefore, we need to be a little more sophisticated.

Notice that in his table only four different values appear: 16,17,18, and 19.  I've made e frequency table with columns for x, f, x-squared, f times x, and f times x-squared:
It is comforting that the sum of the f times x column comes to 177 because that is what is was when I added it in the usual way. In my calculations at the bottom of the page, I get a value of x-bar=17.7 as before and I get a value of s=1.1 for the sample standard deviation.  The calculations all follow the same principals as for the single color.

Your answers for x-bar and s should be similar to what got.  If you've followed my directions, you should all have a sample size of exactly 10.  This is a small sample.  When we learn more about taking samples, it will be shown that a sample size of 30 or more is to be desired.

However, there are several groups and each of you took a sample of 10 that means if you pooled your samples you would have a much larger sample. But the candy is all eaten now and it would be a lot of work to do this all again. If only there were some easy way.....

Well, in fact, there is an easy way.  The groups will now share their data with each other.  Let me show you how this works by showing you an example with some faked data.

I used faked data rather than run the experience 5 more times.  When I faked the data, I fiddled around with the frequencies in a table similar to the one above and obtained values of the sum of the f column, the sum of the f times x column, and the sum of the f times x-squared column form each.  I put the results in the table on the sheet below:
I am being careful to tell you this is faked data because (1) we must practice intellectual honesty and (2) I don't want the M&Ms folks going all commando on me.

Once the data is gathered, the calculations are simple.  The sum of the Sigma f column is 50.  There are 5 groups and each took a sample of 10, so the pooled sample is 50.

The sum of the Sigma f times x column is the sum of all of the f times x entries in all of the groups.  For this fake data, it is 875.  Similarly, the sum of the Sigma f times x-square column is equal to the sum of all of the entries in all of the tables.  With this fake data, it has a value of 15,412.

We calculate x-bar by dividing 875 by 50.  The sum of the pooled data by the total number of items in the pooled data.

Notice when we calculate "Fred," we give him a different name.  We use the variable Sxx.  When I say this out in class, it sounds like I'm saying "sex, sex, sex."

Note that the n value is 50, so the n-1 value is 49.