Friday, July 19, 2013

M&M Activity: Part 2

We are now going to process the data you collected and tabulated in the first part of this assignment.  Recall my data was as follows:
We will begin with the first row, the red M&Ms.  We will be learning how to deal with data in a frequency table.  The first step is to put that data into a frequency table.  There are a variety of ways to do this, but I am going to do this first row by ordering the data and then counting it.

Look first at the top part of the page.  In the first column which is labeled "Raw data," I've simply listed the numbers in the same order as they occured in the original table.  In the column labeled "ordered data," I've listed from the smallest to the largest.  This makes them easier to count.

In the part of the page labeled "Frequency Table," I've set up a five column table with labels x, f, x2(x-squared), fx (f times x), and fx2 (f times x-squared).  In the x column, I put the possible values of x from the lowest that occurs to the highest.  There were 5 bags that contained only one M&M, there were 4 that contained 2, there were no bags that contained 3 (I could've left this row out if I wanted to), and there was one bag that contained 4 M&Ms.

So here f stands for frequency. 

In the x2 column, I've put the squares of the values from the x column.  It kind of makes sense, eh?

In the column labeled fx (f times x), I've put the product of the the value from the f column and the value from the x column: 5 times 1=5, 4 times 2= 8, 0 times 3=0, and 1 times 4 = 4.  I've done a similar thing with the column labeled fx2 (f times x-squared): 5 times 1=1, 4 times 4=16, 0 times 9=0, and 1 times 16 =16.

After filling in this table, I found the sums of the f, fx, and fx2 columns.  They are 10, 17, and 37, respectively. 

I knew before I started that--if I did everything right--the sum of the f column would be 10.  This is because we sampled 10 bags and the sum of the f has to be the sample size.

We can use the sum of the f column and the sum of the fx column to compute the sample mean, x-bar, which is 17/10=1.7.

We need the sum of the fx2 column to calculate the sample standard deviation.  As you may recall, the standard deviation is rather work intensive to calculate.  When you first learns how to calculate standard deviation, you first calculate the sample mean.  Then you put in a new column which consists of the difference between the value of the data item and the value of the sample mean. Then you put in a column which consists of the square of the previous column.  Then you add up that column.  Call the sum Fred (I just like the name) and divide that by the sample size minus 1. 

This is a lot of work just to get Fred. This is awkward as it requires you to calculate x-bar before hand.  However, there is a formula for Fred that removes the awkwardness.  This formula is hard to describe in typing, but I will give it a go. In your left hand, put the sum of the fx2 column.  In your right hand, put what you get when you divide the square of the sum of the fx column by the sum of the f column.  The left hand minus the right hand is Fred.

Don't worry, it's written out on the page above: 37-(17 squared)/10=8.1.  To get the sample standard deviation from this, we go to the next page:
The sample variance is Fred divided by sample size minus 1.  That is 8.1/9=0.90.  The sample standard deviation is the square root of the sample varience. This is 0.949..., but we round it off to one decimal place beyond the original data, making s=0.9.

I've done this for each of the colors of my data:
Your assignment is to take your data and go through this same process.  You will be working in groups, so I would suggest to organize your work in such a way as to have at least two people to independently do the calculations on each. 

The submission for each group may be organized as above.  In whatever way it is organized, it must include
  1. A table for each color of M&Ms with the five columns as above.
  2. A calculation of the sums of the f, fx, and fx2 columns.
  3. A calculation of x-bar (the sample mean) and of s (the sample standard deviation).

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