Scientists typically collect data on a sample of a population and use this data to draw conclusions, or make inferences, about the entire population. (for instance, a “world-wide” ratio of tasters vs. non-tasters being 70% vs. 30%)

Chi-Square Analysis This is a statistical test to determine how the data you observed (from your sample) compares to an expected result (world-wide ratio). Did you get usable data? Is there something wrong with your data? Chi Square helps you answer these questions.

For example, you decide to flip a coin 50 times. You expect a proportion of 50% heads and 50% tails. Based on a 50:50 probability, you predict 25 heads and 25 tails. These are the expected values. (what is your EXPECTED value in the Chaumonty-ville lab?)

You would rarely get exactly 25 and 25, but how far off can these numbers be without the results being significantly different from what you expected? After you conduct your experiment, you get 21 heads and 29 tails (the observed values). Using chi-square will let you know if this is a reasonable/acceptable data set. Or if it is data you need to throw out. Many of you asked me when writing your hypothesis, “Can I say it will be ‘approximately’ 70/30%?”

Chi Square Analysis Equation: X² = chi square value ∑ = sum o = observed data e = expected data The chi-square value is the sum of all the observed data minus the expected, squared….over the expected.

Have you ever wondered why the package of M&Ms you just bought never seems to have enough of your favorite color? Or, why is it that you always seem to get the package of mostly brown M&Ms? What’s going on at the Mars Company? Is the number of the different colors of M&Ms in a package really different from one package to the next, or does the Mars Company do something to insure that each package gets the correct number of each color of M&M? Are you a nerd for wondering this? Yes….but….. Today we will figure this out using a procedure called CHI SQUARE ANALYSIS!!!!

We are going to compare what we observe in a bag of M&Ms to what the Mars Company says their color distribution is per bag. Below is our expected values.

Open your bag of M&Ms -DO NOT EAT ANY….YET -Sort them by color -In your data table write the number of each color in the OBSERVED row. -Write the total number of M&Ms at the end of the row.

WHAT IS EXPECTED????? -For each color of M&M, they told us what % of that color should be in every bag. -USING YOUR BAG’S TOTAL NUMBER, Multiply the decimal percentage for each color and that will give you the number of M&Ms that THEY SAY should be in your bag. -EXAMPLE: If your bag has 56 M&Ms, to find out the expected # of Brown M&Ms, 56 x.13 = 7 So your expected number of Brown M&Ms, according to Mars Co., is 7

Follow the instructions on the side of the chart to fill in the rest of your data….. The last box means the SUM OF ALL THE DIFFERENCES/EXPECTED, SO YOU WILL ADD UP THE #’S IN YOUR LAST FULL ROW AND PUT THAT NUMBER IN AS THE SUM (this is your Chi Square value).

Next, to figure out your “degrees of freedom”, you ADD all of your categories (in this case, colors) and minus that number by 1. In statistics, the number of values in a study that are free to vary. For example, if you have to take ten different courses to graduate, and only ten different courses are offered, then you have nine degrees of freedom. Nine semesters you will be able to choose which class to take; the tenth semester, there will only be one class left to take - there is no choice. What are ‘degrees of freedom’?

Last step…. -Find your degree of freedom. -In that row, find your Chi square value (if the exact number isn’t there…estimate)

If your # is found in this section, Chi square tells us that the difference in your observed vs. expected data is not a significant enough difference to throw the data out, in other words, your data is WITHIN ACCEPTABLE LIMITS!!!!

If your # is found in this section, Chi square tells us that the difference in your observed vs. expected data is too significant to accept the data, in other words, something is wrong- LIKE THEIR STUPID PERCENTAGES OF WHAT GOES IN EACH BAG!!!!!