To date, we have focused on qualitatively describing possible sources of error in our experiments. When you can quantitatively prove your hypothesis, (such as with Hardy-Weinberg) there is a mathematical equation that will help you determine if the possible mistakes in your data were due to acceptable chance, or to unacceptable error.

X 2 = ∑ (o-e)2

e

Let’s try it!

In this equation, the X2

is the “chi square”.

The Greek character sigma (∑) signifies “sum of”

The “o” is the observed, and the “e” is the expected.

The Chi Square

Set up a data table that looks like this in your quadrille.

Gather the type of m & m you prefer.

Count total M&Ms, and calculate expected.

Continue filling out the rest of the table.

Mars M & M has supplied us with the following information: In each bag of M & Ms, the expected percentages are as follows:

DataColor Categories  

Red Blue Brown

Yellow Orange

Green

Total

Observed (o)          

Expected (e)          

Difference(o-e)

         

DifferenceSquared(o-e)2

         

(o-e)2/e          

Σ (d2/e) =X2

         

Type Red Blue Brown Green Orange Yellow

Plain 13% 24% 13% 16% 20% 14%

Peanut Butter

10%20% 10% 20% 20% 20%

You are now ready to read the chi square distribution table, which I’ve given you (and it’s on my webpage too…of course). A sample is below: Determine

how many degrees of freedom you have… which is equal to the total number of categories (colors of M&Ms) you had, minus 1. Since there are 6 colors, your degrees of freedom are 5.

Find the 5 in degrees of freedom column, then follow that number over, until you find your X2 number, or the number closest to it. Look up in the table to the probability number. This is the % probability that your hypothesis is correct.

The number you just found was your Chi Square Number (X2)

If your number falls within the reject hypothesis column…try again, or determine where the errors could have been made!

Were your results what was expected, or were they outside the realm of simple probability, and due to some error? Were the differences in the results you expected due to simple random chance, or was the person who sorted your M & Ms, or machine that sorts “falling down” on the job?

Once you share your results, eat your

M and Ms! Enjoy!

So…now how do you apply this to the Evolution of Taste Lab???

You participate in an experiment that has you tasting papers with chemicals on them.

You are aware of frequencies of taste/non-taste based upon frequencies supplied to you in background. You write a hypothesis based upon the numbers you have in your group.

You carry out your experimental procedures, and collect data and record it.

By your calculations, things did not turn out the way you expected. Did you make an error? Did people “fudge” their data, or were the differences simply due to random chance? Were the unexpected results enough to nullify your hypothesis? How can you determine if the unexpected data were within an acceptable range?

Chi Square

You have PTC test papers and you tasted them, along with class members and family. You collected the data and know that approximately 70% of the world’s population can taste PTC as a bitter substance.

You hypothesize that your AP Biology group data will follow the world’s statistic, and that 70% will be able to taste the PTC.

The following shows your prediction:

Total population tested: 150

Total predicted/expected tasters: .7 x 150 = 105

Results: Tasters actually were 120Does this data support your original hypothesis? Use the Chi Square to determine if the error was reasonable. Set up a data in a table like the one on the next slide, and let’s begin.

Data Categories

Taster Non-taster

Total

Observed (o) 120  30  150

Expected (e) 105 45 150

Difference(o-e) 15 55  

Difference Squared (o-e)2

225  25  

(o-e)2/e 2.14 .55  

Σ (d2/e) = X2      2.14 +.55 = 2.69

In the data table, simply fill out the data that you collected, and do exactly what it tells you to do in the data column. Once you have your Chi Square number, you have to determine what the probability is.

To determine probability, you first need to determine what the degrees of freedom are. That is simply the number of categories (2…tasters and non-tasters, minus 1, so you have one degree of freedom.

You are now ready to read the chi square distribution table, which I’ve given you (and it’s on my webpage too…of course). A sample is below: Find the 1st

degree of freedom row, and follow it over, until you get the closest reading to 2.69.

Then you follow that number up, to the probability row. So, since 2.69 is between 1.64 and 2.71, your probability is between .2 and .1, or 10-20% acceptable. You may accept these results as narrowly within the realm of possibility given your hypothesis. Congratulations!

If, however, your results were within the “reject hypothesis” realm, on the right side of this table, you would have to “tweak” your hypothesis, or collect more data…or start all over again! Bummer!