Chi Square Analysis: Study with Video Lessons, Practice Problems & Examples

The Chi Square Test (χ²) is a statistical method used to determine if the observed results of an experiment align closely with expected results, particularly in genetics. It assesses whether deviations from expected Mendelian ratios, such as 3:1 or 9:3:3:1, are significant. The formula involves summing the squared differences between observed (O) and expected (E) values, divided by the expected values: χ2=∑(O-E)^2E. This test is crucial for validating genetic inheritance patterns.

Chi Square Analysis

So, Chi Square is going to be a statistical test, so we're about to get into a lot of math. I'm really sorry, but it's genetics it has an application in it. So the Chi Square Test is a statistical test, and what it is testing for is whether the expected result that you get is very similar to the observed result. So the expected is what you expected to get and the observed is what you actually see. So the reason we have to have this test is because in genetics, it's never perfect. Right? You, I mean, if you mate organisms together and you get, you know, 2000 offspring, for instance, or flies or something, you're not going to get a perfect 3 to 1 ratio. You're just not. Because there's 2000 offspring. It may be very close, but it's not going to be perfect. I mean, it's just not - life has never worked that way. You also won't get the perfect 9 to 3 to 3 to 1 ratio either. Life isn't perfect. Genetics isn't perfect. And so, if we are doing this experiment, say, we have a bunch of flies, we have 6000 flies, and we counted them all. We have these nice ratios, and it's like, really, it's like 2.96 to 1. You know, is that actually close enough to 3 to 1 to say, okay, this is our normal, this is so close to expected, this is Mendelian inheritance. So that's what the Chi Square Test is for. So it's used to check if your numbers that you got from your experiment are close enough to be expected to say that, to say that, you know, it's Mendelian inheritance. So the important numbers that you have to know to do a Chi Square Analysis are the observed numbers. These are the numbers you actually get in your experiment. And the expected numbers, and these are the numbers that you are expected to get. The perfect ratio, the 3 to 1, the perfect Punnett Square, that is the expected. And so, of course, because we're doing math, there's going to be a formula and it looks like this. Realize it's kind of confusing. Here's your Chi Square, that's what it looks like, that's the notation for it. You don't remember from math, this means sum. And then you have these o's and e's. So what do those stand for? Well, o very clearly means observed number, so these are the numbers you get. And the expected numbers, use green, are the numbers that you, the perfect ratio numbers. So, that's an overview. Let's now move on to the actual practice question and practice using Chi Square, and an actual question that you might get. So let's move on.

χ 2 = ∑ i ( o i - e i ) 2 / e i

Chi Square Analysis

Okay. So in this practice problem, we're going to be walking through the steps to using a Chi Square test. Here's the example question: You have a purple plant, and you think it's heterozygous. You don't know for sure, but you think so. If it's not heterozygous, what could it also be? It could also be homozygous, right? AA or aa, uppercase, lowercase, or just all uppercase. But you think it's heterozygous, and you want to know for sure. So, what you want to do is breed it with a homozygous recessive. This has a fancy name called a test cross. In questions like this, it prompts you to do a test cross, which means mating whatever you have with a homozygous recessive.

So, you have this purple plant, you think it's heterozygous, so you are going to mate it with a white plant that you know is homozygous recessive. After this mating, you produce 120 offspring: 55 are purple and 65 are white. Was your plant heterozygous? How do we actually go about using a Chi Square? Well, the first thing that you have to do is consider these as the observed numbers because you observed this during your experiment. You have 120 offspring, 55 are purple, and 65 are white, so these are your observed values. Remember the formula for chi-square is shown here again, which is Χ2=(O-E2−E. You have these numbers as given in the problem. But the first thing you have to do is determine the expected numbers.

If you cross a heterozygous with a homozygous recessive, first perform a Punnett square. When you mate them together, you get one-half heterozygous (which are purple) and one-half homozygous (which are white). If you have 120 offspring, how many are expected to be purple and white? The Punnett square tells us that half will be purple and half will be white. So half of 120, which is 120 divided by 2, is 60. Here are the expected numbers: 60 purple and 60 white.

So we have our observed: 55 and 65; and our expected: 60 and 60. Now we can use the chi-square formula to calculate this. You actually do it for each class. For purple (which was observed as 55 and expected as 60), when we put it into the formula, O - E is -5, squared is 25, and 25 divided by 60 is approximately 0.42. Because the formula considers the sum, we do the same thing for white: 65 observed, 60 expected, O - E is 5, squared is 25, and 25 divided by 60 is approximately 0.42. Then we add these two values to get our chi-square value, which is approximately 0.84. Do you understand everything in this step?

Now, if you're given a question like this on a test, you will be provided with a chi-square table. You don't have to memorize it. You use the chi-square to determine whether your hypothesis is true. First calculate the degrees of freedom, which equals the number of variables minus 1. In this question, there are two variables (purple and white), so degrees of freedom is 1. Now use your chi-square value, which is 0.84, to find where it fits in the table, between 0.46 and 1.07. These correspond to p-values of 0.50 and 0.30, which means there's a 50% to 30% chance. This indicates the probability of your test results being due to random chance rather than a true effect.

When you determine whether to accept or reject your null hypothesis, it's based on these probabilities. The null hypothesis states that there's no significant difference between observed and expected. So, with a p-value greater than 5% (ours range from 30% to 50%), we accept the null hypothesis, indicating that the observed results (55 purple and 65 white) do not significantly differ from the expected (60 purple and 60 white) results. If we accept the null hypothesis, we conclude with 95% confidence that the purple plant was indeed heterozygous, as there's no statistical evidence to indicate a significant discrepancy from the expected heterozygous outcome. This is crucial in remembering and affirming what you initially sought to test.

Recall that accepting the null hypothesis doesn't mean proving it true beyond all doubt but failing to reject it due to insufficient evidence saying otherwise. Your takeaway is that the purple plant's observed genotype distribution aligns with what's expected for a heterozygous, so your initial assumption stands with 95% confidence. This is how you execute and interpret a Chi-Square test correctly, ensuring you connect your conclusions back to the original research question.

Step 1

Okay. So this question states that you have conducted a monohybrid cross, observing these F2 phenotypes in the cross. It appears you were crossing flowers, likely red or white, and the F2 generation ended with approximately 900 red and 300 white flowers. The question asks which of the following null hypotheses is best for using the Chi-Square Test. The Chi-Square Test is used to determine whether your expected values are the same as your observed values. In this cross, you obtained these offspring, with about 900 red and 300 white flowers. It's asking which of these ratios you expected for this cross, which one you want to test to see if the genetics are functioning accordingly.

A 9 to 3 ratio is what we're going to calculate and write it as a ratio. One of these ratios here must match a known genetic ratio - options are a 3 to 1 ratio, a 2 to 2 ratio, a 9 to 3 to 3 to 1 ratio, or a 3 to 2 ratio. If unsure, there's an obvious one, but we can eliminate some easily. We can discard the 9 to 3 to 3 to 1 because there aren't four phenotypes, only two (red and white), making this option untenable. The next we can discard is the 2 to 2 ratio, representing an equal distribution, which would mean either both phenotypes would be 900 or 300, but this is not what we observe. We see a 900 to 300 ratio, indicating they are not equal, so option b can't fit.

The remaining options are 3 to 1 and 3 to 2. The best approach here is simple division: if you divide 300 into 900, it fits 3 times. Thus, 300 times 3 equals 900, aligning with a 3 to 1 ratio. For the 3 to 2 ratio, we would expect a 900 to 600 ratio, which isn't observed. So, doing the math, dividing 300 by 900 yields 3. Thus, calculating 300 times 3 confirms it equals 900, establishing a 3 to 1 ratio.

Thus, the answer is A. If you observed this phenotype distribution, the null hypothesis that you would want to test is whether your values align with the expected values of a 3 to 1 ratio. With this conclusion, let's continue.

Step 2

Okay. So, using the same exact data from before, this question is asking you, which of the following here represents the degrees of freedom for this problem? So how many degrees of freedom are in this experiment if you were to do a chi-square analysis? Our answers are 1, 2, 3, and 4. So, which one do you think it is? Right. The answer here is actually 1. And the reason that it's 1 is because we have 2 phenotypes. That's red and white. That says white, in case you can't read my horrible handwriting. But remember, the formula for degrees of freedom is the number of phenotypes, which we have 2, minus 1. So the degrees of freedom for this experiment would be a, which is one. So with that, let's now move on.

Step 3

Okay. So in this question, we are going to actually be calculating the Chi square value for this data. So remember the Chi square value is given this, interesting symbol here. Remember, the formula for this is going to be observed, so these values are observed, minus expected, squared. So expected would be what? It was a 9 to 3 or a 3 to 1 ratio. The expected values will be these, an exact 3 to 1 ratio, over expected, and remember this is the sum. So, for the whole thing, for each one of these, we have to do a calculation. Let me disappear. And so, first we'll do red. So there we have, we have (892-900)2∕1900 plus the white which the observed is (294-300)2∕1300. Now, I'm going to give you a second, you can pause it if you want, you can do whatever. Go ahead put this in your calculator, and what do you get? What does this equal? This equals the chi squared value, which is one of these values here. So go ahead put it in your calculator see what you get. Give you a second. I know it takes a little bit of time, so I'll just pause for a second, and give you time to punch that into your calculator. And so, hopefully you have enough time. If not, go ahead and pause it while you finish because I'm about to give the answer. So the answer here is b, this is 0.191, and that is your Chi Square value. So, with that, let's now move on.

Step 4

Alright. So, now that we've calculated the Chi Square value, we know it's 0.191, and we have our degrees of freedom, which we know is a single one, so that's 1. It's saying the next step is to determine the range of p values. So what are our p values for this experiment?

Now in order to do this, you're going to have to have a Chi Squared distribution table. You'll feel free to use the one in the handout, which you probably already have, up. If you don't, go ahead, take your phone, computer, or whatever, open a new tab, just Google "Chi Square Distribution Table". There are a ton of them, wherever they are.

Now, to read this table, the first thing you do is you look at your Degrees of Freedom. There should be a line called "degrees of freedom", and everything down here is going to be listed with different numbers. The one you're interested in is this line here, because it's the degrees of freedom of 1. Now, above here, you're going to have a bunch of different numbers. And what you're looking for is the numbers through which 0.191 sits in the middle.

So, eventually, you're going to get to a table. And now it may not be perfect. If you're using the table from a handout, it'll have the exact numbers. If you're using a table from Google, there might be different numbers, but essentially it's going to be the same thing. And the problem, it won't mess up, no matter what chi square distribution table you use, it's not going to mess up how you solve a Chi Square problem.

So if you're using the table that I provided, we're going to match perfectly. If you're not, it's okay. You're still going to be correct, but the numbers might not match perfectly, but they'll be close enough. So, on the table and the PDF that I provided, you're going to come across numbers 0.15, and I believe 0.46. And if we were to put 0.191, it would fit right in between these numbers. And this is fantastic.

So what you do now, now that you have these two, you kind of circle them if you want. And you go all the way to the bottom, where the p-value sits. And here, what it's going to say is it's going to give you a bunch of different numbers, but you're interested in the one that's lined up in the same column as these two numbers, and that's going to be approximately 0.70 to 0.50. So the answer to this question is a.

Now remember, if you're not using the same table as me, it may not be exactly this. It may be 0.75 to 0.55 or maybe slightly off. But essentially, pick the closest one, which for this problem is going to be 0.70 and 0.50.

Now when you're doing this in a classroom setting or on a quiz or a test, they are going to provide you with the exact same table as everyone else, so there won't be these weird confusions. But for this, I want to give you a chance to practice looking at different chi square distribution tables, so you can figure out how to look at ones differently. But for this question, the answer is 0.70 to 0.50. So let's figure out what that means in the next question.

Step 5

Okay. So, this is kind of the next step of this problem. If a Chi Squared value has led you to receive a p-value range of 0.70 to 0.50, also 70% to 50%, will you accept or reject the null hypothesis? And so, do you remember there are certain cases where the p-values have to be above or below a certain threshold where you determine whether you accept it? Right, that threshold is 5% or 0.05. Now, we obviously got much higher than that, and so if it's larger than that, then what does that mean? That means we accept the null hypothesis. And so, we're going to talk about what accepting the null hypothesis means for this question, next. So with that, let's move on.

Step 6

Okay. So which of the following statements is true when we accept a null hypothesis? We accepted the null hypothesis in the previous question, so what does that mean for our experiment? It means that the observed and the expected values are different? It could mean that we are 95% confident that our observed and expected values are different. We are 95% confident that our observed and expected values are the same. And we are 50% confident that our observed and expected values are the same. So the first thing we know is that when we accept a null hypothesis, we're saying that our observed and our expected values are essentially the same. So automatically, we can go ahead and mark out a and b, which said that this showed it was different. So our choices now are c and d, and the difference between these two is whether we're 95% confident or 50% confident. So which one do you think it is? Okay. The real answer here is 95% confident. We're more than that, so we accepted the null hypothesis. So when we, so what we did is we set our threshold. We said that because we set the threshold at 5%, we are 95% confident, and that's just taken from 100 minus 5 equals 95. Right? So we're not 100% confident because we gave ourselves this, like, 5% range of error to accept the null hypothesis. And so we are 95% confident that the observed and expected values are the same. So in this problem, remember we were looking at red and white flowers in the F2 offspring and we got a certain but I don't remember exactly what it is. It's like 894, 294, something around there. Essentially, we were testing and we were saying, are these close enough to 300 or 900/3100 to say that this problem is, or this trait is a 3 to 1 ratio and, therefore, is Mendelian. Because we didn't know. Right? We just did this experiment. We had no idea we're looking at these red and white flowers. We got this number of offspring and we said, okay. Well, is this a 3 to 1 ratio? Is this close to the Mendelian ratio we would have expected? And we went through all these steps and finally we figured out that, yes, it is because we're 95% confident that the values that we observed, the number of offspring for red and white flowers that we actually got, were close enough to the expected values of 900/3100 to be a 3 to 1 ratio, therefore be Mendelian, so that they are the same. So c is the answer here. So make sure you understand. I realize that this problem is a lot. There's a lot of different steps, so make sure you understand all of these steps because I guarantee you, you will have to solve a Chi-square analysis problem on a test at some point in your genetics career. So make sure you understand what's going on in each one of these steps. So with that, let's now move on.

Here’s what students ask on this topic:

What is the Chi Square Test used for in genetics?

The Chi Square Test (χ²) is used in genetics to determine if the observed results of an experiment align closely with the expected results based on Mendelian inheritance patterns. It helps assess whether deviations from expected ratios, such as 3:1 or 9:3:3:1, are statistically significant. This test is crucial for validating genetic inheritance patterns and ensuring that the observed data is not due to random chance.

How do you calculate the Chi Square value?

To calculate the Chi Square (χ²) value, use the formula:

χ2=∑O-E2/E

Here, O represents the observed values, and E represents the expected values. The formula involves summing the squared differences between observed and expected values, divided by the expected values. This calculation helps determine if the observed data significantly deviates from what was expected.

What are the observed and expected values in a Chi Square Test?

In a Chi Square Test, the observed values (O) are the actual data you collect from your experiment. These are the numbers you count or measure. The expected values (E) are the theoretical numbers you would expect to get based on a specific hypothesis or model, such as Mendelian inheritance ratios (e.g., 3:1 or 9:3:3:1). The test compares these two sets of values to determine if the observed data fits the expected pattern.

Why is the Chi Square Test important in genetics?

The Chi Square Test is important in genetics because it provides a statistical method to validate whether the observed genetic data fits expected Mendelian ratios. Since real-life genetic experiments rarely produce perfect ratios due to random variations, the Chi Square Test helps determine if the deviations are within an acceptable range or if they indicate a significant difference. This is crucial for confirming genetic inheritance patterns and ensuring the reliability of experimental results.

What is the formula for the Chi Square Test?

The formula for the Chi Square (χ²) Test is:

χ2=∑O-E2/E

In this formula, O represents the observed values, and E represents the expected values. The formula involves summing the squared differences between observed and expected values, divided by the expected values. This calculation helps determine if the observed data significantly deviates from what was expected.