Chi-squared Test

Summary

In this informative podcast, Mr. Andersen demystifies the Chi-squared test, a statistical tool used to determine if observed data variations are due to chance or underlying variables. He introduces the concept with a coin flip experiment, explains the null hypothesis, and illustrates how to calculate the Chi-squared value and interpret critical values for decision-making. The podcast also covers degrees of freedom and provides examples with coins and dice to demonstrate the test's application in real-world scenarios, encouraging viewers to apply Chi-squared in their own experiments.

Takeaways

• 📚 The Chi-squared test is a statistical method used to determine if observed data differs from expected data due to chance or a specific variable.
• 🔍 It was developed by Carl Pearson in the early 1900s and is widely used in fields like AP Biology and other sciences.
• 📝 The formula for the Chi-squared test involves summing the differences between observed and expected values, squared and divided by the expected values.
• 🎲 The test uses the concept of 'observed data' and 'expected values', with the latter being theoretical values calculated before an experiment.
• 🤔 The Chi-squared test helps to answer questions like whether a coin flip's outcome is due to chance or a biased coin.
• ❓ The Null Hypothesis in Chi-squared testing states that there is no statistical significant difference between observed and expected frequencies.
• 🔢 'Degrees of freedom' in the test is calculated by the number of outcomes minus one, which affects the critical value used for comparison.
• 📉 Critical values are used to decide whether to accept or reject the Null Hypothesis, with common values determined for specific confidence levels like 0.05.
• 🧩 The test involves comparing the calculated Chi-squared value to a critical value from a Chi-squared distribution chart.
• 🎯 If the Chi-squared value exceeds the critical value, the Null Hypothesis is rejected, indicating a significant difference between observed and expected data.
• 📈 The Chi-squared test can be applied to various scenarios, such as coin flips, dice rolls, or animal behavior studies, to determine if outcomes are due to chance or other factors.

Q & A

What is the primary purpose of the Chi-squared test?

The primary purpose of the Chi-squared test is to determine if the variation in collected data is due to chance or if it's due to one of the variables being tested.

Who developed the Chi-squared test and when was it developed?

The Chi-squared test was developed by Karl Pearson in the early part of the 1900s.

What are the two main components used in the Chi-squared test to compare data?

The two main components used in the Chi-squared test are the observed data (O) and the expected values (E).

What is the Null Hypothesis in the context of the Chi-squared test?

The Null Hypothesis in the context of the Chi-squared test is the assumption that there is no statistical significant difference between the observed values and the expected frequencies.

What are degrees of freedom in a Chi-squared test?

Degrees of freedom in a Chi-squared test refer to the number of values in the final calculation that are free to vary. It is calculated by subtracting one from the number of outcomes being compared.

What is a critical value in the context of the Chi-squared test?

A critical value in the context of the Chi-squared test is the value that determines whether to accept or reject the Null Hypothesis. If the calculated Chi-squared value exceeds the critical value, the Null Hypothesis is rejected.

What does it mean to accept or reject the Null Hypothesis in a Chi-squared test?

Accepting the Null Hypothesis means that the observed data does not show a significant difference from the expected data, suggesting no effect from the variable being tested. Rejecting the Null Hypothesis indicates that there is a significant difference, suggesting the variable has an effect.

How is the Chi-squared value calculated in a test?

The Chi-squared value is calculated by taking the difference between the observed and expected values, squaring the result, and then dividing by the expected value. This is done for each category and the results are summed to get the total Chi-squared value.

What is the significance of the 0.05 value in the context of critical values?

The 0.05 value, or alpha level, is a threshold used in hypothesis testing to determine statistical significance. It represents a 95% confidence level that the results are not due to chance.

Can you provide an example from the script where the Chi-squared test was applied to a coin flip experiment?

In the script, Mr. Andersen applied the Chi-squared test to a coin flip experiment where he flipped a coin 100 times and observed 62 heads and 38 tails. The expected values were 50 heads and 50 tails. The calculated Chi-squared value was compared to the critical value to determine if the coin flip results were due to chance or not.

How does the Chi-squared test help in determining if a coin or dice is biased?

The Chi-squared test helps in determining if a coin or dice is biased by comparing the observed frequencies of outcomes (like heads/tails or dice numbers) to the expected frequencies based on probability. If the calculated Chi-squared value exceeds the critical value, it suggests the coin or dice may be biased.

Outlines

00:00

📊 Introduction to the Chi-squared Test

Mr. Andersen introduces the Chi-squared test, emphasizing its importance in scientific analysis, particularly in AP Biology. He explains that the test is used to determine if variations in data are due to chance or a specific variable being tested. The Chi-squared test, developed by Carl Pearson, involves summing the differences between observed and expected data values. The video aims to demystify the test by explaining its basic concepts, including observed and expected values, and the null hypothesis, which assumes no significant difference between these values. The presenter also introduces the concepts of degrees of freedom and critical values, which are essential for interpreting the test results.

05:03

🎲 Applying the Chi-squared Test with Coin Flips

The presenter illustrates the Chi-squared test with a coin flip experiment, where 50 coins are flipped to determine if the observed number of heads and tails differs significantly from the expected 25 of each. Expected values are calculated based on the probability of each outcome, and observed values are the actual results of the experiment. The Chi-squared formula (O - E)² / E is applied to both outcomes, and the results are summed to get the Chi-squared value. This value is then compared to a critical value from a Chi-squared distribution table to decide whether to accept or reject the null hypothesis. In this case, the Chi-squared value is 0.72, which is lower than the critical value of 3.841 for 1 degree of freedom at the 0.05 significance level, leading to the acceptance of the null hypothesis.

10:04

🎯 Chi-squared Test with Dice Rolls

Expanding on the concept, the presenter uses a dice roll experiment to further explain the Chi-squared test. With 36 dice, the expected value for each number (1-6) is 6. The observed values are the actual numbers rolled. The Chi-squared calculation is performed for each outcome, and the results are summed to obtain a Chi-squared value of 9.6. Degrees of freedom are calculated as the number of outcomes minus one, which in this case is 5. The critical value for 5 degrees of freedom at the 0.05 significance level is 11.070. Since the calculated Chi-squared value does not exceed the critical value, the null hypothesis is accepted, indicating no significant difference between the observed and expected outcomes.

🐛 Chi-squared Test in Animal Behavior Study

The final part of the script poses a question related to an animal behavior study involving pill bugs, examining whether they spend more time in wet or dry conditions. The presenter suggests using the Chi-squared test to analyze the time spent by 10 pill bugs in each condition, with expected values being an equal distribution. The observed values are the actual times recorded. The presenter encourages viewers to apply the Chi-squared test to these values to determine if there is a statistically significant difference between the expected and observed results, inviting them to share their findings in the comments section.

Keywords

💡Chi-squared test

The Chi-squared test is a statistical method used to determine if there is a significant difference between the expected frequencies and the observed frequencies in a study. It is central to the video's theme as it is the main topic being discussed. The script uses the Chi-squared test to analyze various examples, such as coin flips and dice rolls, to illustrate whether the observed outcomes are due to chance or a significant variable.

💡Observed data

Observed data refers to the actual data collected during an experiment or study. In the context of the video, observed data is crucial as it is compared against expected data to determine if the variation is due to chance. For example, when flipping a coin 100 times and getting 62 heads, those 62 heads represent the observed data.

💡Expected values

Expected values are the frequencies of outcomes that one would anticipate in an experiment if the null hypothesis were true, based on probability. The video explains that these values are calculated before an experiment and then compared to the observed data. An example given is flipping a fair coin 100 times where the expected number of heads would be 50.

💡Null hypothesis

The null hypothesis is a fundamental concept in statistical testing, which assumes that there is no significant difference between groups or variables being studied. In the video, the null hypothesis is used to propose that the observed outcomes (like coin flips) are due to chance and not any other variable, which is then tested using the Chi-squared test.

💡Degrees of freedom

Degrees of freedom is a term used in statistics that refers to the number of values in the data set that are free to vary. The video explains that it is calculated by the number of categories or outcomes minus one. For instance, with two outcomes like heads and tails in a coin flip, the degrees of freedom would be one.

💡Critical values

Critical values are the threshold values used in statistical tests to determine the significance of the results. The video mentions that a critical value of 3.841 is used for a 0.05 significance level, and if the calculated Chi-squared value exceeds this, the null hypothesis is rejected.

💡Statistical significance

Statistical significance refers to the likelihood that the observed results are not due to chance. The video uses this concept to explain when to reject or accept the null hypothesis based on whether the Chi-squared value exceeds the critical value. If the Chi-squared value is higher, it suggests that the observed outcomes are statistically significant and not due to chance.

💡Coin flips

The script uses the example of coin flips to demonstrate the application of the Chi-squared test. Flipping a coin 100 times and getting 62 heads is an observed outcome that the test is used to analyze. The expected outcome would be 50 heads and 50 tails, and the Chi-squared test helps determine if the deviation from this expectation is significant.

💡Dice rolls

Dice rolls are another example used in the video to illustrate the Chi-squared test. With 36 dice, the expected outcome is that each number from one to six would appear six times. The observed outcomes are then compared to these expected values to determine if the variation is significant, as shown when many sixes are rolled.

💡Pill bugs

Pill bugs are used in the video as a final example to engage the viewer in applying the Chi-squared test. The script describes a scenario where pill bugs spend time in both wet and dry conditions, and the expected and observed times are compared using the Chi-squared test to determine if there is a preference for one environment over the other.

Highlights

Introduction to the Chi-squared test and its importance in AP biology and science.

Explanation of the Chi-squared test's purpose: to determine if data variation is due to chance or the variable being tested.

Historical development of the Chi-squared test by Carl Pearson in the early 1900s.

Description of the Chi-squared test formula involving observed and expected values.

Use of the Null Hypothesis in the Chi-squared test to assume no statistical difference between observed and expected values.

Example of using the Chi-squared test to analyze the results of flipping a coin 100 times.

Explanation of degrees of freedom in the context of the Chi-squared test.

Importance of critical values in determining whether to accept or reject the null hypothesis.

Demonstration of calculating the Chi-squared value with a coin flipping example.

Application of the Chi-squared test to a dice rolling scenario with 36 dice.

Calculation of Chi-squared value for the dice example and comparison with the critical value.

Discussion on the minimum data requirement for applying the Chi-squared test, suggested to be more than 30 observations per category.

Illustration of the Chi-squared test with a practical example involving the flipping of 50 coins.

Explanation of how expected values can be non-whole numbers in probability scenarios.

Analysis of the Chi-squared test results for the coin and dice examples, concluding with the acceptance of the null hypothesis in both cases.

Invitation to apply the Chi-squared test to a scenario involving pill bugs and their behavior in wet and dry environments.

Encouragement for viewers to practice Chi-squared test problems to improve understanding and application.

Transcripts

Hi. It's Mr. Andersen and welcome to my podcast on the Chi-squared test. Chi-squared

test if you look at the equation lots of students get scared right away. It's really simple

once you figure it out. So don't be scared away, but Chi-squared test especially in AP

biology, especially in science is very important. And it's a way to compare when you collect

data, is the variation in your data just due to chance or is it due to one of the variables

that you're actually testing. And so the first thing you should figure out is what are the,

what do all these variables mean?

So the first one, this right here stands for Chi-squared. And so this was developed way

in the early part of the 1900s by Carl Pearson. Pearson's Chi-squared test. So, what is this

then? That is going to be a sum. So we're going to add up a number of values in a Chi-squared

test. What does the O stand for? Well that's going to be for the data you actually collect.

And so we call that observed data. And then the E values are going to be the expected

values. And so if you're ever doing an experiment, you can actually figure out your expected

values before you start. And then you just simply compare them to your observed values.

Let me give you an example of that with these coins over here.

Let's say I flip a coin 100 times. And I get

62 heads and I get 38 tails. Well is that due to just chance? Or is there something

wrong with the coin? Or the way that I'm flipping the coin? And so the Chi-squared test allows

us to actually answer that. And so what I'm thinking in my head is something called a

Null Hypothesis. And so if we're flipping a coin 100 times. And I think I said 62 head

and 38 tails. Well that would be the observed value that we get in an experiment. But there'd

also be expected values because you know it should be 50 heads and 50 tails. And so you

used something called a null hypothesis in this case where you're saying there's not

statistical significant difference between the observed values and the expected frequencies

that we expect to get and what do we actually find.

And so it's cool, Chi-squared, because we

can actually measure our data, or look at our data and see is there a statistical difference

between those two. The best way to get good at Chi-squared is actually to do some problems.

Before we get to that there's two terms that I have to define. One is degrees of freedom

and then one is critical values. And so the whole point of a Chi-squared test is either

to accept or reject our null hypothesis. And so you have to either exceed or don't exceed

your critical value. But first of all we have to figure out where that number is in this

big chart right here.

First thing is something called degrees of freedom. So since we're comparing outcomes,

you have to have at least two outcomes in your experiment. So in this case if we have

heads and tails, we have two outcomes that we could get, so we'll say that's 2. And then

we simply subtract the number 1 from that to get the degrees of freedom. And so in this

case we have two outcomes minus 1 and so we would have 1 degree of freedom. Now you might

think to yourself why isn' there a zero on this chart? Well, if you just have one outcome

you have nothing to compare it to. So that's an easy way to think about that. So we figured

out that there is one degree of freedom in this case. The next thing you're looking at

is for a critical value. And the critical value that we'll always use in the class is

the 0.05 value. And so that's going to be this column right here. So the first thing

you do is find the 0.05 value and you don't worry about all of the other numbers. So that's

3.841 is something I just know because it means that I'm in the right chart or I'm in

the right column.

A way that I explain this to kids is that you can think of that as being 95% sure that

you're either accepting or rejecting your null hypothesis. And you can see that our

critical values get higher over here. So you can think as we move this way, if we really

want to be sure we'd have to exceed a higher critical value. So what's our null hypothesis

again. Null hypothesis's no statistical difference between observed and expected and so we either

accept or reject that value. So in this case our critical value would be 3.841. And so

when you calculate Chi-squared, if you get a number that is higher than 3.841 then you

reject that null hypothesis. And so there actually is something aside from just chance

that is causing you to get more heads than tails. And if you don't exceed the critical

value then you accept that null hypothesis. And this is usually what ends up happening,

unless you have a variable that's impacting your results. Let's apply this in a couple

of different cases.

So this is my wife here. I asked her to flip a coin and so I asked the statistics teacher

how much data do you have to get before you can actually apply the Chi-squared test? And

Mr. Humberger said something magic about 30. And so I want to exceed that number in each

of these experiments and so this is my wife down here. This is her hand. And what she's

going to do is she's going to, let me get a value you can see, she's going to flip 50

coins. You can see she's really fast so she's flipping 50 coins and then she's sorting them

out. And so if we look at that, the first thing, even before you collect the data is

we could look at the expected values. And so we've got heads or tails. And so if you

flip 50 coins how many do we expect to come up as heads? The right answer would be 25.

And how many would we expect to come up as tails? 25 as well. Now let's say your data

is not as even as that. If you're looking at fruit flies it might be 134 or 133. Well

let's say I flip 51 coins for example instead of 50 then my expected values would be 25.5

and 25.5. So expected values since they're just due to probability don't have to be a

whole number.

If we look at our observed values, well let's look down here. How many heads did we get?

28 heads. And how many tails did we get? So that would just be 22. Okay. So now we're

going to apply Chi-squared and come up with a critical value. And so, what does that mean?

Well let me get this out of the way. So we're going to take our equation which is O minus

E squared over E, and we're going to do that for the heads column and then we're going

to do it for the tails column. So we've also got O minus E squared over E for the tails

column. And so our observed value is going to be 28. So it's 28 minus 25, which is expected,

squared over 25. Now this sum means that we're going to add these two values together so

I'm going to put a plus sign right here. Now we're going to do the tails side. So what's

our observed? It's 22 minus 25 squared over 25. So you can do this in your head. 28 minus

25 is 3, square that is 9. 9 over 25 plus 22 minus 25 is negative 3 squared. It's 9

over 25. And so our answer is 18 over 25 which equals 0.72.

Okay. So that's our Chi-squared value for

this data that we just collected. Now let's go over here to our critical values. Well

we said that we had 1 degree of freedom, because there's two outcomes. 2 minus 1 is 1. So we're

in this right here, this row right here. And then here is our magical 0.05 column and so

our critical value is 3.841. And so if we get a number higher than that we reject our

null hypothesis. We didn't, so we got a value that is lower than that, 0.72 so that means

we have to accept our null hypothesis. That means that my wife did a great job. There's

nothing wrong with the coins. There's not way more heads then there should be and so

we have to accept the null hypothesis that there's no statistical difference between

what we observe and what we expect to see.

So now let's try a little more complex problem. Now we've got dice. So we've got 36 dice.

So let me get this out here. So our expected values, well there are six things you could

get. So we could get a 1, 2, 3, 4, 5 or 6. And so let's play this out. So expected values,

since I have 36 dice here, we would expect to get 6 of each of those numbers coming up.

So I'm just taking 36 total dice divided by 6 so I got 6. But let's see what we get for

observed values. Oh, it looks like we're getting a lot of sixes. So if we look at the observed

values for one here we get 2 ones. We look at the twos, we get 4 of those. For the threes

it looks like 8 threes. For the fours we get 9. For the fives we just get 3. And then for

the sixes, look at all the sixes, so we get 10 right here. Okay. Now we have to figure

out a Chi-squared value. So let me get this out of the way.

And I'm going to stop talking and do the math

and speed up the video a little bit. And so hopefully I don't screw up any of this. So

that is 58 over 6 which is 9.6. So that is our Chi-squared value. It's 9.6 in this case.

Since we added all these up. So now we've got to go over here to our chart. And so first

of all we have to figure out how many degrees of freedom do we have. Well, since there are

6 different outcomes and we take 6 minus 1, so we've got 5. We're in this column of the

0.05 right here so if I read across our critical value is 11.070. And so if we look at that,

did our value go higher than that, no it's only 9.6, it's lower than that, so in this

case since it's 9.6, even though we had all of those sixes we still need to accept our

null hypothesis that there's no statistical significance between or difference between

what we observed and then what we expected.

So now let's leave you with this question. So in the animal behavior podcast as I talk

about that, we're looking at pill bugs and if they spend more time in the wet or if they

spend more time in the dry. And so if you look at the values right here, this is recording

how much time they spend in the wet and how much time they spend in the dry. So what I've

done is we would expect since there are 10 pill bugs we'd have 5 on each side. But since

it looks like they're spending more time on the wet, you can even see them in the video

here spending more time in the wet, I take the average of the wet and the average of

the dry column. And that gives me my wet and my dry and so now I'm not going to show you

how to do this one, but try to apply Chi-squared to figure out if there's a statistical difference

between the expected values of what we expect and what we observed. And you can put your

answer down in the comments. And so I hope that's helpful.