hello AP biology students we're going to cover the last section in unit zero and in this section we're going to cover something called the kai Square equation this is going to round out unit zero and specifically our content about statistics and the math that we're going to be using in this class so first of all don't get scared by this equation once you practice with this a couple times it becomes a lot easier the Ki square is an equation to analyze if the differences in data are due to chance or a variable being tested another way of saying this is the Ki Square analyzes the expected results with the observed results so for an experiment we can have expected results this should happen however we get the observe results this equation looks at and compares the observe what we see and what we take the data from with what we expect to see now what we are able to do with the Ki square is calculate something called the P value the P value is the measure of the evidence against the null hypothesis and remember back the null hypothesis just states there is no difference between the expected deserved results now again this is for a general AP Biology class not for a statistics class but a smaller P value means the test is significant while a larger P value means that test is not significant and it may have occurred just by chance something to remember when calculating this P value is if the P value is less than 0.05 we reject the number null hypothesis and if the P value is more than 0.05 you fail to reject your null hypothesis you might be wondering again why we're using this 0.05 number that comes from having that 95% confidence interval one of the most important parts of this equation that students need to understand is that this x^2 does not equal the P value this x² just represents the kai Square what we're going to calculate the second biggest mistake stud make is thinking that this x s means that you have to square root this side as well as square root this side that's again not the case x s literally means the kai Square value this big e as I like to call it or the summation sign just means that we're going to add up all of the values to the right here the O represents the observed data this is the what we collect and these E's represent expected values this is what we calculate before the experiment so again when we get done calculating everything on this side that number represents the kai Square we use this Kai Square value on something called a Kai square chart this is a Kai square chart and basically what this chart does is it helps us calculate the P values for an experiment these are the P values up above I like to manipulate the Ki square chart a little bit to kind of introduce this concept to my students and show them the critical values here which are along the 0.05 P value now remember if the P value is lower than 0.05 you're going to reject the null hypothesis if the P value is larger than 0.05 then you are going to fail to reject the null hypothesis now your next question while looking at this chart is why do they have different degrees of freedom or different lines here these lines represent different degrees of freedom which are the numbers of outcomes for your experiment minus one the critical value is the cutof value and again we find that using the Ki square chart so again on the left hand side here you can see we have the degrees of freedom and they go down to five now they can go further and further as you add on more variables but your critical values are going to be found here at the 0.05 P value Mark now your next question might be why isn't there a zero value here well if there's only one value or one possible outcome then it's really not an experiment you have to have at least two out comes for instance flipping a coin there are two outcomes heads or tails for our first practice problem for this equation we're actually going to use the heads or tails flipping of a coin that means that the degrees of freedom is going to be on that first line which means our critical value is going to be 3.84 one for our second example problem we're going to look at rolling a dice one dice um which means there's going to be six outcomes so 6 - 1 is 5 so our critical value is 11 .07 all right so let's do that first practice problem how many times do you expect to get heads and tails if you flip a coin 50 times so again since there are only two outcomes you can either be heads or tails it's going to be 25 and 25 however you flip a coin and you get 28 heads and 22 tails is that in the realm of possibility and yes there is a possible chance that you flip 50 heads and zero Tails but we're just saying in the real Poss possibility for our experiment is getting 28 heads and 22 tails in the realm of possibility so again all the kai Square does is it looks at our observed values and our expected values and say hey is this in the realm of possibility or is some other variable affecting the outcomes of this experiment for instance is the coin weighted so now that we have our experiment let's plug in the values again we're going to start out with X2 which is our Kai Square variable Our Big E summation sign which means we're going to add the two variables once we're done going through the equation so for our first variable this is the number of heads that we had we had 28 heads we expected 25 and we put 25 down here as well and we add the Tails variable for Tails we expected 25 but got 22 so it's 22 - 252 over 25 28 - 25 is 3 3^ 2 is 9 over here 2 2 - 25 is -3 -3 2ar is 9 so we're going to place 9 over 25 + 9 over 25 when you add these two values together it's going to be 18 over 25 or 0.72 again one of the most common mistakes I see with students calculating this is they think they have to take the square root of this side and this side remember X2 is your Kai Square so our Ki Square value is 0.72 our second biggest mistake I see students making is that they they think this is the P value this is not the P value this is the kai Square value we're going to use this to figure out what the P value is all right so now we have to find the critical value which one of these numbers highlighted is going to be our critical value remember that we're on the first degree of Freedom here so our critical value is going to be 3.84 one that's again because we had two outcomes for our experiment it's either going to be heads or tails 2 - 1 is 1 so that means we're on the first degree of Freedom our critical value is 3.84 one if our Kai square is larger than this we are going to reject our null hypothesis if our Kai Square value is smaller than this we are going to fail to reject the null hypothesis so now what we have to do is we have to find where this Ki Square value is going to be along our Kai square chart obviously we can tell that 0.72 is not larger than 3.84 it's going to be somewhere between 2.76 and 0.455 now you might be wondering what is the actual P value our Ki square is 0.72 it's going to be a little bit larger here and to the right here but what is the actual P value well again it actually really doesn't matter because all we know is that it is larger than 0.05 as long as the P value for our experiment is larger than 0.05 we're going to fail to reject the null hypothesis it really doesn't matter if it's 0.1 0.5 or 75 as long as it's larger than 0.05 we fail to reject the null hypothesis and that should make sense the null hypothesis states there is no significant difference between the expected and observed data we fail to reject that we can't say that that null hypothesis is wrong looking back at our data yeah this does kind of seem like it falls in the room of possibility if you flip a coin 50 times yeah there's a chance it's 28 and 22 let's do another quick practice problem using the kai Square you roll a dice 36 times now remember a dice has six sides so if you roll a dice 36 times you should expect to get each number six times however these are the values that you get for each of the numbers you can see 1 2 3 4 5 six you should get six of each of the numbers but you get this data again is this data in the realm of possibility so pause the video if you want time to calculate the Ki square but you can see I did it here you can see the value that we got for each of the different variables 1 through six we calculated our Kai square and our Kai square is 18.3 now remember this experiment is different since there are six total outcomes 1 2 3 4 5 6 our degrees of freedom is five 6us one is 5 so we're on this here line which means our critical value is 11.07 our Kai square is 18.3 which means it's over between these two numbers and that means our P value is going to be smaller than Z .05 which means we reject the null hypothesis that means something is going on here some variable is influencing the experiment and we don't expect normally to get these values for rolling a dice all right we have to end this with a joke sup have you heard the latest stats joke probably get it pro probably because okay bad joke