Two classes in a school get a test. The 24 pupils in the first class get a mean mark of 31 out of 40. The 32 pupils in the second class get a mean mark of 36 out of 40. Andrew says that the mean mark for the two classes is 33.5 out of 40. Andrew is not correct. Is the mean mark less than or greater than 33.5 out of 40? Justify your answer.
Well, if we take a look at the question and we see the mean mark that Andrew has actually said, well Andrew says that it’s 33.5. And the way he actually worked this out is he’s actually taken the mean mark of the first class, which was 31. He’s added it to the mean mark of the second class, which was 36. And then, he’s divided it by two. This would work if the class has had the same number of pupils in. However, we can see that actually the classes are not equal. They have different numbers of pupils.
So therefore, what we can say is there are more people in the second class. So therefore, the mean mark should be closer to the mean mark of the second class. So therefore, the true mean would actually be greater than 33.5. Okay, so that actually tells us whether it’s gonna be less than or greater and it actually justifies the answer. What we’re gonna do is actually we’re gonna show what the true mean should be.
So the overall mean is actually going to be equal to 24 multiplied by 31 because that’s the 24 pupils in the first class and their mean mark is 31. So therefore, this will give us the total marks gained by the first class. And then, this is plus 32 multiplied by 36 cause it’s a 32 pupils in the second class and they had a mean mark of 36. Then, this is all divided by 24 plus 32 because this is gonna be the total number of students, so the total number of students in the first class and second class added together, which is gonna be equal to 1896 divided by 56 which is equal to 33.9 to three significant figures.
Okay, so we’ve worked out what the overall mean would be. But to get the marks in this kind of question, that wouldn’t be the end of it because what we now need to do is actually answer the question. So the question was saying, is the mean mark less than or greater than 33.5? So therefore, we can actually say that yes, the mean mark would be greater than 33.5 and that’s because 33.9 is greater than 33.5.
So now, what we can do is actually check with the first method. And actually, yes, it’s giving us the same answer. So you got two methods here: one where we actually talked about the number of pupils in each class and used this to justify the answer which was that the mean is greater than 33.5 and the second method, we actually calculated what the mean would be. And that’s showing that the mean would be greater than 33.5.