Video: Pack 4 • Paper 3 • Question 11

Pack 4 • Paper 3 • Question 11

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Video Transcript

The cumulative frequency graph shows the heights of 50 year seven students. Part a) Estimate the median height.

Now, the thing with cumulative frequency graphs is that they don’t tell us about any individual pieces of data. So from this graph, we don’t know the exact height of any individual student, for example. But what it does tell us is how many students are up to and including certain heights.

And the other thing is that we’ve been asked to estimate the median height. The median height is the height of the student who would be in the middle of the pack if you lined them all up in ascending height order. And I’ve used the word “estimate” because-well, because we don’t know the height of any individual student. So we’re not going to be able to work it out exactly.

So the technique that we’re expecting to use here is to take the number of students in our sample and to divide that number by two and then work out what that height is. We’ve got 50 students: 50 divided by two is 25. So we can use the graph to estimate the height of the 25th student. Now, before we do that, we need to look very carefully at the 𝑦-axis.

Now, between zero and 10, there are 10 little squares. So each one of those little squares represents one student. So 25 on our cumulative frequency is going to be here. And we need to draw a line from 25 across to the curve. When we hit the curve, we then go down and work out the corresponding height. And when we follow that line down, there comes out to be 144.5. So our answer is the median height is approximately 144.5 centimeters.

Now, Simeon says that the minimum height of the students is between 135 centimeters and 140 centimeters. For part b Is he correct? And give a reason for your answer.

Let’s look down at the bottom of our cumulative frequency graph. We can see that the bottom of the graph it actually lines up with the third person. And they would have a height of 135 centimeters. So the third smallest person in the group has a height of 135 centimeters. So three people have a height of 235 centimeters. And the minimum height could be lower than 135 centimeters. So the answer to the question is no, Simeon is not correct.

Now, it is conceivable that all three of those students have a height of exactly 135 centimeters. But Simeon said that the minimum height was between 135 centimeters and 140 centimeters. So it’s possible he’s nearly right, but we don’t know that. And we certainly can’t say that he is correct.

Now, for part c, we’ve got to show that more than 75 percent of the students have a height greater than 140 centimeters. Now, remember the cumulative frequency graph tells us how many people have heights up to a certain value. But we’ve now got to work out how many have a height greater than that value. So let’s look at 140 on the graph.

If we follow the line up to the graph and then across, we can see that nine people have heights up to 140 centimeters. So if we take those nine out of the 50, that tells us that are 50 minus nine students who have heights over 140 centimeters. That’s 41 students. Now, we need to work out what percentage is 41 of 50 or 41 over 50 times 100 gives us 82 percent. So that tells us that 82 percent of students have a height greater than 140 centimeters.

And of course, 82 percent is greater than 75 percent. So we’ve shown that more than 75 percent of the students have a height greater than 140 centimeters. In fact, 82 percent do.

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