The table shows the distribution of marks students attained in a mathematics exams. By drawing a cumulative frequency curve, estimate the median mark attained.
So, what the question asks us to do here is draw a cumulative frequency curve. Now to enable us to do this, what we need to do is find out what the cumulative frequency is for our frequency table. Now, in order to do this, what I’ve done is I’ve added another row to our table and I’ve called this cf or cumulative frequency. Now, if I want to work out what the cumulative frequency is we need to think about what does it mean. Well, if we think of cumulation or cumulate, it means you add up as you go along. But if we want to find the cumulative frequency, we, in fact, add our frequencies up as we move along.
So, our first value is five. Our second value is going to be 15. And that’s because it’s five add 10, so that’s the first two frequencies added together. Then, we have 24, and that’s because we’ve then added on the next frequency which is nine. Then, we’ve got 34 because we’ve added 10 on. Then, we have 46, adding next 12 on. And then finally, if we add the final four on, we’re gonna get 50. And then, if we check this, this should be the same as the total of our frequency. In fact, it is. So, we know that it’s correct.
So now, we’ve got the cumulative frequency for our table. What we can start to do is draw our cumulative frequency graph. So, when we’re plotting a cumulative frequency graph, what we need to remember is that the 𝑦-axis is cumulative frequency, as I’ve shown here, and our 𝑥-axis is gonna be the marks, which I have got here on the bottom of our 𝑥-axis. And remember, we always need to label each of our axes when we’re drawing a cumulative frequency graph.
Now, another thing we need to remember is that whenever you’re plotting a cumulative frequency graph, so we’re gonna plot our marks and our cumulative frequency as our 𝑥- and 𝑦-coordinates, that when we’re looking at our group, cause it’s a grouped frequency table. Then, we need to choose the value that’s the higher bound of each of our groups, so nine, 14, 19, 24, 29, et cetera.
It’s also worth noting here that we must use the values that are for cumulative frequency and not frequency. A common mistake is for students to choose frequency rather than cumulative frequency and plot this. And if you did this, you’d have a shape a bit like this, a bell curve. If you see this in this kinda question, if you see that you’ve drawn this for cumulative frequency, then you know there’s a problem and you know that you’ve, in fact, probably drawn the frequency.
Okay, so now let’s get on and draw what we should draw. So, the first point is at nine, five. Then, we have our next point at 14, 15. Then, we have 19, 24. Then, we’ve got 24, 34. Then, 29, 46. And then finally, 30, 50. And that’s cause 30 is the final value that we’ve, in fact, been given.
So, we’ve now drawn or plotted all of our points. So, now what we need to do is join the points, and you can do that either with a smooth curve or with straight lines. So, I’ve joined them all up. But where does the final point go to at the bottom left? Where do we hit one of the axes? Well, we can see that the lowest possible mark is five. So therefore, what we’re gonna do is we’re gonna draw our final point on the 𝑥-axis at five.
Okay, great, we’ve now plotted our cumulative frequency curve. So, now if we take a look at what the question wants, we want to find the median mark attained. And the median mark is the middle mark, so the median is the middle value. So to find this, what we need to do is, first of all, find the midpoint of our cumulative frequency. So, we take 50, which is our highest value for our cumulative frequency, and we divide it by two. And this is also half of the frequency because we want to find the middle value of all of our values. And when we do this, we get 25.
So, what we’re going to do now is draw a line from 25 on our cumulative frequency, so that’s our 𝑦-axis, across to the cumulative frequency curve we’ve just drawn. And now, to find out what mark is going to be the median mark, what we do is we draw a line vertically down to the 𝑥-axis, so the marks axis. And when we do that, we can see that the value it hits on the 𝑥-axis is 20. So, we’d say this is the median. So therefore, in answer to the question, “What is the median mark attained?”, the answer is 20.