The table shows the marks of 60 students in a mathematics exam. By drawing a cumulative frequency curve, estimate the median mark achieved.
So, looking at our table, 11 students get a score of two. 10 students received a score of six. 13 scored 10. Seven scored 14. Six scored 18. Five scored 22. And eight scored 26. And there was a total of 60 students.
Before we would draw a cumulative frequency curve, we first need to create an ascending cumulative frequency table. And the key word here is ascending, meaning smallest to largest. Because when we look for a median, the median is the middle number, but only when the numbers are in order from least to greatest.
So, for our cumulative frequency table, we have upper boundaries of the mark. So, our marks are two, six, 10, 14, 18, 22, and 26. So, for a cumulative frequency table, the scores accumulate. So, how many times that score happened we keep accumulating together. So, how many students scored less than two? Well, that would be zero. 11 students scored two but not less than two.
Now the next one is how many students scored less than six? Well, we know that 10 students actually had a square of six, but we want less than six. So, less than six would have to be the 11 that scored a two. So again, this is from the 11 students that scored a two. Now for the next one, less than 10, a score of two is less than 10, and a score of six is less than 10. So, 11 students scored a two. 10 students scored a six. So, altogether there’d be 21 students who scored less than 10.
Here are the scores that are less than 14. And here are how many times they’ve happened. So, we add them together and find that 34 students scored less than 14. Here are all the marks that are less than 18 and how many times they happened. So, adding together their frequency, we get 41. We now find that 47 students scored less than 22. 52 students scored less than 26. And all 60 students scored less than 30. And this makes sense because the highest score was 26.
So, why did we choose an upper boundary mark of 30? Well, each score, or mark, are four scores between each other, so two, six, 10, 14, 18, 22, 26, and 30. So, now that we have this ascending cumulative frequency table, we will use it to draw a cumulative frequency curve, and then, in turn, use that to estimate the median mark achieved.
Here on our graph, the horizontal axis, the 𝑥-axis, are the upper boundaries of mark. And on the vertical axis, the 𝑦-axis, we have the cumulative frequency. Zero have a mark less than two. 11 have a mark less than six. 21 have a mark less than 10. 34 have a mark less than 14. 41 have a mark less than 18. 47 have a mark less than 22. 52 have a mark less than 26. And all 60 have a mark less than 30.
So, connecting these marks, we now have this curve. And to identify the median, we draw a horizontal line on the cumulative frequency curve that passes through the order of the median, or halfway the frequency total. For this data, the frequency total is 60. So, we will draw the horizontal line to pass through the order of the median 30, which we’ve drawn here.
The related mark along the 𝑥-axis will now be the median. So, this mark is somewhere between 10 and 14. It looks about halfway maybe a little closer to 14. And remember, we are supposed to estimate the median mark achieved. So, it doesn’t have to be exact. So, halfway between 10 and 14 will be 12. And we said it’d be a little closer to 14. So, our estimate could be 12.8. More than likely, any estimate very close to this would be fine.