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