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Cumulative Frequency

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Tutorial

Source: Tables and Graphs by Katherine Williams

This tutorial covers cumulative frequencies. Cumulative frequencies are like the running total of frequencies. It's where you add up all the values that came before it to get your cumulative frequency. Here we have our frequency chart with the heights of cherry trees. It has the intervals right here and the frequencies here.

In order to find the cumulative frequency, we're going to add up all the frequencies that came before that one-- and including that one. For our first one, we have the frequency of 3. We've seen three values so far. So our cumulative frequency, quite simply, is 3.

For our next interval-- from 65 to 70-- we see another three values. So we're going to add that to what came before. So 3 plus 3 to get 6. For 70 to 75, we have eight more values. So we're going to add up all three of these values-- all the ones that include it and came before it-- to get this cumulative frequency. So 3 plus 3 plus 8 gets us 14.

Now another way we could do it is, if we know our cumulative frequency so far for all this top part is 14, then 75 to 80 adds on another 10 values. So we can do 10 plus 14 to get 24. And similarly, if I had said 3 plus 3 plus 8 plus 10, that would also get us 24. Now we're going to add on another 5 for this 80 to 85 to get 29. And for the 85 to 90, we have another 2, so 31.

Another variation on cumulative frequency is relative cumulative frequency. So this would be the running total for relative frequencies. Now once you have your relative frequencies, you can turn the decimals into percents and graph these. If you're graphing them, it's going to be a line chart with one of the axes going from 0 up to 100 and the intervals going across the other way.

Now when we're doing this particular type of graph that goes from 0% to 100%, it's called an Ogive Plot. So now we're going to be calculating those cumulative relative frequencies. So first we still have the same set of categories-- K through 6 students, 7 through 12 students, adult students, and then at the bottom is marked the total.

The frequency is already filled in. And the relative frequencies were already calculated. And as a reminder, the relative frequency-- this number here-- is coming from the frequency divided by the total. So this 0.330 is 12,617 divided by 38,275. Now in order to take this information and to calculate the relative cumulative frequencies, it's actually pretty simple.

I start by just adding up the values that are lower than that categorical level. So here, we have started with the 0.473. So at that point, it's the only value we have. So our relative cumulative frequency is 0.473. Now for the next level here, to get the relative cumulative frequency for there, I need to add these two spots together-- so 0.473 plus 0.330. So the relative cumulative frequency here is 0.803.

Now for the final one, it's going to be the same thing. I'm going to add the values that came before it. So I'm going to add all three values here and get 1.001. Now this last value is the total. So I should not add that and do it again because I've gone through everything I have. I've included all of the data. So I'm done.

Now typically, the final relative cumulative frequency and the total are going to be exactly the same. And they should be equal to 1. However, because of some rounding error in the calculations of the relative frequency, we get 1.001. That's really close to 1 and it comes just from a rounding error. So we consider that fine. Something like 0.99999 would also be OK. But it should be 1.

Now let's graph our relative cumulative frequencies. So now I have copied over the relative cumulative frequencies. And for the total, we're not going include anything there because that's not going to get graphed. When we go to put it on our graph-- our Ogive Plot-- we're going to need to use percents. So it's easy to transfer those into percents. You multiply by 100.

So here we're going to have-- and I'll write it on this side to make it easier to see-- 47.3%, here, 80.3%, and then 100%. So on the graph, for K-6, I'm going to go up to where I estimate 47.3 to be; for 7-12, up to the estimate for 80; and then for the adult, up to the estimate for 100.

And then we connect these points with lines. And I should do that with a ruler. But that is our graph of our relative cumulative frequencies. So this has been your tutorial on cumulative frequency.