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Cumulative Frequency

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Source: Graph created by Ryan Backman

Hi. This tutorial covers cumulative frequency. Let's take a look at an example. A church is interested in determining how many times a typical congregation member attended a Sunday church service in a month. The church surveyed 30 random members and produced the following data set.

So the data here is presented as a frequency table. As you notice, the variable here is the number of Sunday church services attended in the last month, so 0 through 4. In this specific month, there was 4 Sundays. And then the frequencies for each. So there are 3 people that did not attend any Sunday services, 4 that attended 1, 6 that attended 2 services, 12 that attended 3 Sunday services, and 5 that attended 4.

So let's take a look at another calculation you can make to help summarize the data a little more. So cumulative frequency is the count of all data values equal or less than a specific data value. So what we mean by that is if we fill out cumulative frequency for this frequency distribution, if we want the cumulative frequency for 0, well, that needs to be all of the values equal or less than 0. So since there are no values less than 0, the cumulative frequency for 0 is the same as the frequency for 0, so 3.

Now, the cumulative frequency for 1 church service, now we want to know equal than 1 or less than 1. So in this case, we can include both 0 and 1. So what we do is we add up those 7 people, and 7 is the cumulative frequency for 1. And what that means now is that there are 7 people who attended 1 or fewer church services in the last month.

Now, if we want the cumulative frequency for 2, now what we need to do is add all three of these numbers up. So that'll end up giving you 13, so 13 people attended 2 or fewer Sunday church services. And that would be 25, taking the 13 and adding 12, or taking 12 and adding all the numbers that came before it. And then, finally, that should end up being 30.

Now, the last cumulative frequency in your chart should match the sample size, and it does in this case. So 30 people were surveyed. 30 congregation members were surveyed. So the cumulative frequency for 4 is 30. And that should make sense. All 30 people attended 4 or fewer church services. That accounts for everyone.

All right, the next thing we can do is think about relative frequency. Remember, relative frequency is dividing frequency by the sample size. So we're going to divide each of these by 30. So if we take a look at 3, so we just take 3 divided by 30, and that's going to give you a relative frequency of 0.1. 10% is the same as 0.1. So what that would mean is that 10% of people attended church no times in the last month.

Now, we can continue to calculate relative frequency. So we do 4 divided by 30, so that'd be 0.333 if we round that to the nearest thousandth. This would give me 0.2, 0.4, and 0.167. So this would give me all of the relative frequencies.

Now, we have another calculation called cumulative relative frequency. So we need to use the relative frequency values, so these are the same as what we had just calculated. And now I've added a column for cumulative relative frequency. Now, cumulative relative frequency is calculated the same way that cumulative frequency was, except for now we're just adding up the relative frequencies.

So same as before, the cumulative relative frequency here will be the same as the relative frequency. Now, if we go to the cumulative relative frequency for 1, now we need to add these two values, so 0.233. For 2 times, we'd need to add 0.2 to these values or 0.2 to this, so it would be 0.433. This would then be 0.833.

And you might be a little off due to rounding, but your very last value there should equal 1. 1 is equivalent to 100%. So 100% of the 30 people attended 4 or fewer Sunday church services in the last month. So cumulative relative frequency, you're simply adding up all of the values that came before a specific value.

Cumulative relative frequency can be displayed on a graph using what's called an ogive. And all this is is it's just we have our x variable on the x-axis, so the number of Sunday church services attended, and then we display a cumulative relative frequency here. So your ogive will always start at your first cumulative relative frequency, and it will always end at 1. OK, so in this case, we can see that.

It'll generally have a shape where-- sometimes it'll level off a little bit here, but it will generally have a similar shape here. So an ogive a cumuilative relative frequency graph are a common way where cumulative relative frequencies are used and displayed. So that has been the tutorial on cumulative frequency. Thanks for watching.