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Frequency Tables

Frequency Tables

Author: Ryan Backman

Calculate relative frequency from a frequency table.

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Hi, this tutorial covers frequency tables. So let's start with an example here. A random sample of 20 high school students were asked how many letters they had in their middle name. The following data was collected.

OK, so as you can see, there are 20 numbers here. Each number represents the length of a middle name. So we have the first person had a middle name of length seven letters. Second person had a middle name of five letters long, four letters long, six letters long.

And you can see that some of these values are repeated. So there were, looks like, at least two people that had middle names of seven letters. OK? Several people that had middle names of five letters.

So to avoid some of that repetition, a common way to organize and display a data set is to use a frequency table. Now, let's make sure we have a good idea of what that word frequency means first. So a frequency is the number of times a data point or range of data points occurs. So in this case, we're just going to look at how many times just one data point occurs. And then we'll look at the other case in a minute.

So then a frequency table is a table with two columns. The first column is the list of possible data values. And the second column lists the frequency of each data value. So let's create a frequency table for the middle name data.

All right. So I have the data listed up here and room down here to make the frequency table. So we need two columns. So let's go ahead and do that first.

Now, what I'm in a graph is the possible values of the variable here. So the variable is number of letters. And then in this column, I'm just going to abbreviate frequency as F-R-E-Q. That's a pretty common abbreviation for frequency.

Now, I'm going to list the possible number of letters in this column. So if I look here, the shortest middle name was zero. OK? Which means that they didn't have a middle name. And the largest value was eight. OK? So we had a couple of people with a longer middle name of eight letters.

So what I'm going to do is, in this column, write down the numbers from zero to eight. OK? And then next to each of these numbers, I'm going to figure out what the frequency is. So how many did these data values show up in the data set? OK?

So let's start with zero. So I have three people that had middle names of zero letters or didn't have a middle name. And then I'm just going to cross those off so I don't confuse those later. OK, so in this column, I'm going to put a three next to the zero.

OK, now, if I look at one, there's nobody with a middle name of one letter. So that's going to be zero. OK? If I look for two, OK, generally there aren't many names that have either one or two letters in them, so this one was also zero. OK, now three, so we had one, two, three people with a middle name of three letters.

OK, four, we have one, two, three again. So that's also three. OK, five, this is a pretty common one, I think, so one, two, three, four, five. So five people had a middle name of five letters. Six, we had one, two, I think two.

Seven, one, two. And eight, one, two. So that one was also two.

OK, so in this case, now what I want to make sure is all of my numbers are crossed off. OK? If there was one that wasn't crossed off, then I could go back and change the frequencies.

Now, the other way to check to make sure I have all the numbers is to make sure that your frequencies add up to the sample size. So we knew that we sampled 20 students, so we have to make sure our frequencies add up to 20. So let's do that. So three, six, nine, 14, 16, 18, 20. Good. So that does add up to 20. OK? So we can tell that we do have all of our numbers correctly identified and placed properly into the frequency table. All right.

So that's one way of making a frequency table. Another way to make a frequency table is to use interval categories. So instead of just having how many times each letter comes up, we're going to have each time an interval of letters comes up. So this is still going to be frequency. This is still going to represent number of letters.

But now, instead of just putting the numbers in here, I'm going to do intervals. And I mean do intervals of width three. And I'm just going to write it out first.

And what each of these mean now, so it's zero to less than three. So the numbers I'm counting here are zero, ones, and twos. OK? Then here, three to less than six, I'm counting three in this interval, but I'm not counting six. So this is going to be threes, fours, and fives. And then here is going to be six, sevens, and eights.

And now, we can just use this frequency table to make this. Otherwise, we could go back to our data also. But zeros, ones, and twos, OK, that would just end up being three, frequency of three. Four, fives, and sixes, that'd be three plus three plus five, which is 11. And six through nines-- two, two, and two, so that would be six.

OK? And then same thing, you want to make sure that these values add up to 20, which they do. OK? So those are two ways of making frequency tables using just the data values. And then here using an interval of data values.

OK. Data can also be displayed using a relative frequency table. OK? So let's define now what relative frequency is. So relative frequency is the frequency of a data value divided by the sample size n. So just an easy formula here. Relative frequency equals frequency over n, where n is the sample size. OK?

So let's create a relative frequency table for the middle name data. And I think what's easiest is if we just go back to our frequency table, which is this one, and just add in a new column here. So what I'm going to do is add in a third column here. And my third column is going to be relative frequency. So I'll say relative F-R-E-Q, Freq.

And so what I need to do is I need to take each of these frequencies and divide them by 20. OK? So if I do that, if I take 3 divided by 20, what that'll end up giving me is 0.15. And you could double check that on your calculator.

0 divided by 20 is 0. So this will also be 0. OK, since this frequency is also three, this relative frequency is going to be the same. So it's going to also be 0.15. 0.15. OK, now we need to do 5 divided by 20. And 5 divided by 20 is 0.25, 1/4. Then 2 divided by 20, 0.1, 0.1, 0.1.

OK? Now, if you think about what the sum of these should equal, since these were all added to 20, and we had a sample size of 20, really if you think about the sum, 20 divided by 20 is 1. So these values should all add up to 1.

And if we do that, this would be 0.15, 0.3, 0.45, 0.7, 0.8, 0.9, 1. So those do add up to 1. So this is what you call a relative frequency table. OK? And you don't always have to put both of these together, but you certainly can if you'd like.

OK, so that was a relative frequency table. OK, frequency tables can also be used to display qualitative data. So suppose the same students were also asked about their eye color. Let's call it hair color. Hair color.

The following frequency table was constructed. So hair color was blond, brown, black, red, or other. OK? So notice here instead of the numbers, we just have our categories. And here, we just have the frequency. So three people had blond hair. Eight people had brown hair. Five had black. One red, three other. OK? And these frequencies should also add up to 20.

OK? So frequency tables can be used for both qualitative and quantitative data. And if we wanted to, we could also make this into a relative frequency table. So that is the tutorial on frequency tables. Thanks for watching.

Terms to Know

How often a data value, or range of values, occurs.

Frequency Table

A table showing the values of the data, and their respective frequencies.

Relative Frequency

The percent of the data points that take a particular value. This is obtained by dividing the frequency of each value by the total number of data points.