Frequency Distributions: Videos & Practice Problems

Frequency distributions organize data into classes, showing the frequency of measurements within each class. To create a frequency distribution, calculate the class width using the formula: $(\max -\min )/\mathrm{number\; of\; classes}$. Determine lower and upper class limits, ensuring they do not overlap. Finally, calculate relative frequencies by dividing the frequency of each class by the total number of measurements, then multiplying by 100 to express as a percentage.

Intro to Frequency Distributions

So we talked a lot about how we can use different types of charts and graphs, with their own strengths and weaknesses, to visualize different types of data, whether it's qualitative or quantitative. But one of the very first things you'll have to do before you do any of that is organize your information and data into what's called a frequency distribution. I know that sounds like a scary word, but don't worry because what I'm going to show you in this video is that all this is is it's just a table. It's a table that helps you organize the frequency, the number of measurements that you have, versus different groups of numbers or labels. So let's go ahead and dive right in.

We're going to do an example together. I'll show you some definitions that you need to know, and we'll do some examples. Let's get started. A frequency distribution is just a table. It's a table of values, and it shows the frequency, remember the number of measurements over here in this column, versus chosen groups of numbers or labels.

These are things that are called classes. So, for example, in this example problem, we have this dataset which lists the amount of time in minutes that students spend studying for their exam, and it's listed over here. We're going to construct a frequency distribution using six evenly spaced classes. So what does that mean? Well, basically, what happens is we have this table of values over here, and these six evenly spaced classes are where we see values from 20 to 29, 30 to 39, and so on and so forth.

These are all evenly spaced, and there are six of them if you scroll down. So basically, these things over here are called your classes. In some cases, like this example, the groups are chosen for you. And in other cases, you'll have to come up with them yourself, and we'll see how to do that later on. But all that I'm going to do here is we're going to look through the data values.

And in order to figure out the frequency, the number of measurements, I just have to go figure out how many measurements belong in each one of these classes. So let's go ahead and do that. I'm going to look through my dataset over here. I see numbers that go from 20 all the way up to 75. What I'd like to do is just look through each one of them over here.

I've got my classes over here. And for the 20 to 29 range, all I have to do is just figure out how many of these things belong in that category. If I look through the number I see first is 20. So one of the things I like to do is either tally them or or or cross them out just so I don't double count them later on. It's going to be really helpful when you get, you know, bigger datasets of let's say, numbers.

So 20, I've counted that once. I'm going to put a tally there. Then I count my thirties. 30 to 39, I've got two over here. That's one, two.

I got my forties. That's one, two, and three. So that's three. And I got my fifties. That's two.

And I got my sixties. That's one. And then my seventies. That's one. So now I'm done with all of these things, and I'm just going to replace these tally marks with numbers.

So this is a one over here. This is a two. This is a three, two. And then we got one and one. Alright?

So that is a frequency distribution. We've got my classes over here, and then you've got my frequency over here. That's really all there is to it. Okay? So now a couple of definitions here because you'll notice that each one of these classes has an interval of numbers, 20 to 29.

The lower class limit, which you'll need to know, is basically just the lowest of each one of those numbers, of each one of those classes. So in this case, the lowest numbers are twenty, thirty, forty. Basically, it's just all of the left numbers in this column. So the lower class limit is twenty, thirty, forty, so on and so forth. Alright?

Now if that's the lower class limit, what do you think the upper class limit is? Well, hopefully, you realize that those are going to be the highest numbers. So in other words, those are just going to be the 29, 39, 49, so on and so forth. So these are going to be your upper class limits. Alright?

So we've got 29, 39, and then so on and so forth. Alright? So now something that you may have to know in these problems is not the lowest and highest numbers of each class, but what's the midpoint? And essentially, that is just the middle number in each class. Now, most of the time, you'll have to calculate these, but it's actually pretty simple to calculate.

All you just do is you take the average of the two numbers. So in other words, you just take the lower of each class and the upper and then just divide by two. An example of this would be for the first one, to calculate the class midpoint, it would just be \( \frac{20 + 29}{2} \), and that would give you 24.5, and then so on and so forth. You could calculate them for the rest of them. Alright.

So that's the class midpoint. And the very last thing, the very last definition you'll need to know is something called the class width. So the class width is essentially the difference between two consecutive, and that's the most important word there, consecutive lower or upper class limits. So be really careful here. Right?

The interval of this one class goes from 20 to 29, but the difference between consecutive lower class is from 20 to 30. So in other words, it's basically the class width is 30 minus 20, which is 10. Or another way, you could calculate this is just doing the upper class limits, 39 minus 29, in which you'll just get the same exact number. It's 10. So the class width for this dataset over here is 10, not nine.

So what you don't want to do is you don't want to do the upper minus the lower because that's going to give you nine. That's not what the class width is. If you do that, you're going to get the wrong answer. So just be very, very careful for this. The reason for this, by the way, is that if this was actually 30, then if you were to get a number of 30, you'd have to put it into both of these classes over here.

So you'd have to double count them. And that's why these things are always basically just separated by at least one number here. So 20 to 29, and then the next one starts at 30, not 29. Okay? Alright.

So that's it for the definitions over here. The class width is 10. So now what I want to do is just tal

Use the frequency distribution below to find the class width and class midpoints.

• A. Class width = 10, class midpoints = 10, 20, 30, 40, 50, 60, 70
• B. Class width = 10, class midpoints = 10, 21, 32, 43, 54, 65, 76
• C. Class width = 11, class midpoints = 10, 21, 32, 43, 54, 65, 76
• D. Class width = 11, class midpoints = 10,20, 30, 40, 50, 60, 70

Answer: (C)

The following data set shows the number of overtime hours that 12 employees worked in a month. Construct a frequency distribution, using a lower class limit of 3 and a class width of 4.

• A.
• B.
• C.
• D.

Answer: (A)

Intro to Frequency Distributions Example 1

Everyone, let's go ahead and work out this example together. We're sitting in a cafe one day, and you're counting how many customers are served each hour over a fourteen-hour period. We're going to construct a frequency distribution, and we're actually told what the lower class limit and class widths are. The lower class limit is 15, and the class width is five.

So, essentially, we can figure out what our classes are going to be. We're going to get to the second part in just a second with the relative frequencies. Let's just start off by finding the frequency distribution. So remember, I don't like to draw the box. The first thing you should do is just draw the little T-chart here because you'll fill in the box later. This is going to be customers on the left column, so customers per hour. We're going to come up with the classes for these. And this is going to be frequency. Alright? So, customers per hour.

We're going to use a lower class limit of 15. Right? Those are going to be all the numbers that go in the left column here. And then your class widths, the numbers that go, that sort of separate each consecutive lower class limit, are going to be five. So, in other words, the next one is going to be 20.

The next one is going to be 25, then thirty, thirty-five, and 40. If we go one more, well, if you look at your data set here, what we see is the highest number is 41. So if you go another class and get to 45, there's going to be no data values in this class. So you basically can stop right there at 40. All right?

So, we have a lower class limit of 40, and now we can figure out what the upper class limits are. Remember, you just take this number and subtract one. So, this is going to be 19. This is going to be 24, and then so on and so forth. This is gonna be 29.

Then this is gonna be 34, 39, and then 44. Alright. So now that we have all our classes and we have the upper and lower class limits, now we could just fill in the box because now we know exactly how many columns we need, or how many rows we need. Sorry. So this is going to be these classes over here.

And now we can just go ahead and stick to our normal method of just counting a bunch of data values. All right? Remember, so just mark off each one as you're counting it and just go one by one across the datasets instead of one by one trying to figure out what each of the classes are. It's just more efficient this way. So 15 goes in this category.

24 goes here. 30 goes here. 21 goes here. 27 goes here. 35 goes here.

32 goes here. 31 also goes there, 38 goes here, 41 goes here, 26 goes here, 33 goes in this column, 36 goes to this class, and then 40 finally goes in this class. Alright. So now we can just go ahead and add up all the tallies, and this is going to be 1, 2, 4, 3, 2.

If you add everything up, you should get 14. So this is one plus two is three, plus five, plus nine, plus twelve, and fourteen. Alright? So the total number here is 14 if you add up everything. Just a sanity check that we didn't miss something.

Alright? So this is basically our frequency distribution. We have the frequency versus the classes that we just made. Alright? Are we done?

Well, not quite here because, remember, the second thing that this problem asks us to do is now calculate the relative frequencies of each. Instead of just these numbers, one, two, four, three, two, we're actually going to calculate the percentage of each class relative to the whole data sets. So I'm going to add another column over here and extend the rows because now we're going to calculate a bunch of percentages. Alright? So remember, this is going to be relative frequencies.

Relative frequencies are just you take f divided by n and then multiply it by 100%. Okay? So, in other words, I'm going to take one and divide it by n, which is the total number of measurements. If you look through this dataset, or you just look through the problem, there were 14 numbers that we got. So n equals 14.

Alright? So in other words, one divided by 14 is going to be—it's gonna give you 0.071. Now sometimes these things get represented as percentages, but it's actually perfectly fine to leave them as decimals. In fact, that's most commonly what you're going to see. So, instead of what I'm gonna do here is two over 14, and if you keep on doing this, you're gonna get zero point, one four.

Now I'm not gonna write it out for all of them, but you would basically do the exact same thing. In fact, what's gonna happen is most of these things end up being twos or threes. You can kind of just figure it out. You don't have to recalculate. So this is gonna be 0.14.

This is also gonna be, this one also here is going to be 0.14. All right. If you do this, which is four over 14, this is going to end up being 0.28. And then this three over 14 is going to be 0.21.

Alright. So these are the relative frequencies of each, expressed as decimals, and you can convert them to percentages if you want. But the question actually asks us a final question: what percentage of the day is the cafe serving 30 or more customers per hour? So you have to really read into this question. And, in fact, that's a lot of what this course is going to be, sort of reading into what these questions are asking you.

It's what percentage is that they're serving 30 or more customers per hour. According to our classes, we actually have three classes in which we have frequencies that are 30 customers or above. So the whole idea here is to figure out this question, the percentage: we're basically going to—so 30 or more—30 or more means that we're actually going to be looking at all of this data. It's basically a minimum of 30 and then all the way up to your final class, which is from 40 to 44.

And what we can see here is that you're basically going to take each one of these numbers, that's percentages or decimals, and you're just going to add them all. So, in other words, 0.28 plus 0.21 plus 0.14. If you add this up, you're going to get 0.63, which, if you convert now, this to a percentage, this is going to be 63%. So in other words, 63% of the time, the cafe is serving more than 30 or more customers per hour. That's the percentage relative to the whole day.

Alright. So that is how to do this problem. Thanks for watching. Let me know if you have any questions, and let's move on.

How to Create Frequency Distributions

In the last couple of videos, we were introduced to frequency distributions. Remember, these are just tables that organize the frequency versus different classes of numbers or labels. And in those types of problems, up until now, we were always given exactly what those limits are and how many classes to use, like six or eight or something like that. But in some problems, like the one we're going to work out down here, you're only going to be given the number of classes, like eight or something like that, but you're not actually going to know what those class limits are, and you're going to have to go find them yourself. If that sounds kind of scary, don't worry about it because I'm going to show you a step-by-step process for how to get what those numbers are, and then you can go right back to just counting up the frequency for those classes.

Alright? I just want to warn you, there are slightly different ways of doing this. If there's a preferred method in your class, you should stick to that. But if not, these steps are going to be really helpful. So let's go ahead and take a look.

We're just going to jump right into this problem here. So again, we're working with the amount of time, in minutes, that students spend studying for their exam each week, and we're going to construct the frequency distribution and we're going to use eight classes. So let's go ahead and get started here. So I know I have to count up frequencies versus the classes. How do I actually get those?

Well, let's just go ahead and look at the first step here. The first step is you're going to have to calculate the class width. Remember, that's basically just the number where if you have two lower class limits, the class width is the number that separates the two. It's the number between two consecutive upper or lower class limits or something like that. Now, again, in most problems, you could actually just figure that out. But in this one, we're going to have to actually go calculate it. And here's a way we could do this. You're going to take the maximum minus the minimum number of the data set and divide it by the number of classes you're supposed to use. So let's go ahead and get started here. I'm going to use this formula to calculate this.

This is going to be max over minus min. So in other words, the maximum value of my data set is 115 and the minimum is, five. So, 115 minus five. Let me double let me, undo that. And then divided by the number of classes, which is eight.

When you plug this into your calculator, you should get 13.75. Alright? Now is this the right answer? Is this the class width? The problem with this number here is that, eventually, if I just start, you know, adding 13.75, there's going to be a bunch of weird decimals that pop up in these lower class limits.

So this is where you have a choice. So a lot of times what's going to happen is you're going to round up to either the nearest whole number, so we can round this up either to 14, or you can sometimes round this up to the nearest convenient number. A lot of times, these will be, like, the nearest five or the nearest 10 or something that kinda makes sense given the context of the problem that you're working with. In this case, because we're talking about minutes, another convenient number that we could have used is 15. Right?

So, basically, an hour is cut up into four fifteen-minute chunks. That's a pretty convenient number to use. Here's the thing. You actually could use either one of these. It's perfectly fine.

So there's really no one right way to do this. It's just that I'm going to use 15 in this example because it makes the numbers a little bit easier. So this is 15. That's going to be my class width. So whatever these two numbers are, they're going to be separated by 15.

So that's the first step. We're done with the class width. So now let's move on to the second one. We're going to actually figure out what those lower class limits are. Remember, the lower class limits are basically just all the numbers that go on the left side of those intervals, and I'm going to have to go find them.

So here's what you can do here. The first lower is the most important one. And basically, what you're going to do is pick a number that is either less than or equal to whatever the data minimum is. So in other words, it's going to be a number that's less than or equal to this five. A lot of times, you're just going to see that data minimum just gets used as the lower class limit, and that's exactly what we're going to do here. It's perfectly fine to use five. You may see something like zero sometimes, but five is perfectly fine. So let's go ahead and do, and and, that's five. That's the first lower limits or lower class limits. How do I find the next ones?

Well, to find the next lowers, what you're going to do is take the previous lower and then just add the class width. Remember, this 15 over here is going to be the difference between each consecutive lower class limit. So in other words, this is five, and the next one's going to be 20, and the next one's going to be 35, then 50, then 65, then 80, then 95, and then finally, a 10. Alright? So those are all your lower class limits.

Cool. So now that we're done with the lower class limits, we're going to have to find the upper class limits, which are basically just the numbers that go on the right side of those intervals. How do we find those? Well, basically, what we're going to do here is to find the upper class limits is to find the first upper. You're just going to take the second lower and then subtract one from it.

Here's why. So if I used 20, then if again, if I get a data value that's 20, it's going to go into two classes and I'm going to double count something. What I have to use is take the first the second, the second lower and subtract one. So this is going to be 19. Alright?

And then what happens is I could just do the same exact thing for this one. This is going to be 34. It's probably going to be the next lower minus one. This is going to be 49 and then 64 and then 79 and then so on and so forth. You can kind of see the pattern that's going on here.

It's going to be a hundred and nine. And then finally, this is going to be a 24, again, because everything should be separated by 15. Alright? So these are going to be your upper class limits. All right?

Again, really important here, your class widths are 15. The range between each one of these things is 14. The class widths are 15. All right? Now we're actually done.

We have what all of our classes are for this frequency distribution. And the last step after you're done is now you can just go ahead and find the frequency for each and then tally everything up in its appropriate class. This is exactly what we were just doing before. I'm going to fly through this. We've already done this before.

I'm going to include all the data values that are between fifteen and nineteen oh, sorry. Five and nineteen. And that's going to be one, two. So that's going to be over here. Then I've got, I've got 20 through 24.

This is going to be one, two, and that's it. That's two over here. And I've got 35 through 49. We've got +1, 234. That's +1, 234.

Then I've got 50 to 64. This is going to be +1, 23456. This is going to be +1, 23456. And that's over here. Then I've got 65 through 79.

That's 12. That's two over here. Then I've got 8085. That's going to be, 2 over here. I've got 95 to +1.

I only have 1. And then 110 to +1 is only one. So in other words, the frequencies are two, two, four, six. Then I've got two, two, one, and one. And that's the frequency distribution for this data set.

That last part is a little bit tedious, but you know how to do this. But this is basically what your answers are for the frequency distribution. So that's it. That's how to do these types of problems where you have to make your own classes. Let's go ahead and take a look at some practice.

Thanks for watching.

How to Create Frequency Distributions Example 2

Welcome back, everyone. So we're going to work out this problem together here. This problem shows us the datasets that show the sales in dollars of 15 sales representatives at some company. And what we're going to do in this problem is construct a frequency distribution. What they're not telling in this problem is what the lower class limits are and the class widths.

They're only just telling us the number of classes to use, which is five. And so, in the absence of all that information, we are going to have to use these steps to figure out what those class limits are and the class widths. Okay? So we are going to stick to the steps here. Remember that eventually, I'm just going to end up with a frequency distribution.

I'm going to go ahead and just draw a little table like this. I've got sales over here versus frequency. The first step, remember, is that you have to calculate the class width. I'm going to do that over here. So this is going to be my class width.

I'm going to call this CW just to, you know, abbreviate. But essentially, what we're going to do is you're going to take the maximum of the data values minus the minimum and divide it by the number of classes. We have all those those information. Right? We just have to look at the dataset and figure out what's the max and the min.

Alright. So if you look at the numbers over here, I see some elevens, some tens, and then I only see one number that has a 12 in it. So it's twelve twenty three. Alright. So, the CW, the class width, is going to be twelve twenty-three minus the minimum number.

And if you look again through this dataset over here, you see a couple of nines, a couple of tens. I see some eights, eight forty three, but the number that's the lowest is eight nineteen. There's nothing lower than that. So, twelve twenty-three minus eight nineteen divided by the number of classes, which, again, we're using five classes over here. So, this five is directly, why you use five there.

Okay? When you work this out, what you're going to get here is 80.8. So remember, it's kind of weird to have class widths that involve decimals because then your class widths are going to get all really weird. They're going to have a bunch of decimals in there. So what a lot of times you're going to do is you're just going to round up to the nearest whole number.

In this case, because you have this data that's all over the the place and deals with dollars, there's really not like a convenient number to use. So I'm going to just go ahead and round it to the nearest integer, and I'm just going to use 81. Okay? That's a perfectly valid class width to use. Alright?

So now what happens is, remember, whatever these lower class limits are, I know that the spacing between them is going to be 81. Okay? So now I have to go to the second step, which is figuring out what those lower and upper class limits are. So to do that, so what I'm going to do here so remember, this is the left numbers. The first lower is just going to be a number that is less than or equal to whatever the data minimum is here.

And in this case, because I have numbers that go from 819 to 1223, I'm just actually going to use whatever the minimum is. So in this case, I'm just going to use 819, and that's it. Alright? So that's my first lower. The next lower is going to be this number plus the class width.

So in other words, $819 plus a class width of 81. So if I add this over here, I just get 900. Alright? And if you go ahead and do this for the rest of the data values, you're going to get 981, and you're going to get 1062, and then you're going to get 1143. If you go one more, which you're actually going to get here is 1224, and this number is actually bigger than the biggest data value that you have here.

So in other words, you're going to get a frequency of zero, and you might as well just not even have it. Okay? So these are going to be my classes over here. Alright? So I've got 819, 900, and I got the rest of these numbers here.

Those are the lower class limits. Now I have to figure out the upper class limits. And remember, to do this, all you have to do is find the uppers. You just find the second lower and subtract one. All right.

So this number can't be 900. It has to be one number less than that. So $899. This is going to be 980. This is going to be $1061.

This is going to be $1142. And then what happens here is this is going to be 1223 because this is going to be 1224. Alright? So now that you have your upper and lower class limits, now you actually have defined what your classes are. Now we can go ahead and just count up all the you know, and tally up all the frequencies.

Okay? So this 1223 is going to belong over here. And by the way, now you can go ahead and close the box and also fill out the rest of the rows because now you know how many classes you have. Alright? So this is going to be one.

Second number over here, 1136, is going to fall over here. 819 is going to fall over here. 1089 is going to fall over here. 1011 goes into this class over here. 997 goes into this class.

973 goes into this one. 1025 goes here. 1017 also goes here. 1118 goes here. Then we've got 988 which goes here.

Then we've got 843, which goes here. 1196, which goes here. 1011, which goes here. And then we've got 942, which goes here. Alright?

So in other words, I can just tally up everything and, essentially, I'll get 22. This is going to be +1, 2345. This is going to be 4, and this is going to be 2. Alright? So, hopefully, if you've done this right, you should get 15 numbers.

We've got two, that's 491315. Alright. So that is going to be my frequency distribution. Alright. So that's basically my answer.

I've got my frequency distribution for the classes and that's it for this one. So let me know if you have any questions and I'll see you in the next lecture.

An economist is analyzing the monthly unemployment rates (as a %) across different cities. The lowest was 16% and the highest was 71%. Without constructing a table, find the class width if this data is divided into 7 classes. Then write the lower and upper class limits for each class.

• A. Class width = 7; lower class limits = 16, 23, 30, 37, 44, 51, 58, 65; Upper class limits = 24, 32, 40, 48, 56, 64, 72
• B. Class width = 7; lower class limits = 16, 24, 32, 40, 48, 56, 64; Upper class limits = 24, 32, 40, 48, 56, 64, 72
• C. Class width = 8; lower class limits = 16, 24, 32, 40, 48, 56, 64; Upper class limits = 23, 31, 39, 47, 55, 63, 71
• D. Class width = 8; lower class limits = 16, 24, 32, 40, 48, 56, 64; Upper class limits = 24, 32, 40, 48, 56, 64, 72

Answer: (C)

Here’s what students ask on this topic:

What is a frequency distribution and why is it important in data analysis?

A frequency distribution is a table that organizes data into classes and shows the frequency of measurements within those classes. It is crucial in data analysis because it helps to visualize and understand the distribution of data, making it easier to identify patterns, trends, and outliers. By organizing data into classes, frequency distributions simplify complex data sets, allowing for more effective analysis and interpretation. They are foundational for creating histograms and other graphical representations, which are essential tools for communicating data insights in business and research contexts.

How do you calculate the class width in a frequency distribution?

To calculate the class width in a frequency distribution, use the formula: $\frac{(\max -\min )}{\mathrm{number\; of\; classes}}$. This formula involves subtracting the minimum value from the maximum value in the data set and dividing the result by the number of classes you wish to use. The class width determines the range of values within each class, ensuring that the classes are evenly spaced and do not overlap. It is important to round the class width to a convenient number to avoid decimals, which can complicate the distribution process.

What are lower and upper class limits in a frequency distribution?

Lower and upper class limits are the boundaries that define the range of values within each class in a frequency distribution. The lower class limit is the smallest value that can be included in a class, while the upper class limit is the largest value that can be included. These limits are crucial for ensuring that classes do not overlap, which would lead to double counting of data points. To find the lower class limits, start with a number less than or equal to the minimum data value and add the class width to determine subsequent limits. The upper class limits are found by subtracting one from the next lower class limit.

How do you calculate relative frequencies in a frequency distribution?

Relative frequencies in a frequency distribution are calculated by dividing the frequency of each class by the total number of measurements, denoted as $n$. The formula is: $\frac{f}{n}$, where $f$ is the frequency of the class. This result is then multiplied by 100 to convert it into a percentage. Relative frequencies provide a clearer understanding of the data distribution by showing the proportion of the total data that falls within each class, making it easier to compare different classes and identify significant patterns.

What is the difference between frequency distribution and relative frequency distribution?

A frequency distribution is a table that shows the number of measurements within each class, while a relative frequency distribution expresses these frequencies as percentages of the total number of measurements. The relative frequency distribution provides a more intuitive understanding of the data by showing the proportion of the total data that falls within each class. This is particularly useful for comparing classes and understanding the overall distribution of data. While frequency distribution focuses on raw counts, relative frequency distribution emphasizes the relative importance of each class in the context of the entire data set.