Histograms

Today, we are going to talk about Histograms.
A histogram shows the frequency of an occurrence. A frequency is how often something occurs. A histogram will show a comparison of these frequencies. The height of each bar represents the number of data values in that interval. An interval is a set of numbers which lie between two distinct numbers, those two numbers are also included in the set.
So when we look at creating a histogram, the steps are to 1) collect your date; 2) organize your date in a frequency table. Make sure categories are equal, 3) draw you graph and 4) Count to make sure you have included data points.
So in Step 1 you see our data. It?s very organized. It?s in a nice table.
So in step 2, we have to create a frequency table. Now with frequency tables, it's similar to bar graphs but on the bottom is an interval. So we can see here, we have the interval 0 to 4, 5 to 8, 10 to 14, 15 to 19, and 20 to 24.
Now it says, the categories must be equal. So when we look at 0 to 4, it count that to 0, 1, 2, 3, 4. There are 5 data points that fall in that frequency.
Then 5, 6, 7, 8, 9. Five data points in that frequency.
10, 11, 12, 13, 14. There are five data points in that interval.
15, 16, 17, 18, 19. Five data points in that interval.
20, 21, 22, 23, 24. five data points in that interval.
So now that we see that our categories are equal, we can sit there and calculate it.
So when I look at 0 to 4, I kind of look at it and say there's 1. Then I go look at it. There is only data point that is 0 to 4. So there is a 1 in the frequency table.
Then I look at 5 to 9, and I look over here. 5 and 8 are there. There are 2 data points in the frequency table.
Then I look at 10 to 14. 14, 10 and that is it. So there are 2 data points in the frequency table.
How about 15 to 19. So there's 17, 15, 19, 18, 19, which is 5 data points.
And then 20 to 24, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. So perfect, we have counted the data points and we have the frequency table set up.
So when you look at graphing a histogram, it looks a lot like a bar graph but there's a couple of key difference. One there is an interval at the bottom. And once again, you have to make sure that those intervals are equal. Notice that at the bottom, bars also touch. Because the intervals connect, the bars touch.
But other than that, you create it like a bar graph. From 0 to 4, it goes up one. 5 to 9, it goes up 2. 10 to 14, it goes up 2. 15 to 19, it goes up 5. 20 to 24, it goes up 10.
Remember that your bars must touch and that intervals must be equal.
So now let's look at a couple examples. We see a bar graph here with sixth grade students heights. So we see, 5055, 55-60, 61-70, and 71-75. And we can see the frequency on this side.
So the first question is How many students are in the 71 - 75 inch group?
So if we look right here. And we see across and find 10 students are.
So here, what percent of students are at least 66 inches tall? So when we look at this, that includes 66 and anything above it.
So we are looking at these 2 groups. So when I look here, there are 40, and when I look here, there are 10. Combined, so the Part there are 66 inches taller out of the total is 50. To find the total, we list down all the frequency and we add them up. So when you add up all of these numbers, you get a total of 150.
So quick recap on that. All students that are 66 inches and taller are 50 out of 150 for Part over All. And then x over a hundred for percent. Cross multiply and divide, that gives you 33%.
So let's look at this one over here.
The histogram shows the math quiz sores from a class. A score that is greater than or equal to 70 is passing.
So the question is how many students passed the math quiz. So when we look at the graph here, it's this group, this group and this group. All of these are pass 70. If we total that up, we have 4 here, 3 here and 6 here. And if you add that up, that gives you 13.
How many students failed the math quiz? Well if all of these passed, the remainder failed. So we will draw a little line to separate it. This is the pass side, this is the fail side.
So when we look here, we have 4, and 2 here. A total of 6.
Then the final question, what percent of students scored greater or equal to 70 on the quiz? So now were are asked for percentage. So when we look at the problem as part over whole, it's always over 100. So we can go ahead and put our 100.
So now when we look at part, we are looking for students that scored 70 or greater. We already figured that out. 4 plus 6 plu 3 is 13. And then the total number of students we have all of them. So 13 plus 4 plus 2 equals 19. And then x is your percentage. If you cross multiply and divide, that will give you 68%. So now let's look at a common mistake. A lot of times you will be asked an error. Which graph is correct and which graph is incorrect. The histogram show the number of minutes customers were put on hold before talking to a representative. What is the error? So the first thing I want you to look at is are the bars touching and are the intervals correct? So when you are looking this, a lot of people will say, the intervals are good, the bars are connecting. And then they miss key details. So when you look at this 1 2 3 4, there are 4 numbers. 5 6,there are 2 numbers. 7 8 9, there are 3 numbers. 10 11 12 13 14 15, there are 6 numbers. So we can clearly see that these are not equal. Let's take a look at this. Different ages are listed below, what is wrong with the frequency table. So when you are looking at it as a table, the two things you will see is either the intervals will be off, or the data will not be correct. So I always like to check the data first. So 0 to 20, so I have 1 2 3, and that looks like it. Now, 21 to 40, 1 2 3 4. That looks good. 41 to 60, 1 2 3. That looks good. 61 to 80, 1 2 3 4. So that one looks good. So we see that our mistake is right here. The frequency table showed 4 when it should have shown 3. So key things to remember with histograms. Make sure that you account for all of your data, make sure intervals are equal ( the best way to do it is to say them out loud),