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# Histograms

If you want to analyze numerical data that has been sorted into equal intervals, histograms can be helpful visualization tools. A histogram is a special type of bar graph displaying data in equal intervals where adjacent bars touch one another but do not overlap. Displaying data this way makes it easier to see its frequency distribution and how many pieces of data are in each interval. Here is an example:

In this article, we'll explore how to draw histograms and interpret the data they present. Let's get started!

## Drawing histograms: A step-by-step guide

The first step in drawing a histogram is collecting the raw data we want to depict. For example, the frequency table below shows the admission cost of various theme parks:

Costs (\$) Frequency
7-14 ///// 5
15-22 /////// 7
23-30 //// 4
31-38 0
39-46 // 2

Okay, now we have our data. Next, draw and label a horizontal and vertical axis to serve as the base of our histogram. The intervals from the frequency table above (under the Costs section) should appear on the horizontal axis, with an interval of 2 units on the vertical axis (2, 4, 6, 8, etc.). Be sure to include a title at the top of the histogram and identify what is represented by each axis!

The last step is when we actually draw our bars. Our scale doesn't specifically list any odd numbers, so draw carefully so your bars are halfway between two even numbers instead of being too short or too long. Your bars should also be touching each other, except for the 31-38 interval where we're intentionally graphing zero. The final product should look something like this:

## Interpreting histograms

Drawing histograms is great, but some problems will give us a histogram and ask questions about the data. For instance, the following histogram reveals the highest level of education completed by the employees of a local store:

If we're asked how many of these employees have at least a high school diploma, we would need to add the HS bar to all of the other bars representing further education (associate, bachelor, and graduate). Using the histogram above, we can see that 50 employees finished high school, 20 more earned an associate's degree, 30 more earned a bachelor's, and 10 completed a graduate degree. That means that we're adding $50+20+30+10=110$ employees.

When working with problems like this, it's important to watch out for keywords such as "at least," "in all," and "except" so you're focusing on the right information. For example, the answer would be 200 if we were asked about how many employees the store had "in all" and 190 if we were looking for all employees "except" those with graduate degrees.

## Histograms practice questions

a. Draw a histogram based on the following data:

How Long Middle School Students Spend on Homework Per Night

Time(min) Frequency
0-30 5
31-60 25
61-90 20
91-120 10

We should put the time intervals on the horizontal axis and the frequencies on the vertical axis, using a scale of 5 on the latter since all of our numbers are divisible by 5. Our title can simply be the title of the chart (How Long Middle School Students Spend on Homework Per Night), and we need to specify the time (minutes) for the horizontal axis and the number of students for the vertical axis. Finally, carefully draw bars to correctly indicate the number of students falling into each range.

b. Using the histogram you drew above, how many students spend more than an hour on their homework?

An hour is 60 minutes, so we're looking for the number of students who spend at least 61 minutes on their homework. Two of the intervals above meet that criterion: 61-90 and 91-120. Therefore, we can add each intervals' frequency $\left(20+10\right)$ to get an answer of 30 students.
c. Consider the following histogram:

If Ashton outscored 73 of his classmates on this exam, what could his score have been?

If Ashton outscored 73 of his peers, he needs to have outscored the entire 201-300 interval (5 students), 301-400 interval (18 students), and 401-500 interval (30 students). $5+18+30=53$ students, which means Ashton had to have outscored 20 students in the 501-600 interval as well. However, Ashton cannot have outscored all 40 students in that interval because we know he outscored a total of 73 students. That means Ashton's score must be in the 501-600 interval, and likely toward the midpoint since he outscored 20 of the 40 students in that range. Any score from 501-600 could be correct, but the best guess based on the information we have is 540-560.

Bar Graphs

## Get help with histograms today

Histograms help students interpret data, one of the most important skills to develop in math class. If the student in your life is drawing sloppy graphs they cannot interpret or getting confused by intervals and frequencies, an experienced math tutor can help them identify the root cause of their learning obstacles and address them directly. A 1-on-1 tutor can also provide a private learning environment where it's okay to make mistakes to facilitate the educational process. Contact the friendly Educational Directors at Varsity Tutors today to learn more about the benefits of private instruction and to get signed up.

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