Histogramas

A histogram is a graph used to represent the frequency of continuous data.

It looks similar to a bar chart/graph but a histogram represents quantitative data (information you can measure with numbers: height, shoe size, etc.), whereas a bar graph represents qualitative data (information you can't measure with numbers: eye color, days of the week, etc.).

If you want a review on quantitative vs. qualitative data, check out this chemistry lesson.

An example would be:

The $x$-axis or vertical axis is the bin or interval. The $y$-axis is the frequency. Each bar tells you the frequency of (or how often) the bin/interval appears.

How to Make a Histogram

We usually construct histograms based off of frequency tables.

We put each interval along the $x$-axis and raise the bar up along the $y$-axis to denote the frequency.

If we have the frequency table: $$\begin{array}{|c|c|} \hline \text{Interval}&\text{Frequency}\\ \hline 0-3&4\\ 3-6&3\\ 6-9&2\\ 9-12&1\\ \hline \end{array}$$ we can start by making the intervals.

Then, we raise the interval up to the frequency that corresponds with it.

Just like with frequency tables, you can make your bin width as large or small as you want. It's better to choose a healthy middle point though.

Reading the Graph

Other than constructing a histogram, you will also be asked to gather information from the graph.

Suppose we have the following histogram:

How many data points are there?

We can always figure out how many data points there are by adding all the frequencies together. $$6+3+8=17$$

What fraction of the data is between $5-10$?

To find this, we can take the frequency of the interval we want, $5-10$, and place it over the total frequency. $$\frac{3}{6+3+8}=\frac{3}{17}$$

How many data points are below $10$?

Just add up the frequencies of the intervals that are less than $10$. These would be $0-5$ and $5-10$.

$$6+3=9$$

Finally, there is some data that we can't find when using a histogram. They are a way to quickly visualize data. They are easier to read but the trade off is they can be less accurate.

For example, if you were asked how many $3$'s were in the data using the above histogram, we couldn't answer. We know the frequency for the values in the interval $0-5$, but we don't know how many were exactly $3$.

(Video) Frequency Tables and Histograms

by Anywhere Math

This video by Anywhere Math works through frequency tables and histograms.

In the previous explanation on frequency tables, we constructed a table of shoe sizes in a classroom. $$\begin{array}{|c|c|} \hline \text{Sizes}&\text{Frequency}\\ \hline 4-6&4\\ 6-8&5\\ 8-10&2\\ 10-12&1\\ \hline \end{array}$$ Now, we're going to look at how to turn this into a histogram.

$\large{\text{Summary}}$

Making a Histogram

A histogram is basically just a bar graph. The $x$-axis will be your intervals and the $y$-axis will be the frequencies of those intervals.

Start by making the outline of the graph.

Next, we make a bar for each interval; its height will be the frequency.

Using a Histogram

In this example, we look at interpreting information. Let's look at these Daytona 500 winning speeds:

The first question asked is:

Which interval contains the most data values?

This question is just asking us which interval is the highest. The purple bar goes up to $19$, so the answer is $150-159$.

The next question is:

How many of the winning speeds are $<140$ mph?

In the interval $120-129$ there were $2$. In $130-139$ there were $4$. Therefore, the total number under $140$ mph is $2+4=6$.

The final question is:

How many of the winning speeds are at least $160$ mph?

In the interval $160-169$ there were $7$. In $170-179$ there were $5$. Therefore, the total number at least $160$ mph is $7+5=12$.

What is a Histogram?

A histogram is a type of graph that can be used to show frequencies. They show the distribution of your data, or how the data is spread out. They use the values in a frequency table.

Like frequency tables, histograms can use ranges instead of single values.

Here's an example of a histogram for different terminals at an airport. It shows the frequency of arrivals per minute. Try imagining that data in the graph as data in a frequency table. It's the same exact information, just shown in a different way. Math is cool like that.

If you have a big range of values in your data set, consider dividing the data into ranges or intervals of the same size.

Imagine that you had a lot of extra time on your hands and you decided to measure the heights of all of the black cherry trees in Math Town. Here are your data:

It's always a good idea to take your raw data and put it in a frequency table before you try to make a histogram.

Although your data is already in order from largest to smallest, if you're given a messier dataset with values that are all out of order, STOP. Re-write your data so that the values are in order before you make your frequency table. You'll thank yourself later.

$$\begin{array}{|c|c|} \hline \text{Height Range}&\text{Frequency}\\ \hline 60-64&3\\ 65-69&3\\ 70-74&8\\ 75-79&10\\ 80-84&5\\ 85-89&2\\ \hline \end{array}$$

Note: In this explanation on frequency tables, Alex talks about how when we are working with decimals, the leftmost number in the interval is inclusive, and the rightmost number is exclusive. For this example, ignore that rule, since our decimal data points fit nicely within our intervals.

$60-64$ looks like a range of $4$ values, right? It's not. There are $5$ values in there: $60, 61, 62, 63, 64$. So each of our intervals contains $5$ values, as planned.

Now that your data are prepped, follow these steps to make your histogram:

1. Put frequencies on the vertical axis.
2. Put the lower value of each interval (e.g., "$60$" for the first interval, "$65$" for the second interval, etc.) on the horizontal axis.
3. Draw a bar that spans each interval and reaches the height of the correct frequency.