Histograms
Site: | Mountain Heights Academy OpenCourseWare |
Course: | Mathematics Essentials Q3 v2013 |
Book: | LESSON: Histograms |
Printed by: | Guest user |
Date: | Wednesday, 10 February 2016, 10:37 AM |
You have been learning all about the different ways to display data. In this lesson, you will learn about frequency tables and histograms. Let’s start by looking at frequency tables.
What is a frequency table?
A frequency table is another way of summarizing data. A frequency table depicts the number of times a data value occurs.
A frequency table is created by making a table with three separate columns. One column is designated for intervals. The amount of intervals is determined by the range in data values. If the range in data values is not that great, the intervals will be small. If the range in data values is great, the intervals will be larger. It is important that the intervals are of equal size and do not overlap.
Another column is created for tallied results. This is where you tally the number of times you see a data value from each interval.
In the last column, add the tally marks to determine the frequency results.
Let’s look at how we can apply this information with an example.
Example 1
Twenty people were asked to state the number of hours they sleep each night. The results of the survey are listed below. Create a frequency table to display the data.
7, 8, 6, 9, 10, 12, 5, 7, 8, 9, 10, 11, 12, 7, 6, 7, 8, 10, 11, 9
Step 1: Make a table with three separate columns.
In this case, there is not a wide range in data values, therefore the intervals will be displayed by ones.
Step 2: Looking at the data, tally the number of times a data value occurs.
Step 3: Add the tally marks to record the frequency.
Number of Hours Slept | Tally | Frequency |
5 | I | 1 |
6 | I I | 2 |
7 | I I I I | 4 |
8 | I I I | 3 |
9 | I I I | 3 |
10 | I I I | 3 |
11 | I I | 2 |
12 | I I | 2 |
Now you can see how arranging the data in this way makes it much easier to follow.
Example 2
The data below depicts the amount of time (in minutes) 20 middle school students spent on the computer each day. Arrange the data on a frequency table.
10, 32, 8, 55, 5, 0, 30, 20, 25, 45, 40, 60, 45, 15, 5, 56, 47, 12, 15, 20
Step 1: Make a table with three separate columns.
In this case, there is not a wide range in data values, therefore the intervals will be displayed by ones.
Step 2: Looking at the data, tally the number of times a data value occurs.
Step 3: Add the tally marks to record the frequency.
Number of Minutes on the Computer | Tally | Frequency |
0 – 5 | I I I | 3 |
6 – 10 | I I | 2 |
11 – 15 | I I I | 3 |
16 – 20 | I I | 2 |
21 – 25 | I | 1 |
26 – 30 | I | 1 |
31 – 35 | I | 1 |
36 – 40 | I | 1 |
41 – 45 | I I | 2 |
46 – 50 | I | 1 |
51 – 55 | I | 1 |
56 – 60 | I I | 2 |
Once again, the tally marks in the frequency table can give you a clear picture of the data.
Frequency tables are a great way to record and organize data. Once you have created a frequency table, we can make a histogram to present a visual display of the information in the frequency table.
What is a histogram?
A histogram shows the frequency of data values on a graph. Like a frequency table, data is grouped in intervals of equal size that do not overlap. Like a bar graph, the height of each bar depicts the frequency of the data values. A histogram differs from a bar graph in that the vertical columns are drawn with no space in between them.
Now let’s look at creating a histogram from a frequency table.
Example 1
Create a histogram using the results on the frequency table below.
Number of Hours Slept | Tally | Frequency |
5 | I | 1 |
6 | I I | 2 |
7 | I I I I | 4 |
8 | I I I | 3 |
9 | I I I | 3 |
10 | I I I | 3 |
11 | I I | 2 |
12 | I I | 2 |
To create a histogram:
1. Draw the horizontal and vertical axis.
2. Give the graph the title “Hours Slept Each Night.”
3. Label the horizontal axis “Hours.” List the intervals across the horizontal axis.
4. Label the vertical axis “Frequency.” Since the range in frequencies is not that great, label the axis by ones.
5. For each interval on the horizontal access, draw a vertical column to the appropriate frequency value. On a histogram, there is no space in between vertical columns.
Example 2
Create a histogram to display the data on the frequency table below.
Number of Minutes on the Computer | Tally | Frequency |
0 – 5 | I I I | 3 |
6 – 10 | I I | 2 |
11 – 15 | I I I | 3 |
16 – 20 | I I | 2 |
21 – 25 | I | 1 |
26 – 30 | I | 1 |
31 – 35 | I | 1 |
36 – 40 | I | 1 |
41 – 45 | I I | 2 |
46 – 50 | I | 1 |
51 – 55 | I | 1 |
56 – 60 | I I | 2 |
To create a histogram:
1. Draw the horizontal and vertical axis.
2. Give the graph the title “Minutes Spent on the Computer.”
3. Label the horizontal axis “Minutes.” List the intervals across the horizontal axis.
4. Title the vertical axis “Frequency.” Label the axis by halves (0.5).
5. For each interval on the horizontal access, draw a vertical column to the appropriate frequency value. Recall that on a histogram, there are no spaces in between vertical columns.
Sometimes, you will be given a set of data that you will need to organize. This data will be unorganized. To work with it, you will have to organize it by creating a frequency table. Then you can use that frequency table to create a histogram.
Let’s look at an example.
Example
Fifteen people were asked to state the number of hours they exercise in a seven day period. The results of the survey are listed below. Make a frequency table and histogram to display the data.
8, 2, 4, 7.5, 10, 11, 5, 6, 8, 12, 11, 9, 6.5, 10.5, 13
First arrange the data on a frequency table. Recall that a table with three columns needs to be drawn: one for intervals, one for tallied results, and another for frequency results. The range in values for this set of data is eleven. Therefore, data will be tallied in intervals of three.
Hours of Exercise | Tally | Frequency |
0 – 2 | I | 1 |
3 – 5 | I I | 2 |
6 – 8 | I I I I I | 5 |
9 – 11 | I I I I I | 5 |
12 – 14 | I I | 2 |
Next, the data needs to be displayed on a histogram. Recall that a horizontal and vertical axis needs to be drawn. List the intervals across the horizontal axis. Name this axis “Hours of Exercise.” Label the vertical axis by ones. Title the vertical axis “Frequency.” For each set of intervals, draw vertical columns the appropriate frequency. Color in the vertical columns and ensure that no space is between them. Title the graph “Hours of Exercise.”
Now let’s make some conclusions based on the information displayed in the histogram.
Looking at the histogram above, you can that equal numbers of people reported that they exercise between six and eight and nine and eleven hours each week. Two people stated that they exercise between three and five hours per week. Two people reported that they exercise between twelve and fourteen hours per week. Zero to two is the hours with the least frequency.