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# 7.9: Applications of Histograms

Difficulty Level: Basic Created by: CK-12
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You're collecting data on the amount of minutes that users stay on your website. You have the results for 1,000 users and you want to organize these results in intervals of 3 minutes. You've noticed that the shortest amount of time a user stayed on the site was 30 seconds and the longest amount of time was 15.5 minutes. How would you organize this data?

### Watch This

First watch this video to learn about creating histograms with calculators.

Then watch this video to see some examples.

### Guidance

Technology can also be used to plot a histogram. The TI-83 can be used to create a histogram by using STAT and STAT PLOT on the calculator. The TRACE feature can then be used to find the number of values for each bin, and then a frequency distribution table can be constructed from these values. When constructing a histogram using a TI-83 calculator, it's important that your settings are correct on the WINDOW screen. The setting that determines the bin size is the Xscl setting, so you'll want to make sure that the number for Xscl is correct before creating your histogram.

#### Example A

Scientists have invented a new dietary supplement that is supposed to increase the weight of a piglet within its first 3 months of growth. Farmer John fed this supplement to his stock of piglets, and at the end of 3 months, he recorded the weights of 50 randomly selected piglets.

The following table is the recorded weights (in pounds) of the 50 selected piglets:

120111651101147211610511911493 113 991181089710795 1137584 12010210484 9712169 10010110711877 1051097889 68 74 10387  67 79 90 1099410696 92 88\begin{align*}& 120 \quad 111 \quad 65 \quad 110 \quad 114 \quad 72 \quad 116 \quad 105 \quad 119 \quad 114\\ & 93 \quad \ 113 \quad \ 99 \quad 118 \quad 108 \quad 97 \quad 107 \quad 95 \quad \ 113 \quad 75\\ & 84 \quad \ 120 \quad 102 \quad 104 \quad 84 \quad \ 97 \quad 121 \quad 69 \quad \ 100 \quad 101\\ & 107 \quad 118 \quad 77 \quad \ 105 \quad 109 \quad 78 \quad 89 \quad \ 68 \quad \ 74 \quad \ 103\\ & 87 \quad \ \ 67 \quad \ 79 \quad \ 90 \quad \ 109 \quad 94 \quad 106 \quad 96 \quad \ 92 \quad \ 88\end{align*}

Using the above data set and your calculator, construct a histogram to represent the data. Use a bin size of 10.

The histogram can be created as follows:

#### Example B

Use the TRACE feature to get information about the data in each bar of the histogram that you created in Example A.

First, press TRACE\begin{align*}\boxed{\text{TRACE}}\end{align*}.

The TRACE feature tells you that in the first bin, which is [60-70), there are 4 values.

The TRACE feature tells you that in the second bin, which is [70-80), there are 6 values.

To advance to the next bin, or bar, of the histogram, use the cursor and move to the right. The numbers of values for the subsequent bins are 5, 9, 13, 10, and 3, respectively. The information obtained by using the TRACE feature will enable you to create a frequency table and to draw the histogram on paper.

#### Example C

The following data represents the results of a test taken by a group of students:

955670835966885250776980547568785164556774577353\begin{align*}& 95 \quad 56 \quad 70 \quad 83 \quad 59 \quad 66 \quad 88 \quad 52 \quad 50 \quad 77 \quad 69 \quad 80\\ & 54 \quad 75 \quad 68 \quad 78 \quad 51 \quad 64 \quad 55 \quad 67 \quad 74 \quad 57 \quad 73 \quad 53\end{align*}

Display the results in a histogram using technology and construct a frequency distribution table using a bin size of 10.

The histogram and the numbers of values for the bins are as shown below:

Now the frequency distribution table can be constructed as follows:

Bin Frequency
[4050)\begin{align*}[40-50)\end{align*} 0
[5060)\begin{align*}[50-60)\end{align*} 9
[6070)\begin{align*}[60-70)\end{align*} 5
[7080)\begin{align*}[70-80)\end{align*} 6
[8090)\begin{align*}[80-90)\end{align*} 3
[90100)\begin{align*}[90-100)\end{align*} 1

### Vocabulary

The TRACE feature can be used to find the number of values for each bin of a histogram on a TI-83 calculator. The setting that determines the bin size for a histogram on the TI-83 calculator is the Xscl setting on the WINDOW screen.

### Guided Practice

The following table is the the number of minutes spent practicing per day by a professional violinist for each of the last 80 days:

866710512298838410110012093 99 78101928995102 11010084 778210788 9394103 127111120100105 959991103 79 89 8411296107 1068592105 91 90 10489114106 991018192 88 100 977810386 1071019877 69 89 909810294 103 80 9910910193 86\begin{align*}& 86 \quad 67 \quad 105 \quad 122 \quad 98 \quad 83 \quad 84 \quad 101 \quad 100 \quad 120\\ & 93 \quad \ 99 \quad \ 78 \quad 101 \quad 92 \quad 89 \quad 95 \quad 102 \quad \ 110 \quad 100\\ & 84 \quad \ 77 \quad 82 \quad 107 \quad 88 \quad \ 93 \quad 94 \quad 103 \quad \ 127 \quad 111\\ & 120 \quad 100 \quad 105 \quad \ 95 \quad 99 \quad 91 \quad 103 \quad \ 79 \quad \ 89 \quad \ 84\\ & 112 \quad 96 \quad 107 \quad \ 106 \quad 85 \quad 92 \quad 105 \quad \ 91 \quad \ 90 \quad \ 104\\ & 89 \quad 114 \quad 106 \quad \ 99 \quad 101 \quad 81 \quad 92 \quad \ 88 \quad \ 100 \quad \ 97\\ & 78 \quad 103 \quad 86 \quad \ 107 \quad 101 \quad 98 \quad 77 \quad \ 69 \quad \ 89 \quad \ 90\\ & 98 \quad 102 \quad 94 \quad \ 103 \quad \ 80 \quad \ 99 \quad 109 \quad 101 \quad 93 \quad \ 86\end{align*}

Display the results in a histogram using technology and construct a frequency distribution table using a bin size of 10.

First, press STAT\begin{align*}\boxed{\text{STAT}}\end{align*}, choose Edit on the EDIT menu, and enter the data values into L1. Since there are so many data values, it's a good idea to then use 1-Var Stats to find the minimum and the maximum data values so that the correct settings for the histogram can be entered in the WINDOW screen. To access 1-Var Stats, after returning to the main screen, press STAT\begin{align*}\boxed{\text{STAT}}\end{align*} and choose 1-Var Stats on the CALC menu. After pressing ENTER\begin{align*}\boxed{\text{ENTER}}\end{align*} to choose 1-Var Stats, press ENTER\begin{align*}\boxed{\text{ENTER}}\end{align*} again, and then use the down arrow to find the minimum and maximum values. You should see the following:

Now that we know that the minimum value is 67 and the maximum value is 127, we can enter the settings for the histogram in the WINDOW screen. To do this, press WINDOW\begin{align*}\boxed{\text{WINDOW}}\end{align*} and enter what is shown below:

We are now ready to create our histogram. To do so, press 2ND\begin{align*}\boxed{\text{2ND}}\end{align*} Y=\begin{align*}\boxed{\text{Y=}}\end{align*} ENTER\begin{align*}\boxed{\text{ENTER}}\end{align*}, and make sure you have the following settings:

Finally, press GRAPH\begin{align*}\boxed{\text{GRAPH}}\end{align*}, and you should see the histogram shown below:

At this point, you can use the TRACE feature to find the number of values for each bin. Just press TRACE\begin{align*}\boxed{\text{TRACE}}\end{align*} and use the right arrow to move across the screen.

The numbers for the bins should be 2, 5, 17, 23, 25, 4, and 4, respectively. Notice that 2+5+17+23+25+4+4=80\begin{align*}2+5+17+23+25+4+4=80\end{align*}, which is the number of days for which the data exists. Since we now have the frequency for each bin, the frequency distribution table can be created as follows:

Bin Frequency
[6070)\begin{align*}[60-70)\end{align*} 2
[7080)\begin{align*}[70-80)\end{align*} 5
[8090)\begin{align*}[80-90)\end{align*} 17
[90100)\begin{align*}[90-100)\end{align*} 23
[100110)\begin{align*}[100-110)\end{align*} 25
[110120)\begin{align*}[110-120)\end{align*} 4
[120130)\begin{align*}[120-130)\end{align*} 4

### Practice

Use the histogram shown below to answer the following questions:

1. What is the bin size for the histogram?
2. Which bin(s) have the highest frequency?
3. Which bin(s) have the lowest frequency?
4. What is the total number of data values represented by the histogram?
5. What percentage of the data values are in the bin [60-70)?

Use the histogram shown below to answer the following questions:

1. What is the bin size for the histogram?
2. Which bin(s) have the highest frequency?
3. Which bin(s) have the lowest frequency?
4. What is the total number of data values represented by the histogram?
5. What percentage of the data values are in the bin [15-20)?

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### Vocabulary Language: English

TermDefinition
trace feature The trace feature is a feature on the TI calculator that gives individual values to the five number summary of the box-and-whisker plot.

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