Chapter 2: Descriptive Statistics

2.9 Using Technology for Descriptive Statistics

Learning Objectives

By the end of this section, the student should be able to:

  • use technology to calculate descriptive statistics

This is another section to show technology tools that can be used to help with the statistics concepts taught in this chapter. This section will cover tools to help calculate descriptive statistics: the measures of location, center, and spread.

Statistical software will easily calculate the mean, the median, and the mode, plus the other descriptive statistics values covered in this Chapter. Some graphing calculators can also make these calculations. In the real world, people make these calculations using software and not doing the by-hand calculations.

Graphing Calculator

Similar to all the other technology sections, the main tool introduced will be the TI-83, 83+, 84, or 84+ Graphing Calculator. The graphing calculator can help to cut down on the by-hand calculations that are done with the heavy-duty formulas in statistics to find the values of the descriptive statistics covered in this Chapter. But first, we have to look at how to put data into the calculator.

 

Inputting Data

In many problems you will input data into just one list in the graphing calculator. Other problems it is better to input data into two lists. Let's look at how to do each of these.

Putting Data into One List using the TI-83, 83+, 84, or 84+ Graphing Calculator

**Using one list is good if you don't have many repeated values.

  1. Access statistics mode by pressing stat key.

  2. Select <4:ClrList> to clear data from lists, if desired. Highlight <4:ClrList> and press enter key or press number 4 key. With <ClrList> on the Home Screen, enter list [L1] to be cleared by pressing 2nd key, [L1], enter key. You will see <Done> on the Home Screen.

  3. Press 2nd key, [ENTRY] to display the last instruction on the Home Screen and continue clearing the remaining lists in the same fashion, if desired. So, to clear L2, you would press arrow left key, 2nd key, [L2], enter key. The same can be done for L3 - L6.

  4. After all the wanted lists are cleared, access statistics mode again be pressing stat key.
  5. Highlight over <1:Edit . . .> and press enter key.
  6. Enter data values into <L1>. (You may need to arrow over to it).
  7. Type in a data value and press enter key. For negative numbers, use the negate negative sign key key at the bottom of the keypad.
  8. Continue in the same manner until all data values are entered.

Putting Data into Two Lists using the TI-83, 83+, 84, or 84+ Graphing Calculator

**Two lists are better for data that has repeated data values or if given a frequency table.

  1. Access statistics mode by pressing stat key.

  2. Follow the directions above for clearing data from lists, if desired.

  3. After all the wanted lists are cleared, access statistics mode again be pressing stat key.
  4. Highlight over <1:Edit . . .> and press enter key.
  5. Enter data values into <L1>. (You may need to arrow over to it).
  6. Type in a data value and press enter key. For negative numbers, use the negate negative sign key key at the bottom of the keypad.
  7. Continue in the same manner until all data values are entered in L1.
    • Usually, if using two lists, you are only putting in unique data values in L1, and frequency of the data values will be put in L2.
  8. Now, enter data values (frequencies) into <L2>. (You may need to arrow over to it).
  9. Type in a data value and press enter key. Again, for negative numbers, use the negate negative sign key key at the bottom of the keypad.
    • Note: If a data value appears only once, the frequency is "1"
  10. Continue in the same manner until all data values are entered in L2.

Once you have data put into lists the graphing calculator, you can find all the descriptive statistics you need by going to the 1-Var Stats.

Using the 1-Var Stats in the TI-83, 83+, 84, or 84+ Graphing Calculator

  1. Access statistics mode by pressing stat key.

  2. Navigate to <CALC> by using the arrow right key.
  3. Access <1:1-Var Stats>. Highlight it and press enter key.
  4. Indicate the lists the data is in.
    • If data is just in <L1>, press 2nd key, [L1], enter key after <1-Var Stats> is on the Home Screen.
    • If data is in <L1> and the frequencies are in <L2>, press 2nd key, [L1], comma key, 2nd key, [L2], enter key after <1-Var Stats> is on the Home Screen.
  5. The statistics should be displayed. You can use the arrow down key and arrow up to scroll through the statistics.

    Legend

    • A blank calculator button represents a button press
    • [ ] represents yellow command or green letter behind a key
    • < > represents items on the screen

    Let's look at some examples to use the 1-Var Stats.

     

    Measures of Location

    In 2.4 Box Plots we constructed box plots by hand because we were first calculating the minimum, maximum, [latex]Q_1[/latex], [latex]Q_2[/latex], and [latex]Q_3[/latex] values. Well, with the 1-Var Stats, we can have the graphing calculator find those values, then use them to sketch the box plot (this was mentioned as an option in section 2.8 Using Technology for Charts and Graphs). Remember this example:

    Example

    The following data are the heights of 40 students in a statistics class.

    59; 60; 61; 62; 62; 63; 63; 64; 64; 64; 65; 65; 65; 65; 65; 65; 65; 65; 65; 66; 66; 67; 67; 68; 68; 69; 70; 70; 70; 70; 70; 71; 71; 72; 72; 73; 74; 74; 75; 77

    Find the five-number summary to construct a box plot.

      Let's use the 1-Var Stats to find the five-number summary (minimum, maximum, [latex]Q_1[/latex], [latex]Q_2[/latex], and [latex]Q_3[/latex]). The graphing calculator uses [latex]Med[/latex], for Median, instead of [latex]Q_2[/latex]. Here is a summary of steps using the instructions already provided in this section:

      Find the Five-Number Summary in the TI-83, 83+, 84, or 84+ Graphing Calculator

      Option 1.

      1. Enter data into the list editor (stat key, highlight over <1:Edit . . .> and press enter key). Remember, if you need to clear the lists, follow directions provided above on this page, or when on the List screen, arrow arrow up to the name <L1>, press clear key, then enter key.
      2. Put all the data values into the list <L1>.
      3. Press stat key, navigate to <CALC> by using the arrow right key, highlight <1:1-Var Stats> and press enter key.
      4. With <1-Var Stats> on the Home Screen, press 2nd key, [L1], enter key. 

      Option 2.

      1. Enter data into the list editor (stat key, highlight over <1:Edit . . .> and press enter key). Remember, if you need to clear the lists, follow directions provided above on this page, or when on the List screen, arrow arrow up to the name <L1>, press clear key, then enter key.
      2. Put the unique data values into the list <L1>.
      3. For this option, you'll need to clear the list <L2>, if not done already. Arrow arrow upto the name <L2>, press clear key, then enter key.
      4. Put the frequencies of the data values in <L2>.
      5. Press stat key, navigate to <CALC> by using the arrow right key, highlight <1:1-Var Stats> and press enter key.
      6. With <1-Var Stats> on the Home Screen, press 2nd key, [L1], comma key, 2nd key, [L2], enter key.

      Scroll down to see the five-number summary. For these values, you should see the following in your graphing calculator:

      • [latex]minX=59[/latex]
      • [latex]Q_1=64.5[/latex]
      • [latex]Med=66[/latex]
      • [latex]Q_3=70[/latex]
      • [latex]maxX=77[/latex]

      You can then construct the box plot using the directions from section 2.8 Using Technology for Charts and Graphs.

       

      Measures of Center

      In 2.5 Measures of the Center of the Data calculations were done by-hand to find the mean and the median. Yep, the graphing calculator can find those as well with the 1-Var Stats. Let's revist this example:

      Example

      AIDS data indicating the number of months a patient with AIDS lives after taking a new antibody drug are as follows (smallest to largest):

      3; 4; 8; 8; 10; 11; 12; 13; 14; 15; 15; 16; 16; 17; 17; 18; 21; 22; 22; 24; 24; 25; 26; 26; 27; 27; 29; 29; 31; 32; 33; 33; 34; 34; 35; 37; 40; 44; 44; 47;

      Calculate the mean and the median.

      Again, a summary of steps using the instructions for the 1-Var Stats (already provided in this section) is below. With this example, we could use just <L1>, because there are not many repeated data values, so the instructions below just show that option (though, try it with the frequencies in <L2> if you'd like!).

      Finding the Mean and the Median in the TI-83, 83+, 84, or 84+ Graphing Calculator

      1. Enter data into the list editor (stat key, highlight over <1:Edit . . .> and press enter key). Remember, if you need to clear the lists, follow directions provided above on this page, or when on the List screen, arrow arrow up to the name <L1>, press clear key, then enter key.
      2. Put all the data values into the list <L1>.
      3. Press stat key, navigate to <CALC> by using the arrow right key, highlight <1:1-Var Stats> and press enter key.
      4. With <1-Var Stats> on the Home Screen, press 2nd key, [L1], enter key. 

      For the mean and the median values, you should see the following in your graphing calculator:

      • [latex]\overline{x}= 23.6[/latex]
      • [latex]Med = 24[/latex]

       

      Measures of Spread

      The formulas for finding the measures of spread in 2.7 Measures of the Spread of the Data are very intimidating. In practice, definitely consider using a calculator or computer software to calculate the standard deviation and variance. 

      When using a TI-83, 83+, 84+ calculator, you need to select the appropriate standard deviation. The [latex]is for the population standard deviation, while the [latex]S_x[/latex] is for the sample standard deviation. Unfortunately, the 1-Var Stats does not provide the values of the population or sample variance ([latex]{\sigma}^{2}[/latex] or [latex]s^2[/latex]). For these, we just square the standard deviation values we get from the calculator. Let's revisit this example:

      Example

      In a fifth-grade class, the teacher was interested in the average age and the sample standard deviation of the ages of her students. The following data are the ages for a sample of [latex]n = 20[/latex] fifth grade students. The ages are rounded to the nearest half year:

      9; 9.5; 9.5; 10; 10; 10; 10; 10.5; 10.5; 10.5; 10.5; 11; 11; 11; 11; 11; 11; 11.5; 11.5; 11.5

      A summary of steps using the instructions for the 1-Var Stats is below. Again, with this example, let's use just <L1>, because there are not many repeated data values.

      Finding the Mean and the Standard Deviation in the TI-83, 83+, 84, or 84+ Graphing Calculator

      1. Enter data into the list editor (stat key, highlight over <1:Edit . . .> and press enter key). Remember, if you need to clear the lists, follow directions provided above on this page, or when on the List screen, arrow arrow up to the name <L1>, press clear key, then enter key.
      2. Put all the data values into the list <L1>.
      3. Press stat key, navigate to <CALC> by using the arrow right key, highlight <1:1-Var Stats> and press enter key.
      4. With <1-Var Stats> on the Home Screen, press 2nd key, [L1], enter key. 

      For the mean and the sample standard deviation values, you should see the following in your graphing calculator:

      • [latex]\overline{x}= 10.525[/latex]
      • [latex]\text{Sx}=0.715891[/latex] (or 0.72, rounded to two decimal places)

      If we needed to find the sample variance (or [latex]s^2[/latex]), we would just square [latex]S_x[/latex]. So, [latex]s^2 = (0.72)^2 = 0.5184[/latex]. This is an approximate value because the standard deviation value was rounded. For a more accurate value, square the longer decimal value of [latex]\text{Sx}[/latex].

      What if the data came in a frequency table already, like below.

      Example

      In a fifth-grade class, the teacher was interested in the average age and the sample standard deviation of the ages of her students. The following data are the ages for a sample of [latex]n = 20[/latex] fifth grade students. The ages are rounded to the nearest half year:

      9; 9.5; 9.5; 10; 10; 10; 10; 10.5; 10.5; 10.5; 10.5; 11; 11; 11; 11; 11; 11; 11.5; 11.5; 11.5

      The following frequency table was put together for the data.

      Table 1: Frequency Table of Ages of Fifth Grade Students
      x f
      9 1
      9.5 2
      10 4
      10.5 4
      11 6
      11.5 3
      Well, here we could use 1-Var Stats with <L1> and <L2>. We should anticipate the same answers. Here's a summary of those steps.

      Finding the Mean and the Standard Deviation in the TI-83, 83+, 84, or 84+ Graphing Calculator

      1. Enter data into the list editor (stat key, highlight over <1:Edit . . .> and press enter key). Remember, if you need to clear the lists, follow directions provided above on this page, or when on the List screen, arrow arrow up to the name <L1>, press clear key, then enter key.
      2. Put the unique data values, x, into the list <L1>.
      3. For this option, you'll need to clear the list <L2>, if not done already. Arrow arrow upto the name <L2>, press clear key, then enter key.
      4. Put the frequencies, f, of the data values in <L2>.
      5. Press stat key, navigate to <CALC> by using the arrow right key, highlight <1:1-Var Stats> and press enter key.
      6. With <1-Var Stats> on the Home Screen, press 2nd key, [L1], comma key, 2nd key, [L2], enter key.

      For the mean and the sample standard deviation values, you should see the following in your graphing calculator:

      • [latex]\overline{x}= 10.525[/latex]
      • [latex]\text{Sx}=0.715891[/latex] (rounded to two decimal places)

      If we wanted the population standard deviation for this data, you would use the value [latex]σ_x[/latex].

       

      Grouped Frequency Tables

      In both 2.5 Measures of the Center of the Data and 2.7 Measures of the Spread of the Data, grouped frequency tables were discussed. The difference in grouped frequency tables is they have classes or intervals with the frequency. The by-hand method required us to calculate the midpoint. This is still required to use the calculator, but we can skip having to use the formula. Let's look at this example:

      Example

      Find the mean and standard deviation for the data.

      Table 2: Grouped Frequency Table
      Class Frequency, f
      0–2 1
      3–5 6
      6–8 10
      9–11 7
      12–14 0
      15–17 2

      Before using the calculator, we still are required to calculate midpoints. Remember, the midpoint is [latex]\frac{\text{lower boundary} + \text{upper boundary}}{2}[/latex]. For this example, the midpoint values are 1, 4, 7, 10, 13, and 16. Now, using the 1-Var Stats with <L1> and <L2> is where we have an advantage. 

      Finding Grouped Frequency Statistics in the TI-83, 83+, 84, or 84+ Graphing Calculator

      1. Enter data into the list editor (stat key, highlight over <1:Edit . . .> and press enter key). Remember, if you need to clear the lists, follow directions provided above on this page, or when on the List screen, arrow arrow up to the name <L1>, press clear key, then enter key.
      2. Put the midpoint values into the list <L1>.
      3. For this option, you'll need to clear the list <L2>, if not done already. Arrow arrow upto the name <L2>, press clear key, then enter key.
      4. Put the frequencies, f, of the data values in <L2>.
      5. Press stat key, navigate to <CALC> by using the arrow right key, highlight <1:1-Var Stats> and press enter key.
      6. With <1-Var Stats> on the Home Screen, press 2nd key, [L1], comma key, 2nd key, [L2], enter key.

      For the mean and the sample standard deviation values, you should see the following in your graphing calculator:

      • [latex]\overline{x}= 7.5769[/latex] (rounded to 4 decimal places)
      • [latex]\text{Sx}=3.5005[/latex] (rounded to two decimal places)

      You will see displayed both a population standard deviation, [latex]\sigma_x[/latex], and the sample standard deviation, [latex]S_x[/latex].

      Here are screen captures of the calculator shown for this problem:

      1. Figure 1 is the screen after pressing the STAT button. It displays top menu choices EDIT, CALC, TESTS. EDIT is selected. Below the top menu, option 1: Edit… is selected.
      2. Figure 2 is the List editing screen. The image is showing L1 with entries 1, 4, 7, 10, 13, 16 and L2 with entries 1, 6, 10, 7, 0, 2.
      3. Figure 3 is the screen after pressing the STAT button. It displays top menu choices EDIT, CALC, TESTS. CALC is selected. Below the top menu, option 1: 1–Var Stats is selected.
      4. Figure 4 is the screen after the 1-Var Stats is run. The screen shows 1–Var Stats. Below this, the screen displays x-bar = 7.576923077, upper-case sigma x = 197, upper-case sigma x-squared = 1799, Sx = 3.500549407, and lower-case sigma = 3.432571103, n = 26.

      A TI calculator screen. Showing the screen after pressing the STAT key.
      Figure 1. Screen after pressing STAT button

       

      A TI calculator screen. L1 list shows entries 1, 4, 7, 10, 13, 16. L2 list shows 1, 6, 10, 7, 0, 2.
      Figure 2. List 1 with midpoints and List 2 with frequencies

       

      A TI calculator screen. Showing the CALC screen with 1-Var Stats.
      Figure 3. After pressing STAT button and scrolling to CALC

       

      A TI calculator screen. Showing the 1-Var Stats screen and descriptive statistics values.
      Figure 4. 1-Var Stats displaying x-bar, upper-case sigma x, upper-case sigma x-squared, Sx, lower-case sigma, and n

       

      Helpful Videos for the Graphing Calculator

      Below are links to helpful videos for using the graphing calculator for the concepts covered on this page:

       

      Additional Technology Tools

      In addition to the graphing calculator, there are some additional technology tools that can be used for the concepts covered on this page. Below are links to helpful videos for those tools:

       

      Section Practice

      Listed are 29 ages for Academy Award winning best actors in order from smallest to largest.

      18; 21; 22; 25; 26; 27; 29; 30; 31; 33; 36; 37; 41; 42; 47; 52; 55; 57; 58; 62; 64; 67; 69; 71; 72; 73; 74; 76; 77

      1. Find the 40th percentile.
      2. Find the 78th percentile.
      Solution
      1. The 40th percentile is 37 years.
      2. The 78th percentile is 70 years.

      Twenty-five randomly selected students were asked the number of movies they watched the previous week. The results are as follows:

      Table 3: Frequency Table of Number of Movies Watched
      # of movies Frequency
      0 5
      1 9
      2 6
      3 4
      4 1

      Construct a box plot of the data.

      Santa Clara County, CA, has approximately 27,873 Japanese Americans. Their ages are as follows:

      Table 4: Percentage Table of Age Groups of Japanese Americans
      Age Group Percentage of Community
      0–17 18.9
      18–24 8.0
      25–34 22.8
      35–44 15.0
      45–54 13.1
      55–64 11.9
      65+ 10.3
      1. Construct a histogram of the Japanese American community in Santa Clara County, CA. The bars will not be the same width for this example. Why not? What impact does this have on the reliability of the graph?
      2. What percentage of the community is under age 35?
      3. Which box plot most resembles the information above?
      Three box plots with values between 0 and 100 to answer question about provided data
      Figure 5. Box Plots To Answer Question 3

       

      Long Image Description: Three box plots with values between 0 and 100. Plot i has Q1 at 24, M at 34, and Q3 at 53; Plot ii has Q1 at 18, M at 34, and Q3 at 45; Plot iii has Q1 at 24, M at 25, and Q3 at 54.

       

      Solution
      1. For graph, check student's solution.
      2. 49.7% of the community is under the age of 35.
      3. Based on the information in the table, graph (a) most closely represents the data.

      The following data are the distances between 20 retail stores and a large distribution center. The distances are in miles.

      29; 37; 38; 40; 58; 67; 68; 69; 76; 86; 87; 95; 96; 96; 99; 106; 112; 127; 145; 150

      Use a graphing calculator or computer to find the standard deviation and round to the nearest tenth.

      Solution

      [latex]s = 34.5[/latex]

      Forty randomly selected students were asked the number of pairs of sneakers they owned. Let X = the number of pairs of sneakers owned. The results are as follows:

      Table 5: Frequency Table for the Number of Pairs of Sneakers Owned
      X Frequency
      1 2
      2 5
      3 8
      4 12
      5 12
      6 0
      7 1
      1. Find the sample mean [latex]\overline{x}[/latex]
      2. Find the sample standard deviation, s
      3. Construct a histogram of the data.
      4. Complete the columns of the chart.
      5. Find the first quartile.
      6. Find the median.
      7. Find the third quartile.
      8. Construct a box plot of the data.
      9. What percent of the students owned at least five pairs?
      10. Find the 40th percentile.
      11. Find the 90th percentile.
      12. Construct a line graph of the data
      13. Construct a stemplot of the data

      Following are the published weights (in pounds) of all of the team members of the San Francisco 49ers from a previous year.

      177; 205; 210; 210; 232; 205; 185; 185; 178; 210; 206; 212; 184; 174; 185; 242; 188; 212; 215; 247; 241; 223; 220; 260; 245; 259; 278; 270; 280; 295; 275; 285; 290; 272; 273; 280; 285; 286; 200; 215; 185; 230; 250; 241; 190; 260; 250; 302; 265; 290; 276; 228; 265

      1. Organize the data from smallest to largest value.
      2. Find the median.
      3. Find the first quartile.
      4. Find the third quartile.
      5. Construct a box plot of the data.
      6. The middle 50% of the weights are from [latex]\underline{\hspace{2cm}}[/latex] to [latex]\underline{\hspace{2cm}}[/latex].
      7. If our population were all professional football players, would the above data be a sample of weights or the population of weights? Why?
      8. If our population included every team member who ever played for the San Francisco 49ers, would the above data be a sample of weights or the population of weights? Why?
      9. Assume the population was the San Francisco 49ers. Find:
        • the population mean, [latex]\mu[/latex].
        • the population standard deviation, [latex]\sigma[/latex].
        • the weight that is two standard deviations below the mean.
        • When Steve Young, quarterback, played football, he weighed 205 pounds. How many standard deviations above or below the mean was he?
      10. That same year, the mean weight for the Dallas Cowboys was 240.08 pounds with a standard deviation of 44.38 pounds. Emmitt Smith weighed in at 209 pounds. With respect to his team, who was lighter, Smith or Young? How did you determine your answer?
      Solution
      1. 174; 177; 178; 184; 185; 185; 185; 185; 188; 190; 200; 205; 205; 206; 210; 210; 210; 212; 212; 215; 215; 220; 223; 228; 230; 232; 241; 241; 242; 245; 247; 250; 250; 259; 260; 260; 265; 265; 270; 272; 273; 275; 276; 278; 280; 280; 285; 285; 286; 290; 290; 295; 302
      2. 241
      3. 205.5
      4. 272.5
      5. A box plot with a whisker between 174 and 205.5, a solid line at 205.5, a dashed line at 241, a solid line at 272.5, and a whisker between 272.5 and 302.
        Figure 6. Box Plot of the Published Weights of the San Francisco 49ers
      6. 205.5, 272.5
      7. sample
      8. population
      9. The values are:
        • 236.34
        • 37.50
        • 161.34
        • 0.84 std. dev. below the mean
      10. Young

      A survey of enrollment at 35 community colleges across the United States yielded the following figures:

      6414; 1550; 2109; 9350; 21828; 4300; 5944; 5722; 2825; 2044; 5481; 5200; 5853; 2750; 10012; 6357; 27000; 9414; 7681; 3200; 17500; 9200; 7380; 18314; 6557; 13713; 17768; 7493; 2771; 2861; 1263; 7285; 28165; 5080; 11622

      1. Organize the data into a chart with five intervals of equal width. Label the two columns "Enrollment" and "Frequency."
      2. Construct a histogram of the data.
      3. If you were to build a new community college, which piece of information would be more valuable: the mode or the mean?
      4. Calculate the sample mean.
      5. Calculate the sample standard deviation.
      6. A school with an enrollment of 8,000 would be how many standard deviations away from the mean?
      Solution
      1. Table 6: Enrollment and Frequency of 35 Community Colleges
        Enrollment Frequency
        1000-5000 10
        5000-10000 16
        10000-15000 3
        15000-20000 3
        20000-25000 1
        25000-30000 2
      2. Check student’s solution.
      3. mode
      4. 8628.74
      5. 6943.88
      6. –0.09

      The table below gives the percent of children under five considered to be underweight.

      Table 7: Percent of Underweight Children to Number of Countries
      Percent of Underweight Children Number of Countries
      16–21.45 23
      21.45–26.9 4
      26.9–32.35 9
      32.35–37.8 7
      37.8–43.25 6
      43.25–48.7 1

      What is the best estimate for the mean percentage of underweight children? What is the standard deviation? Which interval(s) could be considered unusual? Explain.

      The population FTES for 2005–2006 through 2010–2011 was given in an updated report. The data are reported here.

      Table 8: Year and Total FTES from 2005-2011
      Year 2005–6 2006–7 2007–8 2008–9 2009–10 2010–11
      Total FTES 1,585 1,690 1,735 1,935 2,021 1,890
      1. Calculate the mean, median, standard deviation, the first quartile, the third quartile and the IQR. Round to one decimal place.
      2. Construct a box plot for the FTES for 2005–2006 through 2010–2011 and a box plot for the FTES for 1976–1977 through 2004–2005.
      3. Compare the IQR for the FTES for 1976–77 through 2004–2005 with the IQR for the FTES for 2005-2006 through 2010–2011. Why do you suppose the IQRs are so different?
      Solution
      • mean = 1,809.3
      • median = 1,812.5
      • standard deviation = 151.2
      • first quartile = 1,690
      • third quartile = 1,935
      • IQR = 245

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