Section 2.2: Tables and Graphs for Quantitative Data

Recall that quantitative variables are numerical. They can be either discrete (things you count) or continuous (things you measure). In this section we will organize both types of numerical data into tables and charts.

Frequency Distributions and Dot Plots

Solution:

Since the student looked back at her phone records,

this is an observational study. It is retrospective, or

case-control, because she is looking at records from

the past. The variable is the number of spam calls she

received, which is quantitative and discrete because

she is counting the calls.

To organize the data, first let’s make a list of the possible number of calls she could receive in a day and put them in order from smallest to largest. This step was unnecessary in section 2.1 when we were working with qualitative data. But here, having the data organized in order makes sense.

Next, recall the definitions of frequency and relative frequency from section 2.1.

To find the frequencies we count the number of times each observation occurred in the data. Table 5 shows the tally marks you would generate going through the data one-by-one. After completing the tally, you can fill in the frequency column.

To add the relative frequency column, divide each frequency by the total.

In Table 6, all decimals were rounded to 4 places. The sum of the relative frequencies is 24/24, which is 1. But, if you add the rounded decimals, you will obtain 0.9999. The difference is due to rounding and is not due to an error. In practice, the sum of the relative frequencies will either be exactly 1 or be slightly off due to rounding.

In section 2.1, we used the frequency table to generate a bar graph. But bar graphs are limited to use with qualitative data. Here we will discuss other options.

Solution:Start with a number line that stretches from 0 (the lowest number of calls) to 7 (the most calls received).

The first values from the set was 2. Place a dot above 2 on the number line.

The next three observations were 1, 1, and 0. Place dots above these numbers on the line. Since 1 is listed twice, begin to stack the dots above the 1.

Continue until you have placed a dot for each data value. Your dot plot should match the final product, seen in figure 2.13.

Figure 2.13: Dot plot for spam call data.

You can see that most commonly the student would get 1 or 2 spam calls per day. The frequency is readily seen as the number of dots above each value on the axis. You can immediately see that there was only one time during the month that she received no telemarketer calls.

STATBELT Utility can be used to create a dot plot. Figure 2.14 is the dot plot for spam calls created using STATBELT Utility.

Figure 2.14: Dot plot for spam call data created using STATBELT Utility.

Sometimes we have so much data that using a dot plot is not helpful. Remember that our goal is to summarize the data to visualize broad trends. If we look at all the data values together it is easy to get lost. This is particularly true when we deal with continuous data. To organize continuous data, we will create dividers among the data and use them to group the data into classes. This will make it easier to generalize the data and allow us to make what we call histograms.

Solution: The classes are the groups 40 – 44, 45 – 49, 50 – 54, 55 – 59, 60 – 64, 65 – 69,

70 – 74, and 75 – 79. No president has been under 40 or over 80 when inaugurated.

The lower-class limits are the lowest possible value in each interval, namely 40, 45, 50, 55, 60, 65, 70, and 75. The lower-class limits are highlighted in red in figure 2.15. The upper-class limits are the largest values in the intervals, namely 44, 49, 54, 59, 64, 69, 74, and 79. They are highlighted in green in figure 2.15. The frequency column is the same as it was in section 2.1. It is highlighted in blue in figure 2.15.

Figure 2.15: Identifying lower-class limits, upper-class limits, and frequency.

The class width is most easily described as the “jump” between two classes. The lower-class limits jump from 40 to 45, 45 to 50, 50 to 55, etc. where each jump is 5 units. Thus, the class width is 5. The definition says we are to find the class width by subtracting successive class limits, so we could find it subtracting 40 from 45 to get 5, or 69 – 64 = 5. But again, it is usually simpler to look at the size of the jump. That is the class width!

The last part of the question asks us to find the relative frequency of the class 55 – 59. From the table we can see that the frequency is 12, meaning that there have been 12 presidents who were between 55 and 59 at inauguration. To find the relative frequency we also need the total. Summing the frequencies in the frequency column we get the total = 46. So, the relative frequency for the 55 – 59 class is 1246≈0.2609. So, 0.26, or 26% of U.S. presidents have been between 55 and 59 years old at inauguration.

Histograms

In section 2.1, we first made frequency and relative frequency tables, then we used bar graphs to visualize the data. When we have quantitative data, we use a histogram instead of a bar graph. A histogram is similar, with a few differences.

*Technically, we should use the class boundaries, which are the midpoints between the lower-class limits of one class and the upper-class limits of the next. But the results are nearly the same and will be undetectable using technology. So, we choose to use the easier option.

Solution: Normally we would not do this by hand, but it is instructive to work through one

example showing the steps involved.

First, create a number line that will allow you to place all the data between the lowest value 40 and the highest value 79. Then use the lower-class limits to mark off increments on the horizontal (x) axis.

Next make a vertical axis that will have a height up to maximum frequency, which in this case is 13.

Draw a rectangular box with base between 40 and 45 and a height equal to 2, since 2 is the frequency for that class. Then draw a box with base between 45 and 50 with height 7, since 7 is the frequency of that class.

Keep going until you have drawn a box for each of the classes. Your histogram should look something like figure 2.16.

Figure 2.16: Histogram for the age of a US President at inauguration.

Of course, the better option is to use technology to create the histogram.

Solution:Click the link and copy the data into STATBELT Utility, or load the “U.S. Presidents Age at Inauguration” sample data. Select Graphs → Histogram and choose “Age at inauguration” for “Column.” Uncheck “Choose start and bin width automatically,” type 40 for “Start at” and 5 for “Bin width,” and select “Frequency” for “Bar height.” Check “Show values above the bars” and click “Open graph in a new window.”

You can see the histogram generated by STATBELT Utility in figure 2.17. Note that the frequencies that we found in creating the frequency table are now listed as the heights of the bars. If you want a relative frequency histogram, follow the same steps but instead select “Relative frequency” for “Bar height.”

Figure 2.17: Histogram for the age of a US President at inauguration, generated by STATBELT Utility.

Solution: To make the frequency table, we first need to determine the classes. The lowest poverty rate was 6.4% and the highest was 19.8%. That means we will start with 5 as the lowest class limit, and add increments of the class width, 2.5, until we get to 19.8. The lower-class limits would then be 5, 7.5, 10, 12.5, 15, and 17.5. Since the data are all rounded to one decimal place, the first class could contain any number up to 7.4. Anything higher then 7.4 (7.5 or more) goes into another class. So, the first class is 5 – 7.4. Similarly, the second class would contain any number up to 9.9, so it would be 7.5 – 9.9. We can add the class width to each of the upper-class limits to generate them. The upper-class limits will be 7.4, 9.9, 12.4, 14.9, 17.4, and 19.9. Make sure that each poverty rate from Table 8 can be placed into one of the classes shown in Table 9 below.

Whether we count the data in each class by hand using the tally method, or sorting the data first, we get the frequencies given in Table 9.

Convert to a relative frequency table by dividing each frequency by 51, shown in Table 10.

To make the histogram, we can either do it by hand using the values in Table 10 or use STATBELT Utility. Using STATBELT Utility, select Graphs → Histogram and choose “Average Poverty rate 2016-2018” for “Column.” Uncheck “Choose start and bin width automatically,” type 5 for “Start at” and 2.5 for “Bin width,” and select “Relative frequency” for “Bar height.” Check “Show values above the bars” and click “Open graph in a new window.”

Figure 2.18: Histogram for the average poverty rate by state, generated by STATBELT Utility.

The histogram is shown in figure 2.18. Note that the relative frequencies are the same as the ones that we calculated in Table 10, but STATBELT Utility shows four decimal places.

Histograms and the Shape of a Distribution

We will use histograms for the rest of this book to describe the “shape” of a quantitative data distribution. There are as many different shapes as there are data sets in the world, but we will focus on four shapes that we will see most commonly. See figure 2.19.

Bell-Shaped

Uniform

Skewed Left

Skewed Right

Figure 2.19: Describing the shape of distributions using histograms.

Bell-shaped is also called “Normal”, and we will study these distributions in detail later in the book. Skewed left is also known as negatively skewed, while skewed right is also called positively skewed.

Solution:The histogram from figure 2.17 is reproduced below. While it is not a perfectly bell-shaped distribution, it is reasonably close, with a very slight skew to the right. We would classify this as bell-shaped.

Solution:The histogram from figure 2.18 is reproduced below. Again, this appears to be bell-shaped.

Figure 2.20: Histogram for High School Grade Point Average.

Solution:Because the “tail” is on the left, we say this data is skewed to the left, or negatively skewed.

Figure 2.21: Histogram of adjusted earnings for superhero movies.

Solution:Because the “tail” is on the right we say this data is skewed to the right, or positively skewed.

Stem-and-Leaf Plots

A Stem-and-Leaf plot breaks each data value into a pair: a stem and a leaf. For example, the number 54 can be seen as a stem = 5 tens and a leaf = 4 ones. When we have multiple numbers with the same stem, we only write the stem once, but write each leaf separately, in order from smallest to largest. So, if we had the data values 54, 57, and 59, we would write this as:

Each different stem will have its own row in the stem-and-leaf plot. For example, if the data were 54, 57, 59, 61,63, 67, 68, 70, and 73, we would add a stem row for the 60’s and a stem row for the 70’s, then write in the leaves. The plot would look like this:

You can see the frequency of each stem by counting the number of leaves. This makes a stem-and-leaf plot like a sideways frequency histogram, with one advantage: You can see the original data in the stem-and-leaf plot. For example, by looking at the plot you can see that the number 71 was not among the data, but 70 was. Stem-and-leaf plots are usually best to use for reasonably small data sets. Too much data will make the leaves stretch too far and become difficult to understand. In those cases, a histogram is likely a better choice.

Solution:The stems will consist of numbers from 6 through 19. The first row contains only 6.4, so we put a 4 into the leaf.

The next values are 7.1 and 7.9, so we put a 1 and 9 into the stem row for 7.

Continuing in this way we complete the plot, given in figure 2.22.

Figure 2.22: Stem-and-leaf plot for average poverty in the US 2016-2018.

Follow the instructions below to make the stem and leaf plot using STATBELT Utility. Note that the stem digit was set to “Ones” because the leaf was the digit to the right of the decimal point. See figure 2.23.

Figure 2.23: Stem-and-leaf plot for average poverty in the US 2016-2018, using STATBELT Utility.

Figure 2.24: Stem-and-leaf plot for test scores on a statistics exam.

Solution:a) The leaf that was repeated most often was the 4 in the stem row for 8. That means that the score 84 appeared 5 times in the data, which was the highest frequency.

b)The were no scores in the 50’s because the stem row for 5 is empty.

c)There was no score exactly equal to 95. There were two 93s, and the next score was a 96.

d)The longest row of leaves was in the row with stem 9, representing grades in the 90’s. There are 12 leaves, meaning that there were 12 scores in the 90’s on the exam. We say the modal stem, or modal class, was grades in the 90’s because they had the highest frequency.

e)There were 26 scores in the rows with stem 8, 9, or 10. That means that 26 grades were 80 or higher.

f)Looking at the stem-and-leaf plot turned sideways, it appears to be skewed left.

Time-Series Plots

Figure 2.25: Time-series plot for the number of states observing Juneteenth by year.

Solution: Figure 2.26 is a blow up of the time-period and is probably a little easier to read. In the year 2000, there were 5 states that observed Juneteenth, but only one that observed it as a paid holiday (Texas). In 2003, there were 16 states observing Juneteenth, but still only one as a paid holiday. In 2022 all 50 states and the District of Columbia observe Juneteenth. Additionally, there are 24 states observing it as a paid holiday, with Connecticut making 25 in 2023.

Figure 2.26: Blow up of time-series plot for the number of states observing Juneteenth by year.

In the previous example, we chose “Year” for “Time column” and checked both value columns, the number of states observing Juneteenth and the number observing it as a paid holiday. You can open the Juneteenth data file (opens in a new tab) in STATBELT (download CSV) to try it yourself!

Solution: Early in the pandemic, the highest death rates per 100,000 belonged to Black and Hispanic people. Before vaccines were available, COVID affected Black, Latinx, Indigenous, and other people of color the most. This could have been due to a lack of access to health care, job or living conditions, or other factors. However, the death rates for those groups dropped below the death rate per 100,000 for Whites late in 2021 during the Delta variant wave. By early 2022, Black, Hispanic, and White had similar death rates. The Asian ethnic group appears to have maintained the lowest death rates throughout the pandemic. Vaccination rates can only partially explain that, as vaccinations did not exist until December 2020.

Figure 2.27: Time-series plot for the number COVID deaths by ethnicity. Source: CDC Health Equity (opens in a new tab)