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Omniguides by NOX
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1.2 Graphing Quantitative Data

Section Overview

Describing Qualitative Data

Categorical data is important in its own right, but there are generally a lot more things you can do with quantitative data. The following types of measures are primarily what we use to describe distributions of qualitative data:

  • Shape — One way we can describe a distribution’s shape is through its symmetry. A distribution can be symmetric, which means the data points, or observations, are equally concentrated on both sides of the median. In other words, both sides are mirror images of one another. Otherwise, the distribution is either right-skewed (the observations are more closely concentrated on the left side of the median than on the right) or left-skewed (the observations are more closely concentrated on the right side of the median than on the left).

Another description of shape is the number of peaks a distribution has. Distributions with one peak, two peaks, or many peaks are called unimodal, bimodal, and multimodal respectively. The mode of the distribution is its highest peak.

  • Center — Although there are measures of center that we’ll discuss in later sections, one important measure of center is the median, or the point in the distribution where half of the data in the distribution is smaller and the other half of the data is larger. If the observations are numbered from least to greatest, then the median can be more easily picked out. Further, if the distribution has an odd number of observations, then the median is the observation in dead center. If the distribution has an even number of observations, then the median is the average of the two “innermost” observations.
  • Spread — One common measure of center is the range, or the numerical value that comes from taking the difference between the largest observation and the smallest observation. Similarly to the center of a distribution, there are additional measures of spread that we’ll discuss in sections to come.
  • Unusual Features — We can also describe a distribution in terms of clusters (observations closely grouped together) and gaps (areas of empty space in a distribution). Lastly, we have outliers, data points that differ greatly from most of the other data points in a distribution, usually lying on the outer reaches of the graph (see what we did there?) Although our eyes can help us identify potential outliers, we won’t know until the next section if these extreme observations actually fit the bill.
Source gap (S0021): Source terminology is retained: the heading and opening paragraph say “qualitative” within this section about quantitative data. The Spread bullet on source page 13 calls the range a “measure of center”.

Dotplots

One way to graph a distribution is to create a dotplot, a graph where the observations are represented by dots and are placed over their numerical values displayed on the bottom axis. See Interactive 1.2.1 for a better idea of what a dotplot looks like. With that said, let’s dive into the example problem that accompanies this dotplot:

Example 1.2.1

You are the (American) soccer coach for the Peakpoint Middle School Screaming Wombats. The table below displays the data on the number of goals the Wombats have scored all 24 games of the season.

Number of Goals Scored per Game

Number of goals scored per game, in the source table’s row order
4101311
029635
144513
154642

Describe the distribution of data using the measures of a) shape, b) center, c) spread, and d) any unusual features you find.

Interactive 1.2.1 Example 1.2.1

Soccer-ball dotplot of number of goals scored: one at 0, six at 1, two at 2, three at 3, five at 4, three at 5, two at 6, none at 7 or 8, one at 9, and one at 10. Captured annotation labels point to Right-Skew and Peak, Median (between 3 and 4), Clusters and Gaps (gap near 7 and 8), and Outliers? (near 9 and 10). Open full-size image S0023
Source gap (S0023): The screenshot records the four annotation labels “Right-Skew and Peak”, “Median”, “Clusters and Gaps”, and “Outliers?” with their pointers. No opened annotation-panel text is captured.
Example 1.2.1 — recorded solution

a) Shape: Looking at the dotplot, we see that the distribution is bimodal with peaks at 1 goal and 4 goals, with 1 being the mode. The distribution is also right-skewed, with most of the observations located near 0.

b) Center: There is an even number of soccer games played, so the median will be the average of the two innermost observations. The dotplot automatically orders the observations from least to greatest, so we simply count up to the 12th observation, 3 goals. The other innermost observation is the 13th observation, 4 goals. The number between these two values is the median, with a value of 3.5 goals.

c) Spread: The range of the distribution is the the difference between the largest observation and the smallest observation, or

10−0=10 goals10-0=10\text{ goals}

d) Unusual features: Most of the values are clustered near the first half of the graph, leaving a gap between the cluster and the observations with a value of 9 and 10. These values could also be potential outliers.

Stemplots

We can also use stemplots, also called stem-and-leaf plots, to graph distributions. At first glance, a stemplot looks a bit like a dotplot turned on its side, but with the numerical values of the observations themselves populating the graph instead of dots. The stem in the stemplot consists of every digit from a distribution’s observations save for the smallest one, which make up the leaves, ordered from least to greatest. Any well-made stemplot should also include a key that lets other people know what the stems and leaves represent. This is especially important when your observations consist of non-integer values.

Stemplots are also ideal for comparing two distributions. These types of graphs are called back-to-back stemplots. An example of a back-to-back stemplot can be seen in Interactive 1.2.2.

Example 1.2.2

The speed of two internet browsers, Net Navigator and Waterwolf, were compared on a number of test computers. The results for each browser are displayed in the following table in milliseconds:

Waterwolf Speeds (ms)
96137116123142
11812511100126
13498125135143
121140112122
Net Navigator Speeds (ms)
123153142157161
16481164124149
146128166165157
169130152137

Compare the distributions using the measures of a) shape, b) center, c) spread, and d) any unusual features you find.

Interactive 1.2.2 Example 1.2.2

Back-to-back stemplot, Waterwolf at left and Net Navigator at right, stems 8 through 16. Captured labels Shape, Outlier?, and Median point to the distribution shapes, the Net Navigator 81 ms observation, and the Net Navigator 152 ms leaf. The key text at lower left is partly clipped, with 134 milliseconds visible. Open full-size image S0025
Waterwolf leavesStemNet Navigator leaves
81
8 69
010
8 6 6 211
6 5 5 3 2 1123 4 8
7 5 4130 7
2 0142 6 9
3152 3 7 7
161 4 4 5 6 9
Source gap (S0025): The source stemplot’s key is clipped at the left edge; only the ending “represents a Waterwolf …r speed of 134 milliseconds” is visible. The annotation labels “Shape”, “Outlier?”, and “Median” are shown without opened explanatory panels.
Example 1.2.2 — recorded solution

a) Shape: When you create a stemplot for the values in the table, it is apparent that the Waterwolf distribution is roughly symmetric and unimodal with its peak located at the 120 ms mark. The Net Navigator distribution, is left-skewed and also unimodal, with its peak located at the 160 ms mark.

b) Center: There is an odd number of observations for both distributions, so the median will be in dead center. When the observations for each distribution are ordered from least to greatest, we find the median for the Waterwolf distribution to be 123 ms and the median for the Net Navigator distribution to be 152 ms.

c) Spread: The range for the Waterwolf distribution is 163−96=67 ms163-96=67\text{ ms} and the range for the Net Navigator distribution is 169−81=88 ms169-81=88\text{ ms}.

d) Unusual features: Most of the values for the Waterwolf distribution are clustered around the middle of the graph, while most of the values for the Net Navigator distribution are clustered around the bottom of the graph. There are no notable gaps or potential outliers for the Waterwolf distribution. For the Net Navigator distribution, however, there is a gap between the cluster of values and the potential outlier, the 81 ms observation.

Source gap (S0025): Source inconsistencies in Waterwolf values: the table contains “11” in its second row, third numeric column. The stemplot instead has two 6 leaves on stem 11, a 3 leaf on stem 15, and no 3 leaf on stem 14. The recorded range uses 163 − 96 = 67 ms, although neither the table nor the stemplot shows 163. All three source representations are retained.

Histograms

The last type of graph we’ll cover in this section are histograms. Histograms look similar to bar graphs, but are used for quantitative variables instead of categorical variables. Each bar in a histogram represents a class, or an equal subdivision of the distribution’s range. The number of classes you choose for your histogram depends on how large you decide to make each class. There is no exact formula for picking the size or number of classes, however. Pick too many or too few classes, and it could result in you or others having a hard time interpreting the distribution, especially without the original data set to rely on. When you pick your class size, you can display the count or percent of the observations that fall within each class in either a frequency table or relative frequency table.

Example 1.2.3

To create hype for its upcoming summer blockbuster, a movie studio has decided to partner with a large restaurant chain, Patty Prince, in order to sell toys with the kid’s meal. There are 30 toys to collect in all. The following table shows the number of different figures 39 randomly selected children have collected:

Number of Different Toys Collected per Child
1543124
536111
07281
11013172
10283105
395212
35963
7354

Create a histogram and describe the a) shape, b) center, c) spread, and d) unusual features of the distribution.

ClassCount
0 to < 5 or 0 to 417
5 to < 10 or 5 to 912
10 to < 15 or 10 to 147
15 to < 20 or 15 to 192
20 to < 25 or 20 to 240
25 to < 30 or 25 to 291
ClassPercent
0 to < 5 or 0 to 443.6%
5 to < 10 or 5 to 930.8%
10 to < 15 or 10 to 1417.9%
15 to < 20 or 15 to 195.1%
20 to < 25 or 20 to 240
25 to < 30 or 25 to 292.6%
Example 1.2.3 — recorded solution

Before creating a histogram, we have to decide on the size and number of our classes. With a size of 5, we could have a total number of 6 classes: 0 to 5, 5 to 10, 10 to 15, 15 to 20, 20 to 25, and 25 to 30. However, notice that the classes are actually a size of 6 if you include the first and last numbers.

We can either change the names of the classes to be:

0 to < 5, 5 to < 10, 10 to < 15, 15 to < 20, 20 to < 25, and 25 to < 30, or

0 to 4, 5 to 9, 10 to 14, 15 to 19, 20 to 24, and 25 to 29

Once we have decided on our size and number of classes, we have to record the counts or percents of observations out of the total that fall under each class.

We can do this in either a frequency table or relative frequency table.

After we have created our frequency table of choice, we can create our histogram. The widths of each bar are equal, since they are based on the class size. The height of each bar graph is the count or percent of each class.

a) Shape: The graph is right-skewed and unimodal with its peak located at the 0 to < 5 class.

b) Center: 39 is an odd number of observations, so the median will be in dead center. When the observations for each distribution are ordered from least to greatest, we find the median for the distribution to be the 20th observation, or the one where the child has 5 toys collected.

Even if we don’t have the original data set, if the 0 to < 5 class class has 17 (or 43.6%) observations and the 5 to < 10 class has more than the 3 (or 6.4%) observations remaining to reach the median, the median would still fall somewhere in the 5 to < 10 class.

c) Spread: The range for the distribution is 28−0=28 toys collected28-0=28\text{ toys collected}.

d) Unusual features: There is a small gap between the cluster at the left side of the graph where the majority of the observations are found and the potential outlier of 28 toys collected.

To better visualize this example problem, refer to Gallery 1.2.1

Source gap (S0032): Source wording and percentage are retained: the median explanation says “3 (or 6.4%) observations”. No corrected percentage is substituted.

Pop Quiz 1.2.1

Question 1 of 5

A histogram graphing a frequency table has a horizontal axis from 40 to 50, with a class size of 2. Each bar has a height of: 5, 7, 14, 10, and 19. The median falls in the…

Question 2 of 5

If a distribution has a range of 15 and over 100 observations, the best graph to use would be a:

Question 3 of 5

If a distribution has a range of 75, over 100 observations, and you wanted to know the exact value of each observation, the best graph to use would be a:

Question 4 of 5

Draw a stemplot with stems 4, 5, 6, and 7 each with 1, 2, 4, and 3 leaves respectively in this order: 8, 2, 6, 0, 4, 5, 8, 2, 3, 9. Which of the following values below was omitted?

Question 5 of 5

From question 4, which of the following values below was included?

Source gap (S0033): The five quiz captures show the questions and choices without any checked-answer or feedback state. No source answer key is available.
Glossary and deeper reading · 14 refreshers

These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.

  • Quantitative variable

    A quantitative variable records a numerical count or measurement for each individual. Differences between its values have a meaningful numerical interpretation. For example, the number of books borrowed and the mass of a backpack are quantitative variables.

    Read more: Quantitative variable in the captured Statistics reading →

  • Distribution of data

    A distribution describes the values a variable takes and how often those values occur. For quantitative data, useful features include shape, center, spread, and unusual observations. For example, a distribution of commuting times shows which times are common and how widely they vary.

  • Symmetric distribution

    A symmetric distribution has approximately matching shapes on either side of its center. Symmetry alone does not establish that a distribution is Normal. For example, a uniform distribution over a centered interval is symmetric but has a flat rather than bell-shaped density.

  • Observation

    An observation is a recorded value or set of values for an individual or occasion. The individual may be a person, object, or event rather than a human subject. For example, one plant’s measured height is an observation in a plant-growth study.

  • Unimodal distribution

    A unimodal distribution has one main peak. The peak marks a region where observations or probability are especially concentrated. For example, a histogram of heights might have one prominent peak around the most common height interval.

  • Bimodal distribution

    A bimodal distribution has two main peaks. Two peaks can suggest different groups, but the graph alone does not establish their cause. For example, combining commute times from nearby and distant neighborhoods may produce two peaks.

  • Median

    The median is the middle of an ordered numerical data set. For an even number of observations, it is the average of the two middle values under the usual convention. For example, the median of 2, 4, and 20 is 4.

  • Range of data

    The range of a data set is its maximum value minus its minimum value. It measures spread and is sensitive to extreme observations. For example, values from 3 to 11 have a range of 8.

    Read more: Range of data in the captured Statistics reading →

  • Outlier

    An outlier is an observation unusually far from the overall pattern. The 1.5-IQR rule flags possible outliers below the first quartile minus 1.5 IQR or above the third quartile plus 1.5 IQR. For example, a very long commute may be flagged for investigation without automatically being a recording error.

    Read more: Outlier in the captured Statistics reading →

  • Dotplot

    A dotplot places a dot for each observation above its numerical value. Repeated values are usually shown by stacking dots. For example, three observations equal to 4 produce a stack of three dots above 4.

  • Stemplot

    A stemplot splits each numerical observation into a stem and a leaf. A key specifies the place values, so the original observations can be recovered. For example, a key of 3 | 7 = 37 means that stem 3 with leaf 7 represents 37.

  • Back-to-back stemplot

    A back-to-back stemplot compares two data sets using shared stems. Leaves for one group go on one side and leaves for the other group go on the opposite side. For example, two classes’ test scores can be compared while retaining their individual values.

  • Bar graph

    A bar graph uses separate bars to compare category counts or proportions. The bar lengths or heights encode the values, and category order need not be numerical. For example, separate bars can compare the number of votes for three lunch options.

    Read more: Bar graph in the captured Statistics reading →

  • Categorical variable

    A categorical variable records a group or label for each individual. Numeric labels still count as categories when arithmetic on them has no meaningful interpretation. For example, favorite school subject is categorical, even if each subject receives a numbered code.

    Read more: Categorical variable in the captured Statistics reading →

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