An award-winning example

Here’s an example of how to create a histogram for all you movie lovers out there (especially those who love old movies). The Academy Awards started in 1928, and one of the most popular categories for this award is Best Actress in a Motion Picture. Table 7-1 shows the winners of the first eight Best Actress Oscars, the years they won (1929–1936), their ages at the time of winning their awards, and the movies they were in. From the table, you see that the ages cover a range from 22 to 63 — much wider than you may have thought it would be.

To find out more about the ages of the Best Actress winners, I expanded my data set to the period 1929 to 2021. The age variable for this data set is numerical, so you can graph it using a histogram. From there you can answer questions like: What do the ages of these actresses look like? Are they mostly young, old, or in between? Are their ages all spread out, or are they similar? Are most of them in a certain age range, with a few outliers (either very young or very old actresses, compared to the others)? To investigate these questions, a histogram of ages of the Best Actress winners is shown in Figure 7-1.

Notice that the age groups are shown on the horizontal (x) axis. They go by groups of five years each: 20–25, 25–30, 30–35, … , 80–85. The percentage (relative frequency) of actresses in each age group appears on the vertical (y) axis. For example, about 27 percent of the actresses were between 25 and 30 years of age when they won their Oscars.

Creating appropriate groups

For Figure 7-1, I used groups of five years each in the preceding example because increments of five create natural breaks for years and because this grouping provides enough bars to look for general patterns. You don’t have to use this particular grouping, however; you have a bit of creative license when making a histogram. (However, this freedom also allows others to deceive you, as you see in the later section, “Detecting misleading histograms.”) Here are some tips for setting up your histogram:

• Each data set requires different ranges for its groupings, but you want to avoid ranges that are too wide or too narrow.
• If a histogram has really wide ranges for its groups, it places all the data into a very small number of bars that make meaningful comparisons impossible.
• If the histogram has very narrow ranges for its groups, it looks like a big series of tiny bars that cloud the big picture. This can make the data look very choppy with no real pattern.
• Make sure your groups have equal widths. If one bar is wider than the others, it may contain more data than it should.

One idea that may be appropriate for your histogram is to take the range of the data (largest minus smallest) and divide by 10 to get 10 groupings.

Handling borderline values

In the Academy Award example, what happens if an actress’s age lies right on a borderline? For example, Olivia de Havilland was 30 years old in 1947 when she won the Oscar for To Each His Own. Does she belong in the 25–30 age group (the lower bar) or the 30–35 age group (the upper bar)?

As long as you are consistent with all the data points, you can either put all the borderline points into their respective lower bars or put all of them into their respective upper bars. The important thing is to pick a direction and be consistent. In Figure 7-1, I went with the convention of putting all borderline values into their respective lower bars — which puts Olivia de Havilland’s age in the second bar, the 25–30 age group of Figure 7-1.

Clarifying the axes

The most complex part of interpreting a histogram for the reader is to get a handle on what’s being shown on the x and y axes. Having good descriptive labels on the axes helps. Most statistical software packages label the x-axis using the variable name you provide when you enter your data (for example, “age” or “weight”). However, the label for the y-axis isn’t as clear. Statistical software packages often label the y-axis of a histogram by writing “frequency” or “percent” by default. These terms can be confusing: frequency or percentage of what?

Clarify the y-axis label on your histogram by changing “frequency” to “number of” and adding the variable name. To modify a label that simply reads “percent,” clarify by writing “percentage of” and the variable. For example, in the histogram of ages of the Best Actress winners shown in Figure 7-1, I labeled the y-axis “Percentage of actresses in each age group.” In the next section, you see how to interpret the results from a histogram. How old are those actresses anyway?

4 Suppose you have a loaded die. You roll it several times and record the outcomes, which are shown in the following figure.

1. Make a relative frequency histogram of these results.
2. You can make a relative frequency histogram from a frequency histogram; can you go in the other direction?

Checking out the shape of the data

One of the features that a histogram can show you is the shape of the data — in other words, the manner in which the data fall into the groups. For example, all the data may be exactly the same, in which case the histogram is just one tall bar; or the data may have an equal number in each group, in which case the shape is flat.

Some data sets have a distinct shape. Here are three shapes that stand out.

• Symmetric: A histogram is symmetric if you cut it down the middle and the left-hand and right-hand sides resemble mirror images of each other.

  Figure 7-2a shows a symmetric data set; it represents the amount of time each of 50 survey participants took to fill out a certain survey. You see that the histogram is close to symmetric.

• Skewed right: A skewed-right histogram looks like a lopsided mound, with a tail going off to the right.

  Figure 7-1 from earlier in the chapter, showing the ages of the Best Actress Award winners, is skewed right. You see on the right side that there are a few actresses whose ages are older than the rest.

• Skewed left: If a histogram is skewed left, it looks like a lopsided mound with a tail going off to the left.

  Figure 7-2b shows a histogram of 17 exam scores. The shape is skewed left; you see a few students who scored lower than everyone else.

Following are some particulars about classifying the shape of a data set:

• Don’t expect symmetric data to have an exact and perfect shape. Data hardly ever fall into perfect patterns, so you have to decide whether the data shape is close enough to be called symmetric.

  If the shape is close enough to symmetric that another person would notice it, and the differences aren’t enough to write home about, I’d classify it as symmetric or roughly symmetric. Otherwise, you classify the data as nonsymmetric. (More sophisticated statistical procedures exist that actually test data for symmetry, but they’re beyond the scope of this book.)

• Don’t assume that data are skewed if the shape is nonsymmetric. Data sets come in all shapes and sizes, and many of them don’t have a distinct shape at all. I include skewness on the list here because it’s one of the more common nonsymmetric shapes, and it’s one of the shapes included in a standard introductory statistics course.

  If a data set does turn out to be skewed (or close to it), make sure to denote the direction of the skewness (left or right).

As you can see in Figure 7-1, the actresses’ ages in the histogram are skewed right. Most of the actresses were between 20 and 50 years of age when they won, with about 27 percent of them between the ages of 25 and 30. A few actresses were older when they won their Oscars; about 8 percent were between 60 and 65 years of age, and not more than 2 percent (total) were over 70 (if you add the percentages from the last two bars in the histogram). The last few bars on the right side are what give the data a shape that is skewed right.

Measuring center: Mean versus median

A histogram gives you a rough idea of where the “center” of the data lies. The word center is in quotes because many different statistics are used to designate center. The two most common measures of center are the average (the mean) and the median. (For details on measures of center, see Chapter 5.)

To visualize the average age (the mean), picture the data as people sitting on a teeter-totter. Your objective is to balance it. Because data don’t move around, assume the people stay where they are and that you can move the pivot point (which you can also think of as the hinge or fulcrum) anywhere you want. The mean is the place the pivot point has to be in order to balance the weight on each side of the teeter-totter.

The balancing point of the teeter-totter is affected by the weights of the people on each side, not by the number of people on each side. So the mean is affected by the actual values of the data, rather than the amount of data.

The median is the place where you put the pivot point so you have an equal number of people on each side of the teeter-totter, regardless of their weights. With the same number of people on each side, the teeter-totter wouldn’t balance in terms of weight unless the teeter-totter had people with the same total weight on each side. So the median isn’t affected by the values of the data, just their location within the data set.

The mean is affected by outliers, values in the data set that are away from the rest of the data, on the high end and/or the low end. The median, being the middle number, is not affected by outliers.

Viewing variability: Amount of spread around the mean

You also get a sense of variability in the data by looking at a histogram. For example, if the data are all the same, they are all placed into a single bar, and there is no variability. If an equal amount of data is in each group, the histogram looks flat, with the bars close to the same height; this shows a fair amount of variability.

The idea of a flat histogram indicating some variability may go against your intuition, and if it does, you’re not alone. If you’re thinking a flat histogram means no variability, you’re probably thinking about a time chart, where single numbers are plotted over time (see the section, “Tackling Time Charts,” later in this chapter). Remember, though, that a histogram doesn’t show data over time — it shows all the data at one point in time.

Equally confusing is the idea that a histogram with a big lump in the middle and tails sloping sharply down on each side actually has less variability than a histogram that’s straight across. The curves looking like hills in a histogram represent clumps of data that are close together; a flat histogram shows data equally dispersed, with more variability.

Variability in a histogram is higher when the taller bars are more spread out around the mean and lower when the taller bars are close to the mean.

For the Best Actress Award winners’ ages shown earlier in Figure 7-1, you see many actresses are in the age range from 25 to 30, and most of the ages are between 20 and 50 years, which is quite diverse. Then you have some outliers, those few older actresses (two of them include Jessica Tandy for Driving Miss Daisy and Katharine Hepburn for On Golden Pond); their ages spread the data out farther, increasing its overall variability.

The most common statistic used to measure variability in a data set is the standard deviation, which, in a rough sense, measures the average distance that the data lie from the mean. The standard deviation for the Best Actress Award age data is 12.23 years. (See Chapter 5 for all the details on standard deviation.) A standard deviation of 12.23 years is fairly large in the context of this problem, but the standard deviation is based on average distance from the mean, and the mean is influenced by outliers, so the standard deviation will be as well (see Chapter 5 for more information).

In the later section, “Interpreting a boxplot,” I discuss another measure of variability, called the interquartile range (IQR), which is a more appropriate measure of variability when you have skewed data.

Q. The police checked the speeds of cars after the city painted lines on a certain section of a street where the road narrows. The speeds are organized in the histogram shown in the following figure. Describe what this histogram tells you about the speeds of the cars.

A. The speeds of the cars in this data set range from 28 miles per hour (lowest) to 41 mph (highest). Most of the cars traveled from 30 to 35 mph (you can tell by noting that those bars have the highest frequencies). A few cars drove faster than the rest (noted by the few short bars at the upper end), which indicates a skewed-right shape. The average speed seems to center around 32 mph (but this is hard to tell without doing more analysis).

5 An ATM machine asks customers who use the “fast cash” option to choose an amount in $50 increments from $100 to $500. Results from a recent sample of customer withdrawals are shown in the following figure. Discuss the shape, center, and spread of the data.

6 A histogram of a sample of rods made by Rowdy Rod’s is shown in the following figure. The rods should be 100 inches in length. Discuss the company’s accuracy (in terms of meeting the length specification) by interpreting the shape, center, and spread of the data.

7 A histogram of the amount of money 317 households spent on fruits and vegetables in a year is shown in the following figure (based on a random sample). Discuss the shape, center, and spread. What do these three characteristics say about how much families spend on fruits and vegetables? (Such analysis is called interpreting your results in the context of the problem.)

8 An investor monitors the percentage return for a particular group of stocks in Portfolio A over a one-year period. The one-year percentage returns for these stocks are shown in the following figure.

1. Some of the values on the x-axis are negative numbers. What does this mean?
2. On any histogram in general, when (if ever) can the x- and/or y-axis contain negative values?

9 You make two histograms from two different data sets (see the following figures), with each one containing 200 observations. Which of the histograms has a smaller spread, the first or the second?

Q. The police checked the speeds of cars after the city painted lines on a certain section of a street where the road narrows. The speeds are organized in the histogram shown in the following figure. Which is greater: the mean of this data or the median of this data?

A. The mean is greater. The data is skewed to the right because there a few more outlying large values than there are smaller values. So the mean gets pulled out toward the large values to keep that seesaw-type balance.

10 The incomes of last year’s new graduates of a certain large and very successful program are shown in the following figure.

1. Discuss the implications for graduates of this program.
2. Estimate where the median salary is in this data set.
3. Do you see any issues that anyone who tries to interpret this data should take into account?

Missing the mark with too few groups

Although the number of groups used for a histogram is up to the discretion of the person making the graph, there is such a thing as going overboard, either by having way too few bars, with everything lumped together, or by having way too many bars, where every little difference is magnified.

To decide how many bars a histogram should have, take a good look at the groupings used to form the bars on the x-axis and see whether they make sense. For example, it doesn’t make sense to talk about exam scores in groups of 2 points; that’s too much detail — too many bars. On the other hand, it doesn’t make sense to group actresses’ ages by intervals of 20 years; that’s not descriptive enough.

Figures 7-4 and 7-5 illustrate this point. Each histogram summarizes observations of the amount of time between eruptions of the Old Faithful geyser in Yellowstone Park. Figure 7-4 uses six bars that group the data by ten-minute intervals. This histogram shows a general skewed-left pattern, but with 222 observations, you are cramming an awful lot of data into only six groups; for example, the bar for 75–85 minutes has more than 90 pieces of data in it. You can break it down further than that.

Figure 7-5 is a histogram of the same data set, where the time between eruptions is broken into groups of three minutes each, resulting in 19 bars. Notice the distinct pattern in the data that shows up with this histogram, which wasn’t uncovered in Figure 7-4. You see two distinct peaks in the data: one peak around the 50-minute mark, and one around the 75-minute mark. A data set with two peaks is called bimodal;Figure 7-5 shows a clear example.

Looking at Figure 7-5, you can conclude that the geyser has two categories of eruptions: one group that has a shorter waiting time, and another group that has a longer waiting time. Within each group, you see the data are fairly close to where the peak is located. Looking at Figure 7-4, you couldn’t say that.

If the interval for the groupings of the numerical variable is really small, you see too many bars in the histogram; the data may be hard to interpret because the heights of the bars look more variable than they should. On the other hand, if the ranges are really large, you see too few bars, and you may miss something interesting in the data.

Watching the scale and start/finish lines

The y-axis of a histogram shows how many individuals are in each group, using counts or percents. A histogram can be misleading if it has a deceptive scale and/or inappropriate starting and ending points on the y-axis.

Watch the scale on the y-axis of a histogram. If it goes by large increments and has an ending point that’s much higher than needed, you see a great deal of white space above the histogram. The heights of the bars are squeezed down, making their differences look more uniform than they should. If the scale goes by small increments and ends at the smallest value possible, the bars become stretched vertically, exaggerating the differences in their heights and suggesting a bigger difference than really exists.

An example comparing scales on the vertical (y) axes is shown in Figures 7-5 and 7-6. I took the Old Faithful data (time between eruptions) and made a histogram with vertical increments of 20 minutes, from 0 to 100; see Figure 7-6. Compare this to Figure 7-5, with vertical increments of five minutes, from 0 to 35. Figure 7-6 has a lot of white space and gives the appearance that the times are more evenly distributed among the groups than they really are. It also makes the data set look smaller, if you don’t pay attention to what’s on the y-axis. Of the two graphs, Figure 7-5 is more appropriate.

Q. In an earlier practice question, you see data from a die that’s clearly loaded (not fair). However, someone (the gambler’s lawyer, perhaps?) could make that same die appear fair by setting up the histogram a certain way. Explain how the following histogram (made from that same data) makes the die appear to be fair.

A. This histogram combines the results of rolling a 1 and 2, rolling a 3 and 4, and rolling a 5 and 6, so there are three bars on the histogram now, not six. This histogram is misleading because when you combine the results into three groups of two outcomes each, the differences that make the die loaded don’t show up. The lack of precision works to the advantage of the person who created the graph.

11 Suppose your friend believes their gambling partner plays with a loaded die (not fair). They show you a graph of the outcomes of the games played with this die (see the following figure). Based on this graph, do you agree with your friend? Why or why not?

12 The first month’s telephone bills for new customers of a certain phone company are shown in the following figure. The histogram showing the bills is misleading, however. Explain why, and suggest a solution.

Q. Make a vertical boxplot from this data set of exam scores: 43, 54, 56, 61, 62, 66, 68, 69, 69, 70, 71, 72, 77, 78, 79, 85, 87, 88, 89, 93, 95, 96, 98, 99, 99.

A. From an example question in Chapter 5, you found the five-number summary for this data set to be 43, 68, 77, 89, and 99, which completes Step 1. Steps 2 and 3 say to draw a vertical number line (the y-axis) that spans 43 to 99 and mark the five-number summary values. In Step 4 you draw a box with 68 and 89 marking the edges, and in Step 5 you draw a line for the median at 77.

   To start Step 6, you calculate . Any large outliers would have to be greater than , and any small outliers would have to be less than . Because none of the exam scores fall outside of those values, there are no outliers. According to Step 7, you draw the remaining lines from the quartiles to the maximum (99) and minimum (43). See the completed boxplot in the following picture.

13 Suppose you have measured the height in inches of 11 randomly selected adults, and in order from lowest to highest they look like this: 59, 61, 65, 66, 67, 67, 69, 70, 72, 73, 75. Make a horizontal boxplot of these heights.

14 Suppose you have surveyed 14 randomly selected adults for their typical commute times to work, and the responses look like this: 16, 8, 35, 17, 13, 15, 15, 5, 16, 25, 20, 20, 12, 10. Make a vertical boxplot of these times.

Checking the shape with caution!

A boxplot can show whether a data set is symmetric (roughly the same on each side when cut down the middle) or skewed (lopsided). A symmetric data set shows the median roughly in the middle of the box. Skewed data show a lopsided boxplot, where the median cuts the box into two unequal pieces. If the longer part of the box is to the right of (or above) the median, the data is said to be skewed right. If the longer part is to the left of (or below) the median, the data is skewed left.

As shown in the boxplot of the data in Figure 7-7, the ages are skewed right. The part of the box to the left of the median (representing the younger actresses) is shorter than the part of the box to the right of the median (representing the older actresses). That means the ages of the younger actresses are closer together than the ages of the older actresses. Figure 7-3 shows the descriptive statistics of the data and confirms the right skewness: the median age (33 years) is lower than the mean age (36.78 years).

If one side of the box is longer than the other, it does not mean that side contains more data. In fact, you can’t tell the sample size by looking at a boxplot; it’s based on percentages, not counts. Each section of the boxplot (the minimum to , to the median, the median to , and to the maximum) contains 25 percent of the data no matter what. If one of the sections is longer than another, it indicates a wider range in the values of data in that section (meaning the data are more spread out). A smaller section of the boxplot indicates the data are more condensed (closer together).

Although a boxplot can tell you whether a data set is symmetric (when the median is in the center of the box), it can’t tell you the shape of the symmetry the way a histogram can. For example, Figure 7-8 shows histograms from two different data sets, each one containing 18 values that vary from 1 to 6. The histogram on the left has an equal number of values in each group, and the one on the right has two peaks at 2 and 5. Both histograms show the data are symmetric, but their shapes are clearly different.

Figure 7-9 shows the corresponding boxplots for these same two data sets; notice they are exactly the same. This is because the data sets both have the same five-number summaries — they’re both symmetric with the same amount of distance between , the median, and . However, if you just saw the boxplots and not the histograms, you might think the shapes of the two data sets are the same, when indeed they are not.

Despite its weakness in detecting the type of symmetry (you can add a histogram to your analyses to help fill in that gap), a boxplot has a great upside in that you can identify actual measures of spread and center directly from the boxplot, where on a histogram you can’t. A boxplot is also good for comparing data sets by showing them on the same graph, side by side.

All graphs have strengths and weaknesses; it’s always a good idea to show more than one graph of your data for that reason.

Measuring variability with IQR

Variability in a data set that is described by the five-number summary is measured by the interquartile range (IQR). The IQR is equal to , the difference between the 75th percentile and the 25th percentile (the distance covering the middle 50 percent of the data). The larger the IQR, the more variable the data set is.

From Figure 7-3, the variability in age of the Best Actress winners as measured by the IQR is years. Of the group of actresses whose ages were closest to the median, half of them were within 13 years of each other when they won their awards.

Notice that the IQR ignores data below the 25th percentile or above the 75th, which may contain outliers that could inflate the measure of variability of the entire data set. So if data are skewed, the IQR is a more appropriate measure of variability than the standard deviation.

Picking out the center using the median

The median, part of the five-number summary, is shown by the line that cuts through the box in the boxplot. This makes it very easy to identify. The mean, however, is not part of the boxplot and can’t be determined accurately by just looking at the boxplot.

You don’t see the mean on a boxplot because boxplots are based completely on percentiles. If data are skewed, the median is the most appropriate measure of center. Of course, you can calculate the mean separately and add it to your results; it’s never a bad idea to show both.

Investigating Old Faithful’s boxplot

The relevant descriptive statistics for the Old Faithful geyser data are found in Figure 7-10.

You can predict from the data set that the shape will be skewed left a bit because the mean is lower than the median by about four minutes. The IQR is minutes, which shows the amount of overall variability in the time between eruptions; 50 percent of the eruptions are within 21 minutes of each other.

A vertical boxplot for length of time between eruptions of the Old Faithful geyser is shown in Figure 7-11. You confirm that the data are skewed left because the lower part of the box (where the small values are) is longer than the upper part of the box.

You see the values of the boxplot in Figure 7-11 that mark the five-number summary and the information shown in Figure 7-10, including the IQR of 21 minutes to measure variability. The center as marked by the median is 75 minutes; this is a better measure of center than the mean (71 minutes), which is driven down a bit by the left-skewed values (the few that are shorter times than the rest of the data).

Looking at the boxplot (Figure 7-11), you see there are no outliers denoted by asterisks. However, note that the boxplot doesn’t pick up on the bimodal shape of the data that you see previously in Figure 7-5. You need a good histogram for that.

Denoting outliers

Looking at the boxplot in Figure 7-7 for the Best Actress ages data, you see a set of what the computer software defines at outliers (ten in all) on the right side of the data set, marked by a group of asterisks (as described in Step 8 in the earlier section, “Making a boxplot”). Four of the asterisks lie on top of one another because four actresses were the same age, 61, when they won their Oscars.

The computer uses certain criteria to determine what it decides is an outlier. You can verify these outliers by applying the rule described in Step 6 of the section, “Making a Boxplot.” The IQR is 13 (from Figure 7-3), so you take years. Add this amount to and you get years; subtracting this amount from , you get years. So an actress whose age was below 8.5 years (that is, 8 years old and under) or above 58.5 years (that is, 59 years old or over) is considered to be an outlier.

Of course, the lower end of this boundary (8 years) isn’t relevant because the youngest actress was 21 (Figure 7-3 shows the minimum is 21). So you know there aren’t any outliers on the low end of this data set.

However, ten outliers, as defined by Minitab, are on the high end of the data set, where the 59-and-over actresses’ ages are. Table 7-2 shows the information on all ten outliers in the Best Actress ages data set.

Making mistakes when interpreting a boxplot

It’s a common mistake to associate the size of the box in a boxplot with the amount of data in the data set. Remember that each of the four sections shown in the boxplot contains an equal percentage (25 percent) of the data; the boxplot just marks off the places in the data set that separate those sections.

In particular, if the median splits the box into two unequal parts, the larger part contains data that’s more variable than the smaller part, in terms of its range of values. However, there is still the same amount of data (25 percent) in the larger part of the box as there is in the smaller part.

Another common error involves sample size. A boxplot is a one-dimensional graph with only one axis representing the variable being measured. There is no second axis that tells you how many data points are in each group. So if you see two boxplots side by side and one of them has a very long box and the other has a very short one, don’t conclude that the longer one has more data in it. The length of the box represents the variability in the data, not the number of data values.

When viewing or making a boxplot, always make sure the sample size (n) is included as part of the title. You can’t figure out the sample size otherwise.

Q. Here is a boxplot of the most recent test scores from a class of 36 students.

1. Describe the shape of the distribution. Are there any outliers?
2. How many students are represented by the box in the middle of the boxplot?

A. Keep in mind that a boxplot is constructed from just the positions of the five values from the five-number summary and not much else. Be careful when making interpretations.

1. The fact that both the line and the part of the box in the lower half of the boxplot stretch out longer than the same parts in the upper half leads you to believe that the data is also stretched out to smaller values. This means the data is likely skewed left. Also, it seems there is one outlier around the value of 40. (Yikes! Maybe that student wasn’t studying as hard as you are.)
2. The middle box of any boxplot will contain 50 percent of the data. That’s everybody from (the 25th percentile) to (the 75th percentile). But that’s not enough to know how many students are in there. You also need the sample size. Fortunately, it’s given that there are students in the class. That means there are students represented by the middle box in the boxplot.

15 Which of the following two boxplots contains more data: the one containing the heights of men or the one containing the heights of women?

Watching the scale and start/end points

The scale on the vertical axis can make a big difference in the way the time chart looks. Refer to Figure 7-12 to see the original time chart of the ages for the Best Actress Academy Award winners from 1929 to 2021 in increments of ten years. You see a fair amount of variability, as discussed previously.

In Figure 7-12, the starting and ending points on the vertical axis are 0 and 100, which creates some extra white space on the top and bottom of the picture. I could have used 10 and 90 as my start/end points, but this graph looks reasonable.

Now, what happens if I change the vertical axis? Figure 7-13 shows the same data, with start/end points of 20 and 80. The increments of 10 years appear longer than the increments of 10 years shown in Figure 7-12. Both of these changes in the graph exaggerate the differences in ages even more.

How do you decide which graph is the best one for your data? There is no perfect graph, no right or wrong answer; but there are limits. You can quickly spot problems just by zooming in on the scale and the start/end points.

Simplifying excess data

A time chart of the intervals between eruptions for the Old Faithful data is shown in Figure 7-14. You see 222 dots on this graph; each one represents the time between one eruption and the next, for every eruption during a 16-day period.

This figure looks very complex; data are everywhere, there are too many points to really see anything, and you can’t find the forest for the trees. There is such a thing as having too much data, especially nowadays when you can measure data continuously and meticulously using all kinds of advanced technology. I’m betting they didn’t have a student standing by the geyser recording eruption times on a clipboard, for example!

To get a clearer picture of the Old Faithful data, I combined all the observations from a single day and found its mean; I did this for all 16 days, and then I plotted all the means on a time chart in order. This reduced the data from 222 points to 16 points. The time chart is shown in Figure 7-15.

From this time chart, I see a little bit of a cyclical pattern to the data; every day or two, it appears to shift from short times between eruptions to longer times between eruptions. While these changes are not definitive, it does provide important information for scientists to follow up on when studying the behavior of geysers like Old Faithful.

A time chart condenses all the data for one unit of time into a single point. By contrast, a histogram displays the entire sample of data that was collected at that one unit of time. For example, Figure 7-15 shows the daily average time between eruptions for 16 days. For any given day, you can make a histogram of all the eruptions observed on that particular day. Displaying a time chart of average times over 16 days accompanied by a histogram summarizing all the eruptions for a particular day would be a great one-two punch.

Q. The following figure shows the revenues of a company taken over time. Each dot represents the revenue for that year, in millions of dollars.

1. What’s the time period over which this data set was collected?
2. Describe what the line graph tells you about the revenues of this company over the time period.
3. What do you need to take into account in order to properly interpret revenues (or any variable reported in dollars) over time?

A. Be aware of the impact of inflation over time on data reported in dollars. Some line graphs adjust for inflation and some don’t. Look at the fine print.

1. The time period is approximately 1970 to 2000.
2. Revenues increased a little in the 1970s and then began a more steady increase in the ’80s until around 1989, when the company broke the $50 million barrier. The company experienced a big jump around 1990–1991, with a very strong and steady increase each year since. In 2000, the company’s revenue was up to $225 million and rising.
3. You should take inflation of the dollar over time into account. The revenues may look larger later on, but the value of the dollar has decreased over time as well. Some line graphs adjust for inflation.

16 Check out the sales of a particular car across the United States over a 60-day period in the following figure.

1. Can you see a pattern to the sales of this car across this time period?
2. What are the highest and lowest numbers of sales and when did they occur?
3. Can you estimate the average of all sales over this time period?

17 After a heart attack, Bob decides that it is a good time to get in shape, so he starts exercising each day and plans to increase his exercise time as he goes along. Look at the two line graphs shown in the following figures. One is a good representation of his data, and the other should get as much use as Bob’s treadmill before his heart attack.

1. Compare the two graphs. Do they represent the same data set, or do they show totally different data sets?
2. Assume both graphs are made from the same data. Which graph is more appropriate and why?

18 The line graph in the following figure shows one company’s revenues over time. Explain why this graph is misleading and what you can do to fix the problem.