Tallying personal expenses

When you spend your money, what do you spend it on? What are your top three expenses? According to the U.S. Bureau of Labor Statistics Consumer Expenditure Survey, the top six sources of consumer expenditures in the U.S. were housing (33.9 percent), transportation (17.0 percent), food (12.8 percent), personal insurance and pensions (11.1 percent), healthcare (5.9 percent), and entertainment (5.6 percent). These six categories make up over 85 percent of average consumer expenses. (Although the exact percentages change from year to year, the list of the top six items remains the same.)

Figure 6-1 summarizes the U.S. expenditures in a pie chart. Notice that the “other” category appears a bit large in this chart (13.7 percent). However, with so many other possible expenditures out there (including this book), each one would only get a tiny slice of the pie for itself, and the resulting pie chart would be a mess. In this case, it is too difficult to break “other” down further. (But in many other cases, you can.)

Ideally, a pie chart shouldn’t have too many slices because a large number of slices distracts the reader from the main point(s) the pie chart is trying to relay. However, lumping the remaining categories into one slice that’s one of the largest in the whole pie chart leaves readers wondering what’s included in that particular slice. With charts and graphs, doing it right is a delicate balance.

Bringing in a lotto revenue

State lotteries bring in a great deal of revenue, and they also return a large portion of the money received, with some of the revenues going to prizes and some being allocated to state programs such as education. Where does lottery revenue come from? Figure 6-2 is a pie chart showing the types of games and their percentage of revenue as recently reported by Ohio’s state lottery. (Note that the slices don’t sum to 100 percent exactly due to a slight rounding error.)

You can see by the pie chart in Figure 6-2 that 49.3 percent of the lottery sales revenue comes from the instant (scratch-off) games. The rest comes from various lottery-type games in which players choose a set of numbers and win if a certain number of their numbers match those chosen by the lottery.

Notice that this pie chart doesn’t tell you how much money came in, only what percentage of the money came from each type of game. About half the money (49.3 percent) came from instant scratch-off games; does this revenue represent 1 million dollars, 2 million dollars, 10 million dollars, or more? You can’t answer these questions without knowing the total amount of revenue dollars.

I was, however, able to find this information on another chart provided by the lottery website: The total revenue (over a 10-year period) was reported as “1,983.1 million dollars” — which you also know as 1.9831 billion dollars. Because 49.3 percent of sales came from instant games, they therefore represent sales revenue of $977,668,300 over a 10-year period. That’s a lot of (or dare I say a “lotto”) scratching.

Ordering takeout

It’s also important to watch for totals when examining a pie chart from a survey. A newspaper I read reported the latest results of a “people poll.” They asked, “What is your favorite night to order takeout for dinner?” The results are shown in the pie chart in Figure 6-3.

You can clearly see that Friday night is the most popular night for ordering takeout (and that result makes sense), with decreasing demand moving from Saturday through Monday. The actual percentages shown in Figure 6-3 really only apply to the people who were surveyed; how close these results mimic the population depends on many factors, one of which is sample size. But unfortunately, sample size is not included as part of this graph. (For example, it would be nice to see “n = XXX” below the title, where n represents sample size.)

Even though you have a good sample selected (i.e. it’s representative of the target population and is random), without knowing the sample size, you can’t tell how accurate the information is. Which results would you find to be more accurate: those based on 25 people, 250 people, or 2,500 people? When you see “10 percent,” you don’t know if it’s 10 out of 100, 100 out of 1,000, or even 1 out of 10. To statisticians, is not the same as , even though they both represent 10 percent. (Don’t tell that to mathematicians — they’ll think you’re nuts!)

Pie charts often don’t include the total sample size. Always check for the sample size, especially if the results are very important to you; don’t assume it’s large! If you don’t see the sample size, go to the source of the data and ask for it.

Projecting age trends

The U.S. Census Bureau provides an almost unlimited amount of data, statistics, and graphics about the U.S. population, including the past, the present, and projections for the future. It often makes comparisons between years in order to look for changes and trends.

One recent Census Bureau population report looked at what it calls the “older U.S. population” (by the government’s definition, this means people 65 years old or over). Age was broken into the following groups: 65–69 years, 70–74 years, 75–79 years, 80–84 years, and 85 and over. The Bureau calculated and reported the percentage in each age group for the year 2020 and made projections for the percentage in each age group for the year 2050.

I made side-by-side pie charts for the years 2020 versus 2050 (projections) to make comparisons; you can see the results in Figure 6-4. The percentage of the older population in each age group for 2020 is shown in one pie chart, and alongside it is a pie chart of the projected percentage for each age group for 2050 (based on the current age of the entire U.S. population, birth and death rates, and other variables).

If you compare the sizes of the slices from one graph to the other in Figure 6-4, you see that the slices for corresponding age groups are larger for the 2050 projections (compared to 2020) among the older age groups, and the slices are smaller for the 2050 projections (compared to 2020) among the younger age groups. For example, the 65–69 age group decreases from 30 percent in 2020 to a projected 25 percent in 2050; while the 85-and-over age group increases from 14 percent in 2020 to 19 percent projected for 2050.

The results from Figure 6-4 indicate a shift in the ages of the population toward the older categories. From there, the medical and social research communities can examine the ramifications of this trend in terms of healthcare, assisted living, social security, and so on.

The operative words here are if the trend continues. As you know, many variables affect population size, and you need to take those into account when interpreting these projections into the future. The U.S. government always points out caveats like this in its reports; it is very diligent about that.

The pie charts in Figure 6-4 work well for comparing groups because they are side by side on the same graph, they use the same coding for the age groups in each chart, and their slices are in the same order for both charts as you move clockwise around them. They aren’t all scrambled up on each chart so you have to hunt for a certain age group on each chart separately.

Q. A hardware store wants to know what percentage of its customers are women. The manager takes a random sample of 76 customers who enter the store and records their gender. Twenty-two customers are females; the rest are males. I summarize the results in the following pie chart.

1. Describe the results.
2. How can this pie chart be improved?

A. Apparently, the DIY craze is popular with women, too.

1. The results of the pie chart show that the percentage of female customers appears to be around 1/3 (or around 33 percent).
2. You can improve the chart by showing the exact percentages in each slice. (The actual percentages are females: 28.9 percent; males: 71.1 percent.)

2 Suppose Lewis, a restaurant owner, keeps track of data on when his customers patronize his restaurant: breakfast, lunch, dinner, or other times. For a month, he takes time to check off which category each customer falls into. He records data on 1,000 customers for the month. The pie chart in the following figure shows his results.

1. What does this information tell him?
2. Can you spot a problem with the “other” category? How can this study be improved in the future?

5 Suppose as part of a driver’s education program, students have to observe drivers in the real world and see how consistently they come to a complete stop at intersections. The students sit at an intersection for four hours and record whether each driver comes to a complete stop, rolls through the stop sign slowly, or runs the stop sign altogether. You can see the data from the study in the following pie chart.

1. Interpret the results.
2. Do you see any issues with this pie chart?
3. Is it a big deal if you don’t know the sample size?
4. Can you make generalizations about all drivers from this data?

8 An employer/employee study website conducts a survey that asks employers and employees if they think surfing non-work-related websites compromises employee productivity. I summarize the results with the following pie charts.

1. Interpret these results.
2. What important information is missing from these pie charts?

Tracking transportation expenses

How much of their income do people in the United States spend on transportation to get to and from work? It depends on how much money they make. The Bureau of Transportation Statistics (did you know such a department existed?) recently conducted a study on transportation in the U.S., and many of its findings are presented as bar graphs like the one shown in Figure 6-5.

This particular bar graph shows how much money is spent on transportation by people in different household-income groups. It appears that as household income increases, the total expenditures on transportation also increase. This makes sense, because the more money people have, the more they have available to spend.

But would the bar graph change if you looked at transportation expenditures not in terms of total dollar amounts, but as the percentage of household income? The households in the first group make less than $5,000 a year and have to spend $2,500 of that income on transportation. (Note: The label reads “2.5,” but because the units are in thousands of dollars, the 2.5 translates into $2,500.)

This $2,500 represents 50 percent of the annual income of those who make $5,000 per year; the percentage of the total income is even higher for those who make less than $5,000 per year. The households earning $30,000–$40,000 per year pay $6,000 per year on transportation, which is between 15 percent and 20 percent of their household income. So, although the people making more money spend more dollars on transportation, they don’t spend more as a percentage of their total income. Depending on how you look at expenditures, the bar graph can tell two somewhat different stories.

Another point to check out is the groupings on the graph. The categories for household income as shown aren’t equivalent. For example, each of the first four bars represents household incomes in intervals of $5,000, but the next three groups increase by $10,000 each, and the last group contains every household making more than $50,000 per year. Bar graphs using different-sized intervals to represent numerical values (such as Figure 6-5) make true comparisons between groups more difficult. (However, I’m sure the government has its reasons for reporting the numbers this way; for example, this may be the way income is broken down for tax-related purposes.)

One last thing: Notice that the numerical groupings in Figure 6-5 overlap on the boundaries. For example, $30,000 appears in both the fifth and sixth bars of the graph. So, if you have a household income of $30,000, which bar do you fall into? (You can’t tell from Figure 6-5, but I’m sure the instructions are buried in a huge report in the basement of some building in Washington, D.C.) This kind of overlap appears quite frequently in graphs, but you need to know how the borderline values are being treated. For example, the rule may be “Any data lying exactly on a boundary value automatically goes into the bar to its immediate right.” (Looking at Figure 6-5, that puts a household with a $30,000 income into the sixth bar rather than the fifth.) As long as they are being consistent for each boundary, that’s okay. The alternative, describing the income boundaries for the fifth bar as “20,000 to $29,999.99,” is not an improvement. Along those lines, income data can also be presented using a histogram (see Chapter 7), which has a slightly different look to it.

Making a lotto profit

That lotteries rake in the bucks is a well-known fact; but they also shell them out. How does it all shake out in terms of profits? Figure 6-6 shows the recent sales and expenditures of a certain state lottery.

In my opinion, this bar graph needs some additional info from behind the scenes to make it more understandable. The bars in Figure 6-6 don’t represent similar types of entities. The first bar represents sales (a form of revenue), and the other bars represent expenditures. The graph would be much clearer if the first bar weren’t included; for example, the total sales could be listed as a footnote.

Tipping the scales on a bar graph

Another way a graph can be misleading is through its choice of scale on the frequency/relative frequency axis (that is, the axis where the amounts in each group are reported) and/or its starting value.

By using a “stretched-out” scale (for example, having each half-inch of a bar represent 10 units versus 50 units), you can stretch the truth, make differences look more dramatic, or exaggerate values. Truth-stretching can also occur if the frequency axis starts out at a number that’s very close to where the differences in the heights of the bars start; you are, in essence, chopping off the bottom of the bars (the less exciting part) and just showing their tops — emphasizing (in a misleading way) where the action is. Not every frequency axis has to start at zero, but watch for situations that elevate the differences.

A good example of a graph with a stretched-out scale is shown in Chapter 2, regarding the results of numbers drawn in the “Pick 3” lottery. (You choose three one-digit numbers and if they all match what’s drawn, you win.) In Chapter 2, the percentage of times each number (from 0–9) was drawn is shown in Table 2-2, and the results are displayed in a bar graph in Figure 2-1a. The scale on the graph is stretched and starts at 465, making the differences in the results look larger than they really are; for example, it looks like the number 1 was drawn much less often, whereas the number 2 was drawn much more often, when in reality there is no statistical difference between the percentage of times each number was drawn. (I checked.)

Why was the graph in Figure 2-1a made this way? It might lead people to think they’ve got an inside edge if they choose the number 2 because it’s “on a hot streak,” or they might be led to choose the number 1 because it’s “due to come up.” Both of these theories are wrong, by the way, because the numbers are chosen at random; what happened in the past doesn’t matter. In Figure 2-1b you see a graph that’s been made correctly. (For more examples of where your intuition can go wrong with probability and what the scoop really is, see Probability For Dummies by Deborah Rumsey, also published by Wiley.)

Alternatively, by using a “squeezed-down” scale (for example, having each half-inch of a bar represent 50 units versus 10 units), you can downplay differences, making results look less dramatic than they actually are. For example, maybe a politician doesn’t want to draw attention to a big increase in crime from the beginning to the end of their term, so they may have the number of crimes of each type shown where each half-inch of a bar represents 500 crimes versus 100 crimes. This squeezes the numbers together and makes differences less noticeable. Their opponent in the next election would go the other way and use a stretched-out scale to emphasize a crime increase in dramatic fashion, and voilà! (Now you know the answer to the question, “How can two people talk about the same data and get two different conclusions?” Welcome to the world of politics.)

With a pie chart, however, the scale can’t be changed to over-emphasize (or downplay) the results. No matter how you slice up a pie chart, you’re always slicing up a circle, and the proportion of the total pie belonging to any given slice won’t change, even if you make the pie bigger or smaller.

Pondering pet peeves

A recent survey of 100 people with office jobs asked them to report their biggest pet peeves in the workplace. (Before going on, you may want to jot down a couple of yours, just for fun.) A bar graph of the results of the survey is shown in Figure 6-7. Poor time management looks to be the number-one issue for these workers (I hope they didn’t do this survey on company time).

If you take a look at the percentages shown for each pet peeve listed, you see they don’t sum to 1. That tells you that each person surveyed was allowed to choose more than one pet peeve (like that would be hard to do); perhaps they were asked to name their top three pet peeves, for example. For this data set and others like it that allow for multiple responses, a pie chart wouldn’t be possible (unless you made one for every single pet peeve on the list).

Note that Figure 6-7 is a horizontal bar graph (its bars go side to side) as opposed to a vertical bar graph (in which bars go up and down, as shown earlier in Figure 6-6). Either orientation is fine; use whichever one you prefer when you make a bar graph. Do, however, make sure that you label the axes appropriately and include proper units (such as gender, opinion, or day of the week) where appropriate.

Q. Following are a pie chart and bar graph (respectively) of scores from a quiz of ten questions, where the data shows the number of questions answered correctly. Name one advantage each has over the other.

A. The pie chart shows everything as part of a whole, so you can make relative comparisons, and you know it all sums to 100 percent. The bar graph, however, makes it easier to compare the groups to each other. (And if the pie chart doesn’t show the percentages, you have a much harder time estimating the percent in each group.)

9 The following figure shows a frequency bar graph of 500 people who make up three categories (1: support a smoking ban; 2: oppose a smoking ban; and 3: no opinion).

1. Make a relative frequency table of this data.
2. Use the relative frequency table to make a bar graph of this data.
3. Interpret the results. (How do people in the sample feel about the smoking ban?)

10 Suppose a health club asks 30 customers to rate the services as very good (1), good (2), fair (3), or poor (4). You can see the results in the following bar graph. What percentage of the customers rated the services as good?

11 A polling organization wants to find out what voters think of Issue X. It chooses a random sample of voters and asks them for their opinions of Issue X: yes, no, or no opinion. I organize the results in the following bar graph.

1. Make a frequency table of these results (including the total number).
2. Evaluate the bar graph as to whether or not it fairly represents the results.

12 Suppose that a random sample of 270 graduating seniors are asked what their immediate priorities are, including whether or not buying a house is a priority. The results are shown in the following bar graph.

1. The bar graph is misleading; explain why.
2. Make a new bar graph that more fairly presents the results. Note that 100 said Yes, 90 said No, and 80 said Undecided.

13 A car dealership specializing in minivan sales conducts a survey to find out more about who their customers are. One of the variables the company measures is gender; the results of this part of the survey are shown in the following bar graph.

1. Interpret these results.
2. Explain whether or not you think the bar graph is a fair and accurate representation of this data.

14 A survey is conducted to determine whether 20 office employees of a certain company would prefer to work at home, if given the chance. The overall results are shown in the first bar graph, and the results broken down by gender are presented in the second.

1. Interpret the results of each graph.
2. Discuss the added value in including gender in the second bar graph. (The second bar graph in this problem is called a side-by-side bar graph and is often used to show results broken down by two or more variables.)
3. Compare the side-by-side bar graph with the two pie charts that you make for Problem 1. Which of the two methods is best for comparing two groups, in your opinion?

1 A pie chart is a graphical version of what type of table: a frequency table or a relative frequency table?

2 What are three ways in which a bar graph can mislead you?

3 Why is it so important to include the sample size (n) somewhere with your pie chart?

4 Can you make a pie chart out of the following data table?

5 Name two important items to add to your pie chart, other than the names of the slices and the percentage for each slice.

6 What is the impact of changing the scale on a bar graph?

7 The starting point on a bar graph can affect how the bar graph looks. How?

8 Is it wrong to have an “other” category in a pie chart? Explain your answer.

9 Three-dimensional pie charts are a great way to show categorical data. True or False?

10 You want to avoid having too many slices on your pie chart. Why is that?