According to EPA data, the gas mileage for compact SUVs in the 2013 model year has a mean of approximately 22 mpg and a standard deviation of about 3 mpg. One SUV gets 25 mpg. Thus, its standardized score is z = (25 - 22)/3 = 1. It is one standard deviation above the mean. Another SUV gets 20.5 mpg. Thus, its standardized score is z = (20.5 - 22)/3 = -0.5. It is one-half of a standard deviation below the mean.

The standardized scores give you a way to compare relative standing of values on different lists where the distributions might have roughly similar shapes.

According to EPA data, 4-cylinder 2013 model year cars have CO2 emissions that average 333 ppm (parts per million) with a standard deviation of 51 ppm; while 6-cylinder cars made that year average 431 ppm with a standard deviation of 44 ppm. Which vehicle has higher CO2 emissions relative to other cars with the same number of cylinders; the 4-cylinder Honda Civic that emits 284 ppm or the 6-cylinder Toyota Camry that emits 358 ppm?

Answer

Consider the following three variables from data that was collected from a sample of n = 198 Stat 100 students:

• Variable #1: Heights (inches)
• Variable #2: Grade Point Average
• Variable #3: Number of Tattoos

Figure 4.1. Histogram of Height (Mean = 66.3 inches & Median = 66 inches)

The Heights Variable is a great example of a histogram that looks approximately like a normal distribution as shown in Figure 4.1. Since a normal distribution is a type of symmetric distribution, you would expect the mean and median to be very close in value. With this example, the mean is 66.3 inches and the median is 66 inches.

Figure 4.2. Histogram of GPA (Mean = 3.25 & Median = 3.3)

The GPA Variable that gives the Grade Point Averages of these 198 Stat 100 students is slightly skewed left and could only very roughly be said to follow a normal distribution as shown in Figure 4.2. Notice the upper tail where the data is clumped. This can be partially explained by the fact that GPAs at Penn State cannot exceed 4.0. However, the mean and median are still pretty close, and using the normal curve (to calculate percentiles for example) should give very rough approximations. It is likely that the GPA variable would look more like a normal curve if the data were restricted to a more homogeneous group with a similar number of credit hours taken.

Figure 4.3. Number of Tattoos (Mean = .23 & Median = 0)

The Tattoo Variable is not normally distributed at all as shown in Figure 4.3. The major problem with this variable is that it is extremely skewed to the right since most people have no tattoos at all. Also, the graph has gaps because this variable is discrete with only a few values in the data set. Thus, the normal curve should not be used to make even rough approximations for data about the number of tattoos.

Recall the variable heights used in Example 4.3. Since the histogram shows that this data is normally distributed, the empirical rule can be applied. The mean and standard deviation (SD) for this sample is 66.3 inches and 4 inches, respectively. Below are the calculations for the sample of heights.

Mean ± 1(SD) = 66.3 ± 4 inches = (62.3 to 70.3 inches)
Mean ± 2(SD) = 66.3 ± 2(4) inches = 66.3 ± 8 inches = (58.3 to 74.3 inches)
Mean ± 3(SD) = 66.3 ± 3(4) inches = 66.3 ± 12 inches = (54.3 to 78.3 inches)

Because the sample of heights is normally distributed, one can say that approximately

• 68% of the heights lie between 62.3 and 70.3 inches
• 95% of the heights lie between 58.3 and 74.3 inches
• 99.7% of the heights lie between 54.3 and 78.3 inches

One would expect it to be very unusual for someone in this sample to be smaller than 54.3 inches or taller than 78.3 inches. Since 68% of the heights are within one standard deviation of the mean, the remaining 32% would fall outside of that. Further, since the distribution is symmetric we would have 16% (half of the 32%) falling below 62.3 inches and another 16% falling above 70.3 inches.

A histogram of the highway gas mileage for the 171 compact SUVs sold in the United States and tested by the EPA in 2013 is shown in Figure 4.5. The mean mileage was 22.20 mpg with a standard deviation of 2.85 mpg. General Motors' 2013 Encore compact SUV got 28 mpg. What percentage of the compact SUVs got worse mileage than the Encore?

Figure 4.5 Histogram of Highway Mileage for 2013 compact SUVs

To solve this, we first have to compute the standard score of the value of interest which is found by:

\(z=\dfrac{(28 - 22.2)}{2.85} ≈ 2.04\) (this says that 28 mpg is 2.04 standard deviations above the mean).

Next, we look at Table 8.1 to find that this standard score corresponds to approximately the 98th percentile of a normal distribution. Thus, the Encore gets better mileage than about 98% of the 2013 compact SUVs (and hence worse mileage than about 2% of them).

The Gallup World Poll takes random samples of the adults in 132 different countries. In many of those countries, Gallup asks respondents to try and think about an overall evaluation of their lives and to specify how satisfied they are on a four-point scale (very dissatisfied, dissatisfied, satisfied, or very satisfied). Figure 4.6 shows a comparative bar graph giving the results from two surveys, one taking a random sample of adults in the United States and one using a random sample of adults in China. Bar graphs are often used to show the results of data for categorical variables and, as in Figure 4.6 can be used to compare categorical variables in two or more circumstances.

Figure 4.6 Life Satisfaction Results from Two Countries (Gallup World Poll)

Each day the Gallup Poll takes a random sample of about 500 American adults nationally and asks them about a variety of issues including how much money they spent the day before not counting the purchase of a home or car or paying normal household bills like for electrical or phone service. Figure 4.7shows a line graph of the data presented in two ways: both as a 3-day rolling average (the dark green line) and as a 14-day rolling average (the light green line). For example, each point on the dark green line represents the average results of the amount spent by the 1500 American adults who had responded to the survey over the 3-day period leading up to the day of the survey. This type of line graph is called a time series plot because the points represent the variable being measured across time.

Figure 4.7 Time Series Plot of Consumer Spending from February 2008 to February 2015 (Gallup Poll)

When looking at a time series plot like Figure 4.7 it is important to examine and interpret some key basic features:

• Is there a long-term trend? For example, in the consumer spending data, we can see a long-term generally upward trend since the end of the recession in June 2009. One note of caution when looking at economic data that extends over decades in time; check if they have been adjusted for inflation. An apparent upward trend may be nothing more than reflecting a change in the value of the dollar.
• Are there seasonal components? While temperature data is dramatically affected by regular seasonal cycles, many other variables change in predictable patterns because of people's behavioral changes in certain months or seasons. For example, have a close look at Figure 4.7. You should be able to see a bump in consumer spending each year associated with the holidays in December. There are other cyclic effects in this data. If you look really closely at the 3-day averages, you can see that there is increased spending on weekends compared with weekdays.
• What is the nature of the random fluctuations? We know that every measurement is subject to natural variability and that averages will be more reliable if they are based on larger sample sizes. Have a look at Figure 4.7 and see how the 3-day averages based on surveys of about 1500 people show random fluctuations much larger than the 14-day averages based on surveys of about 7000 people.

An experiment was carried out to see how Blood Alcohol Content (BAC) as measured by a breathalyzer change with the number of 12-ounce beers you drink (the experiment is discussed in the Electronic Encyclopedia of Statistics Examples and Exercises). In the experiment, 16 subjects each drew a number out of a hat. For example, if the number was a 3, then that subject drank 3 beers. A half-hour after finishing the last assigned beer a police officer used a breathalyzer, like the ones they use in the field, to measure the subject's BAC level. Figure 4.8 shows a scatterplot of the results. Each point represents a different subject. For example, one subject drank 6 beers and had a BAC of 0.10; over the legal limit for driving.

Figure 4.8 Number of 12-ounce Beers Consumed versus BAC for 16 Subjects

Scatterplots are used for displaying the relationship between two measurement variables. Examining Figure 4.8, we can see a clear trend - there is an obvious positive association between the number of beers and the BAC - increases in one variable are associated with increases in the other. The data here were based on a randomized experiment and the causal nature of this particular relationship is quite well established. Positive associations are reflected in a cloud of points in the scatterplot that goes "uphill" as you move from left to right. A negative association, like the one we would see if we plotted the weights of cars versus their gas mileage, shows a cloud of points going "downhill".