Chapter 2. Section 3.2 -- Numerical Data Descriptions: Mean and Standard Deviation

Introduction

When the data we are interested in are something, we commonly summarize the data not only graphically but also numerically. As with our graphical descriptions, we would like our numerical descriptions to address the important characteristics of the data set—its shape, center and spread. Consider once again the data table that gave information about some of the 25 top-grossing movies in the United States in November 2010.

In reviewing this table we see, in particular, that the third column and the fifth column each contain values of a quantitative variable. In order to get a “picture” of these data, we might want to determine how much, on average, a top-grossing film earned, or whether the running time for The Lord of the Rings: The Return of the King was unusually long. These are questions that can be answered using numerical descriptive measures.

This is a test.

Source: The Internet Movie Database *as of February 2011

And another test of the system of inserting blocks here.

2.1 Locating the Center: Mean

We will begin our discussion by considering a numerical description with which you are probably familiar, the mean. The mean of a set of quantitative data is simply its arithmetic average, the very same average you learned to calculate in elementary school. The mean is found by summing all data values, then dividing the result by the number of values. We indicate this symbolically using formulas:

You are probably thinking that these two formulas are remarkably similar; indeed, they are nearly identical. That is because what we do is exactly the same; what we say depends on whether we are working with a population or a sample.

In working with a population, we denote the mean by µ, the Greek letter mu. The upper case N is the population size (the number of values we are adding), and the x’s are the individual data values, labeled from 1 to N. If we are calculating the mean of a sample, we indicate the result by , which we call (not very creatively) x-bar, with the lower case n the sample size, and the x’s labeled from 1 to n.

interrupter

While this might seem unnecessarily “picky” at first glance, the notation indicates two important distinctions that we must keep in mind:

• We denote the mean µ of a population of size N and the mean of a sample of size n differently, because the information they provide is different in nature. A population has only one mean, and if we calculate it correctly, we have the mean. When we calculate a sample mean, we are generally interested in using it to estimate the population mean. So while each sample has only one mean, if we select a different sample, we are likely to get a different sample mean. Recall that when we calculate a numerical descriptive measure for a population, we are finding a parameter; if we calculate a numerical descriptive measure for a sample, we are finding a statistic. It is easy to remember which is which—calculating from a population, you have a parameter; calculating from a sample, you have a statistic. Further, we use Greek letters for parameters and English ones for statistics. So µ is a parameter, while is a statistic.
• We denote an individual data value as x and the mean of a sample as . It is probably easier to see why the notation must be different here. Anyone who owns a vehicle has certainly felt the “pinch at the pump” and has witnessed the volatile gas prices, especially in the last few years. Certainly you have experienced a difference between the price of gasoline on a certain day (an x), and the average price of gasoline for a sample of days over several years (an ).

2.2 Calculating the mean

Now that we have talked a lot about means, let’s actually calculate one. Here are the national average retail prices ($/per gallon) of regular grade gasoline for a sample of ten months since 2000 according to the United States Department of Energy:

Calculating the sample mean, we find that

dollars per gallon. So while the average prices went up and down over the years, the average of these averages was about $2.10. We might further observe that 6 of the data values were less than the mean and 4 were greater than the mean.

This calculation was fairly simply, even by hand, given the dollars and cents values and the sample size of 10. What happens when numbers are not so nice or we have a large data set? In that case, we use statistical software to do our calculation.

2.3 Determining Spread: Standard Deviation

Using the mean to locate the center of a distribution gives us one important piece of information, but it doesn’t tell us everything we might want to know. If your instructor reports that the class average on a statistics test is 75%, you wonder whether most students scored in the 70s, or whether anyone made 100% on the test. Knowing how the scores are spread out provides more information about the distribution of data. The range, the difference between the largest value and the smallest value, is the easiest way to describe the spread. For the gasoline data, the range of the sample of is $3.80 – $1.40=$2.40 per gallon.

The range tells us how far the maximum value is from the minimum value, but not how the values are distributed within the range. Are the values spread uniformly over this interval? Are most of the values clustered near the mean? To answer questions like these, we use the standard deviation, a number that describes, on average, how the data set is spread away from its center (as measured by the mean).

Let’s start with the formula for the standard deviation of a sample, which we denote by s, and then “de-construct” it.

The numerator of the fraction consists of a sum of squares; each term being squared is the difference between an individual measurement (an x) and the sample mean (). We call each of these differences the measurement’s deviation from the mean. We then divide this sum by n – 1, one less than the sample size. Finally, we take the principal (positive) square root of the resulting number.

To illustrate the procedure, we use a small sample with small whole number values: 1, 2, 2, 3, 4, 4, 5. The mean for this sample is 3. The accompanying table displays the measurements, the deviations and the squared deviations.

Adding the values in the last column, dividing by n − 1, and taking the square root, we find that (correct to two decimal places).

2.4 Standard deviation calculation FAQ's

This calculation might raise several questions in your mind.

Q: If we are concerned about deviations from the mean, why don’t we just add the deviations rather than their squares?

A: If you check the sum of the deviations, you will see that it is 0. This is always the case (which is not very informative), so we square the deviations, making the resulting values all nonnegative. Further, if the deviations are small in absolute value, the squares are also small in absolute value. If the deviations are large in absolute value, the squares are even larger.

Q: If we want to know what is happening “on average,” why don’t we divide by n rather than n – 1?

A: When we have data from a sample rather than from the whole population, dividing by n underestimates how the data varies. Statisticians have determined that an appropriate correction for this error is to divide by the slightly smaller n – 1, which makes the result a bit larger.

Q: Once we have this “average,” why do we take its square root?

A: Before taking the square root, the quantity is measured in square units—the square of the units in which the data were originally measured. It is more helpful to our understanding to have the spread measured in the same units as the original measurements.

As with the mean, we generally use statistical software to calculate sample standard deviation.

Question 2.7 Standard deviation of gas prices

Use statistical software to calculate the standard deviation for our sample of gasoline prices, repeated below.

Gasoline Prices Sample Data

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

2.5 Interpreting the standard deviation

Once we’ve calculated standard deviation, how should we interpret the number? Notice that the quantities involved in the standard deviation fraction are all non-negative. And the principal square root of a non-negative number is also non-negative. In fact, the standard deviation is only ever zero when all of the data values are identical. In that case, the mean is the same as each data value, so there is absolutely no deviation from the mean.

Figure 2.1: Side-by-side dotplots of four groups of data (A, B, C, and D).

The more spread out that the data points are from the mean, the larger the standard deviation will be. Here is an example which illustrates this concept. The side-by-side dotplots in Figure 2.1 show four groups of data, A, B, C, and D, where each data set has ten values.

The mean for each of the four data sets is five. Although the mean is the same for each group, the spread in each data set varies considerably. It is easy to see, for instance, that the data values in group D are the most spread out from the mean, that group B has a very small amount of spread from the mean, and that there is no spread in group A. Using statistical software to compute the standard deviations we see numerically that groups A, B, C, and D have increasing standard deviations, as shown in the table below.

2.6 Empirical (68-95-99.7) Rule

Knowing the mean and standard deviation can give us a good idea of how our set of data is distributed. In particular, when our data is roughly bell-shaped, the Empirical Rule tells us that:

• about 68% of all measurements lie with 1 standard deviation of the mean;
• about 95% of measurements lie within 2 standard deviations of the mean;
• about 99.7% (nearly all) of the measurements are within 3 standard deviations of the mean.

Figure 2.2: Margin of victory (in runs) for games played by the Boston Red Sox in the 2010 season.

The histogram in shows the margin of victory (in runs) for the regular season games played by the Red Sox during the 2010 season. Also in you can see...

From the histogram it is clear that for the majority of the Red Sox’s games, they either won or lost games by no more than 4 runs. It was very rare for them to lose by more than 12 runs (in fact it only happened once), as was it for them to win by more than 8 runs. Since the data set is roughly bell-shaped, the Empirical Rule allows us to say even more.

The mean margin of victory for the Red Sox in 2010 was 0.429 runs, with a standard deviation of 4.174 runs. Thus, according to the Empirical Rule,

• 68% of the margins of victory should lie within 4.174 runs of 0.429, that is, between 0.4259 – 4.174 and 0.4259 + 4.174 runs, so between –3.7481 and 4.5999 runs. It turns out that indeed this was approximately true. In 112 of the 162 games that the Red Sox played during the 2010 regular season, their margin of victory was between -3.7481 and 4.5999 runs. Therefore 69% of the margin of victories fell within this range.
• 95% of the margins of victory should lie within 2 × 4.174 runs of 0.4259, that is, between 0.4259 – 8.348 and 0.4259 + 8.348 runs, so between –7.064 and 8.7739 runs. Again, what “should have happened”, essentially happened. Their margin of victory was between -7.064 and 8.7739 runs in 92.59% (150 of their 162 games).
• 99.7% of the margins of victory should lie within 3 × 4.174 runs of 0.4259, that is, between 0.4259 – 12.552 and 0.4259 + 12.552 runs, so between –12.1261 and 16.696 runs. In 99.3% (161 out of 162) of the games, the Red Sox had a margin of victory between -12.1261 and 16.696 runs.

2.7 Bell-shaped curves

What exactly do we mean by “bell-shaped”? The distribution should be roughly symmetric, with its peak near the center of its range, and the heights of the boxes decreasing as you move to left and to the right of the peak. To see this more clearly, consider the histogram in Figure 2.2. Although it is clearly not a perfect fit, the curve approximates the overall pattern of the distribution and resembles a bell, with its base sitting on the horizontal axis.

The graph at right is a histogram that we saw earlier in the chapter, displaying the distribution of petal lengths in centimeters for a sample of Setosa iris. We can see from the overlaid curve that this distribution is also bell-shaped.

The petal lengths graphed in this histogram are given in the following table:

Answer the following questions about 40tBxBgtJuS+A= this sample of data to make sure you've grasped all the concepts presented so far in this section.

2.8 z-scores

Frequently people are interested in comparing measurements not merely within a single distribution but between two different distributions. Perhaps the classic example of such a comparison is the argument among baseball fans about who is the greatest homerun hitter of all time. When baseball stars played in different eras and under different conditions, how can we compare their performances? One way to consider the question is to decide which player was more outstanding in his own era.

In order to do this, we use a statistical measure called a z-score. A z-score uses the standard deviation as a “ruler” to measure how far a measurement is away from the mean. The formula for calculating a z-score is

Because the z-score formula includes subtracting the mean from a measurement (the “deviation from the mean” once again), if a measurement is larger than the mean, its z-score will be positive; if the measurement is smaller than the mean, its z-score will be negative. Thus, the z-score tells us both the number of standard deviations the measurement is from the mean, and the side of the mean on which it lies. Further, the more standard deviations a measurement is away from the mean, regardless of direction, the more unusual that measurement is with respect to the rest of the data. Thus, a z-score of –2.1 indicates a more unusual measurement than a z-score of 0.85.

2.9 Using z-scores

Suppose we want to determine if Babe Ruth or Hank Aaron was more outstanding, given the time that they played baseball. From 1914-1935 Babe Ruth played almost exclusively for the Red Sox and the Yankees, while Hank Aaron played for the Milwaukee (later Atlanta) Braves and the Milwaukee Brewers between 1954 and 1976.

We’ll compare the yearly homerun production of Babe Ruth and Hank Aaron by looking at how each player’s “at bats” per homerun compared to his contemporaries. For the years Babe Ruth played, from 1914-1935, the league average AB/HR was 123.49, with a standard deviation of 87.78. Babe Ruth’s AB/HR career average value was 17.00. (These calculations omit Ruth’s first year of play since he only played five games that year.) For the years Hank Aaron played, from 1954-1976, the league average AB/HR was 42.14, with a standard deviation of 6.92. Hank Aaron’s AB/HR career average value was 18.51. Calculating z-scores, we find that

while .

So Ruth’s AB/HR value was 1.21 standard deviations below the mean for his era, and Aaron’s AB/HR value was 3.42 standard deviations below the mean for his era. The negative values in this case indicating fewer at bats required per home run hit than the average for the era. Thus, we conclude that Aaron’s performance was more outstanding compared to his contemporaries, and, by this measure, he was the better homerun hitter.

2.10 How do outliers affect the mean and standard deviation

The Cleveland Cavalier total players’ payroll exceeded $51,000,000 in 2010-2011. Although the majority of players’ salaries were between roughly ½ million and 4 million dollars, there were a few exceptions. In particular, Antawn Jamison collected an incredible $13.35 million dollars, not too bad for a year’s work. Figure 3.2.4 shows a dotplot of the annual salary for each of the 14 players on the Cavs’ roster.

Figure 2.3: Dotplot of salaries for Cleveland Cavalier players, 2010-2011 season.

Using software, we find that the mean salary for a Cav’s player was $3.68 million with a standard deviation of $3.77 million. Do these values change much if we delete the high outlier, $13.35 million? The answer is yes. The revised mean and standard deviation, based on the remaining players’ salaries, are considerably lowered. The mean is now $2.93 million and the standard deviation is $2.64 million. Deleting one high salary resulted in having the mean players’ salary decrease by almost $1 million, which is substantial. If you think about how the mean and the standard deviation are computed, it might not be so surprising since both formulas take into account the value of every data measurement.

A measure is called resistant if is not influenced by extremely high or low data values. Both the mean and standard deviation are not resistant to outliers. When describing a data set, the moral of the story is to graph your data first. If the graphs are severely skewed or have outliers, the mean and standard deviation might not reflect an accurate description of the data set’s center and spread. In the next section we look at another measure of center and spread which are, in general, resistant to outliers.