Summarizing and Graphing Your Data

Arithmetic mean

The arithmetic mean, also commonly called the mean (or the average), is the most familiar and most often quoted measure of central tendency. Throughout this book, whenever we use the two-word term the mean, we’re referring to the arithmetic mean. (There are several other kinds of means besides the arithmetic mean, which we describe later in this chapter.)

The mean of a sample is often denoted by the symbol m or by placing a horizontal bar over the name of the variable, like . The mean is obtained by adding up the values and dividing by the sample size — meaning how many there are. (If you are using software for this, make sure missing values are excluded, or the equation will not compute.) Here’s a small sample of numbers — the diastolic blood pressure (DBP) values of seven study participants (in mmHg) arranged in increasing numerical order: 84, 84, 89, 91, 110, 114, and 116. For the DBP sample:

You can write the general formula for the arithmetic mean of N number of values contained in the variable X in several ways:

See Chapter 2 for a refresher on mathematical notation and formulas, including how to interpret the various forms of the summation symbol ∑ (the Greek capital sigma). In the rest of this chapter, we use the simplest form, meaning the form without the i subscripts that refer to specific elements of an array, whenever possible.

Some statistical books use the notation such that capital and capital N refer to census parameters, and lowercase versions of those to refer to sample statistics. In this book, we make it clear each time we present this notation whether we are talking about a census or a sample.

Median

Like the mean, the median is a common measure of central tendency. In fact, it could be argued that the median is the only one of the three that really takes the word central seriously.

The median of a sample is the middle value in the sorted (ordered) set of numbers. By definition, half of the numbers are smaller than the median, and half are larger. The median of a population frequency distribution function (like the curves shown in Figure 9-2) divides the total area under the curve into two equal parts: Half of the area under the curve (AUC) lies to the left of the median, and half lies to the right.

Consider the sample of diastolic blood pressure (DBP) measurements from seven study participants from the preceding section. If you arrange the values in order from lowest to highest mmHg, you can list them as 84, 84, 89, 91, 110, 114, and 116. There are seven values, and 91 is the fourth of the seven sorted values, so that is the median. Three DBPs in the sample are smaller than 91 mmHg, and three are larger than 91 mmHg. If you have an even number of values, the median is the average of the two middle values. So imagine that you add a value of 118 mmHg to the top of your list, so you now have eight values. To get the median, you would make an average of the fourth and fifth value, which would be (91 + 110)/2 = 100.5 mmHg (don’t be thrown off by the 0.5).

Statisticians often say that they prefer the median to the mean because the median is much less strongly influenced by extreme outliers than the mean. For example, if the largest value for DBP had been very high — such as 150 mmHg instead of 116 mmHg — the mean would have jumped from 98.3 mmHg up to 103.1 mmHg. But in the same case, the median would have remained unchanged at 91. Here’s an even more extreme example: If a multibillionaire were to move into a certain state, the mean family net worth in that state might rise by hundreds of dollars, but the median family net worth would probably rise by only a few cents (if it were to rise at all). This is why you often hear the median rather than mean income in reports comparing income across regions.

Mode

The mode of a sample of numbers is the most frequently occurring value in the sample. One way to remember this is to consider that mode means fashion in French, so the mode is the most popular value in the data set. But the mode has several issues when it comes to summarizing the centrality of observed values for continuous numerical variables. Often there are no exact duplicates, so there is no mode. If there are any exact duplicates, they usually are not in the center of the data. And if there is more than one value that is duplicated the same number of times, you will have more than one mode.

So the mode is not a good summary statistic for sampled data. But it’s useful for characterizing a population distribution, because it’s the value where the peak of the distribution function occurs. Some distribution functions can have two peaks (a bimodal distribution), as shown earlier in Figure 9-2d, indicating two distinct subpopulations, such as the distribution of age of death from influenza in many populations, where we see a mode in young children, and another mode in older adults.

Considering some other “means” to measure central tendency

Several other kinds of means are useful measures of central tendency in certain circumstances. They’re called means because they all calculated using the same approach. The difference is that each type of mean adds a slightly different twist to the basic mathematical process.

GEOMETRIC MEAN

The geometric mean (often abbreviated GM) can be defined by two different-looking formulas that produce exactly the same value. The basic definition has this formula:

We describe the product symbol Π (the Greek capital pi) in Chapter 2. This formula is telling you to multiply the values of the N observations together, and then take the Nth root of the product. Using the numbers from the earlier example (where you had DBP data on seven participants, with the values 84, 84, 89, 91, 110, 114, and 116 mmHg), the equation looks like this:

Even with technology, this formula is computationally challenging. By using logarithms (which turn multiplications into additions and roots into divisions), you can develop a numerically stable alternative formula, which is:

This formula may look complicated, but it really just says, “The geometric mean is the antilog of the mean of the logs of the values in the sample.” In other words, to calculate the GM using this formula, you take the log of each value in your sample, then average all those logs together, and then take the antilog of that average. You can choose to use either natural or common logarithms, but make sure that whatever you choose, you use same type of antilog. (Flip to Chapter 2 for the basics of logarithms.)

Standard deviation, variance, and coefficient of variation

The standard deviation (usually abbreviated SD, sd, or just s) of a set of numerical values tells you how much the individual values tend to differ from the mean in either direction (see “Locating the center of your data” for a discussion of the mean). The SD is calculated as follows:

This formula is saying that you calculate the SD of a set of N numbers by first subtracting the mean from each value () to get the deviation () of each value from the mean. Then, you take the square each of these deviations and add up the terms. After that, you divide that number by N – 1, and finally, you take the square root of that number to get your answer, which is the SD.

For the sample of diastolic blood pressure (DBP) measurements for seven study participants in the example used earlier in this chapter, where the values are 84, 84, 89, 91, 110, 114, and 116 mmHg and the mean is 98.3 mmHg, you calculate the SD as follows:

Several other useful measures of dispersion are related to the SD:

• Variance: The variance is just the square of the SD. For the DBP example, the variance .
• Coefficient of variation: The coefficient of variation (CV) is the SD divided by the mean. For the DBP example, percent.

Range

The range of a set of values is the minimum value subtracted from the maximum value:

Consider the example from the preceding section, where you had DBP measurements from seven study participants (which were 84, 84, 89, 91, 110, 114, and 116 mmHg). The minimum value is 84, the maximum value is 116, and the range is 32 (equal to ).

Centiles

The basic idea of the median is that ½ (half) of your numbers are less than the median, and the other ½ are greater than the median. This concept can be extended to other fractions besides ½.

A centile (also referred to as percentile) is a value that a certain percentage of the values are less than. For example, ¼ of the values are less than the 25th centile (and ¾ of the values are greater). The median is just the 50th centile. The 25th, 50th, and 75th centiles are called the first, second, and third quartiles, respectively, and are used often. There are other sets of centiles, such as deciles, which break at every ten percentiles, that are used less often.

As we explain in the earlier section “Median,” if the sorted sequence of your numerical variable has no middle value, you have to calculate the median as the average of the two middle numbers. The same situation comes up in calculating centiles, but there are different ways that statistical software does the calculation. Fortunately, the different formulas they use give nearly the same result.

The inter-quartile range (IQR) is the difference between the 25th and 75th centiles (the first and third quartiles).

Skewness

Skewness refers to the left-right symmetry of the distribution. Figure 9-3 illustrates some examples.

Figure 9-3b shows a symmetrical distribution. If you look back to Figures 9-2a and 9-2c, which are also symmetrical, they look like the vertical line in the center is a mirror reflecting perfect symmetry, so these have no skewness. But Figure 9-2b has a long tail on the right, so it is considered right skewed (and if you flipped the shape horizontally, it would have a long tail on the left, and be considered left-skewed, as in Figure 9-3a).

How do you express skewness in a summary statistic? The most common skewness coefficient, often represented by the Greek letter γ (lowercase gamma), is calculated by averaging the cubes (third powers) of the deviations of each point from the mean and scaling by the SD. Its value can be positive, negative, or zero.

Here is how to interpret the skewness coefficient (γ):

• A negative γ indicates left-skewed data (Figure 9-3a).
• A zero γ indicates unskewed data (Figures 9-2a and 9-2c, and Figure 9-3b).
• A positive γ indicates right-skewed data (Figures 9-2b and 9-3c).

Notice that in Figure 9-3a, which is left-skewed, the γ = –0.7, and for Figure 9-3c, which is right-skewed, the γ = 0.7. And for Figure 9-3b — the symmetrical distribution — the γ = 0, but this almost never happens in real life. So how large does γ have to be before you suspect real skewness in your data? A rule of thumb for large samples is that if γ is greater than , your data are probably skewed.

Kurtosis

Kurtosis is a less-used summary statistic of numerical data, but you still need to understand it. Take a look at the three distributions shown in Figure 9-4, which all have the same mean and the same SD. Also, all three have perfect left-right symmetry, meaning they are unskewed. But their shapes are still very different. Kurtosis is a way of quantifying these differences in shape.

A good way to compare the kurtosis of the distributions in Figure 9-4 is through the Pearson kurtosis index. The Pearson kurtosis index is often represented by the Greek letter k (lowercase kappa), and is calculated by averaging the fourth powers of the deviations of each point from the mean and scaling by the SD. Its value can range from 1 to infinity and is equal to 3.0 for a normal distribution. The excess kurtosis is the amount by which k exceeds (or falls short of) 3.

One way to think of kurtosis is to see the distribution as a body silhouette. If you think of a typical distribution function curve as having a head (which is near the center), shoulders on either side of the head, and tails out at the ends, the term kurtosis refers to whether the distribution curve tends to have

• A pointy head, fat tails, and no shoulders, which is called leptokurtic, and is shown in Figure 9-4a (where ).
• An appearance of being normally distributed, as shown in Figure 9-4b (where ).
• Broad shoulders, small tails, and not much of a head, which is called platykurtic. This is shown in Figure 9-4c (where ).

A very rough rule of thumb for large samples is that if k differs from 3 by more than , your data have abnormal kurtosis.

Log-normal distributions

Because a sample is only an imperfect representation the population, determining the precise shape of a distribution can be difficult unless your sample size is very large. Nevertheless, a histogram usually helps you spot skewed data, as shown in Figure 9-6a. This kind of shape is typical of a log-normal distribution (Chapter 25), which is a distribution you often see when analyzing biological measurements, such as lab values. It’s called log-normal because if you take a logarithm (of any type) of each data value, the resulting logs will have a normal distribution, as shown in Figure 9-6b.

Because distributions are so important to biostatistics, it’s a good practice to prepare a histogram for every numerical variable you plan to analyze. That way, you can see whether it’s noticeably skewed and, if so, whether a logarithmic transformation makes the distribution normal enough so you can use statistics intended for normal distributions on your data.

If you can’t find any transformation that makes your data look even approximately normal, then you have to analyze your data using nonparametric methods, which don’t assume that your data are normally distributed.

Bar charts

One simple way to display and compare the means of several groups of data is with a bar chart, like the one shown in Figure 9-7a. Here, the bar height for each group of patients equals the mean (or median, or geometric mean) value of the enzyme level for patients at the clinic represented by the bar. And the bar chart becomes even more informative if you indicate the spread of values for each clinical sample by placing lines representing one SD above and below the tops of the bars, as shown in Figure 9-7b. These lines are always referred to as error bars, which is an unfortunate choice of words that can cause confusion when error bars are added to a bar chart. In this case, error refers to statistical error (described in Chapter 6).

But even with error bars, a bar chart still doesn’t provide a picture of the distribution of enzyme levels within each group. Are the values skewed? Are there outliers? Imagine that you made a histogram for each subgroup of patients — Clinic A, Clinic B, Clinic C, and Clinic D. But if you think about it, four histograms would take up a lot of space. There is a solution for this! Keep reading to find out what it is.

Box-and-whiskers charts

The box-and-whiskers plot (or B&W, or just box plot) plot uses very little space to display a lot of information about the distribution of numbers in one or more groups of participants. A box plot of the same enzyme data used in Figure 9-7 is shown in Figure 9-8a.

Looking at Figure 9-8a, you notice the box plot for each group has the following parts:

• A box spanning the interquartile range (IQR), extending from the first quartile of the variable to the third quartile, thus encompassing the middle 50 percent of the data.
• A thick horizontal line, drawn at the median, which is also the 50th centile. If this falls in the middle of the box, your data are not skewed, but if it falls on either side, be on the lookout for skewness.
• Lines called whiskers extending out to the farthest data point that’s not more than 1.5 times the IQR away from the box, and terminate with a horizontal bar on each side.
• Individual points lying outside the whiskers, which are considered outliers.

Box plots provide a useful visual summary of the distribution of each subgroup for comparison, as shown in Figure 9-8a. As mentioned earlier, a median that’s not located near the middle of the box indicates a skewed distribution.

Some software draws the different parts of a box plot according to different rules, so you should always check your software’s documentation before you present a box plot so you can describe your box plot accurately.

Software can provide various enhancements to the basic box plot. Figure 9-8b illustrates two such embellishments you may consider using:

• Variable width: The widths of the bars can be scaled to indicate the relative size of each group.
• Notches: The box can have notches that indicate the uncertainty in the estimation of the median. If two groups have non-overlapping notches, they probably have significantly different medians.