Statistics of central tendency
All of the tests in the first part of this handbook have analyzed nominal variables. Data from a nominal variable is summarized as a percentage or a proportion. For example, 76.1 percent (or 0.761) of the peas in one of Mendel's genetic crosses were smooth, and 23.9 percent were wrinkled. If you have the percentage and the sample size (556, for Mendel's peas), you have all the information you need about the variable.
The rest of the tests in this handbook analyze measurement variables. Summarizing data from a measurement variable is more complicated, and requires a number that represents the "middle" of a set of numbers (known as a "statistic of central tendency" or "statistic of location"), along with a measure of the "spread" of the numbers (known as a "statistic of dispersion"). The arithmetic mean is the most common statistic of central tendency, while the variance or standard deviation are usually used to describe the dispersion.
The statistical tests for measurement variables assume that the probability distribution of the observations fits the normal (bell-shaped) curve. If this is true, the distribution can be accurately described by two parameters, the arithmetic mean and the variance. Because they assume that the distribution of the variables can be described by these two parameters, tests for measurement variables are called "parametric tests." If the distribution of a variable doesn't fit the normal curve, it can't be accurately desribed by just these two parameters, and the results of a parametric test may be inaccurate. In that case, the data are usually converted to ranks and analyzed using a non-parametric test, which is less sensitive to deviations from normality.
The normal distribution
Many measurement variables in biology fit the normal distribution fairly well. According to the central limit theorem, if you have several different variables that each have some distribution of values and add them together, the sum follows the normal distribution fairly well. It doesn't matter what the shape of the distribution of the individual variables is, the sum will still be normal. The distribution of the sum fits the normal distribution more closely as the number of variables increases. The graphs below are frequency histograms of 5,000 numbers. The first graph shows the distribution of a single number with a uniform distribution between 0 and 1. The other graphs show the distributions of the sums of two, three, or four random numbers.
|Histograms of sums of random numbers.|
|Histograms of sums of random numbers.|
As you can see, as more random numbers are added together, the frequency distribution of the sum quickly approaches a bell-shaped curve. This is analogous to a biological variable that is the result of several different factors. For example, let's say that you've captured 100 lizards and measured their maximum running speed. The running speed of an individual lizard would be a function of its genotype at many genes; its nutrition as it was growing up; the diseases it's had; how full its stomach is now; how much water it's drunk; and how motivated it is to run fast on a lizard racetrack. Each of these variables might not be normally distributed; the effect of disease might be to either subtract 10 cm/sec if it has had lizard-slowing disease, or add 20 cm/sec if it has not; the effect of gene A might be to add 25 cm/sec for genotype AA, 20 cm/sec for genotype Aa, or 15 cm/sec for genotype aa. Even though the individual variables might not have normally distributed effects, the running speed that is the sum of all the effects would be normally distributed.
If the different factors interact in a multiplicative, not additive, way, the distribution will be log-normal. An example would be if the effect of lizard-slowing disease is not to subtract 10 cm/sec from the total speed, but instead to reduce the speed by 10% (in other words, multiply the total speed by 0.9). The distribution of a log-normal variable will look like a bell curve that has been pushed to the left, with a long tail going to the right. Taking the log of such a variable will produce a normal distribution. This is why the log transformation is used so often.
|Histograms of the product of four random numbers, without or with log transformation.|
|Histograms of the product of four random numbers, without or with log transformation.|
The figure above shows the frequency distribution for the product of four numbers, with each number having a uniform random distribution between 0.5 and 1. The graph on the left shows the untransformed product; the graph on the right is the distribution of the log-transformed products.
Different measures of central tendency
While the arithmetic mean is by far the most commonly used statistic of central tendency, you should be aware of a few others.
Arithmetic mean: The arithmetic mean is the sum of the observations divided by the number of observations. It is the most common statistic of central tendency, and when someone says simply "the mean" or "the average," this is what they mean. It is often symbolized by putting a bar over a letter; the mean of Y1, Y2, Y3,… is Y.
The arithmetic mean works well for values that fit the normal distribution. It is sensitive to extreme values, which makes it not work well for data that are highly skewed. For example, imagine that you are measuring the heights of fir trees in an area where 99 percent of trees are young trees, about 1 meter tall, that grew after a fire, and 1 percent of the trees are 50-meter-tall trees that survived the fire. If a sample of 20 trees happened to include one of the giants, the arithmetic mean height would be 3.45 meters; a sample that didn't include a big tree would have a mean height of about 1 meter. The mean of a sample would vary a lot, depending on whether or not it happened to include a big tree.
In a spreadsheet, the arithmetic mean is given by the function AVERAGE(Ys), where Ys represents a listing of cells (A2, B7, B9) or a range of cells (A2:A20) or both (A2, B7, B9:B21). Note that spreadsheets only count those cells that have numbers in them; you could enter AVERAGE(A1:A100), put numbers in cells A1 to A9, and the spreadsheet would correctly compute the arithmetic mean of those 9 numbers. This is true for other functions that operate on a range of cells.
Geometric mean: The geometric mean is the Nth root of the product of N values of Y; for example, the geometric mean of 5 values of Y would be the 5th root of Y1×Y2×Y3×Y4×Y5. It is given by the spreadsheet function GEOMEAN(Ys). The geometric mean is used for variables whose effect is multiplicative. For example, if a tree increases its height by 60 percent one year, 8 percent the next year, and 4 percent the third year, its final height would be the initial height multiplied by 1.60×1.08 ×1.04=1.80. Taking the geometric mean of these numbers (1.216) and multiplying that by itself three times also gives the correct final height (1.80), while taking the arithmetic mean (1.24) times itself three times does not give the correct final height. The geometric mean is slightly smaller than the arithmetic mean; unless the data are highly skewed, the difference between the arithmetic and geometric means is small.
If any of your values are zero or negative, the geometric mean will be undefined.
The geometric mean has some useful applications in economics, but it is rarely used in biology. You should be aware that it exists, but I see no point in memorizing the definition.
Harmonic mean: The harmonic mean is the reciprocal of the arithmetic mean of reciprocals of the values; for example, the harmonic mean of 5 values of Y would be 5/(1/Y1 + 1/Y2 + 1/Y3 + 1/Y4 + 1/Y5). It is given by the spreadsheet function HARMEAN(Ys). The harmonic mean is less sensitive to a few large values than are the arithmetic or geometric mean, so it is sometimes used for highly skewed variables such as dispersal distance. For example, if six birds set up their first nest 1.0, 1.4, 1.7, 2.1, 2.8, and 47 km from the nest they were born in, the arithmetic mean dispersal distance would be 9.33 km, the geometric mean would be 2.95 km, and the harmonic mean would be 1.90 km.
If any of your values are zero, the harmonic mean will be undefined.
I think the harmonic mean has some useful applications in engineering, but it is rarely used in biology. You should be aware that it exists, but I see no point in memorizing the definition.
Median: When the Ys are sorted from lowest to highest, this is the value of Y that is in the middle. For an odd number of Ys, the median is the single value of Y in the middle of the sorted list; for an even number, it is the arithmetic mean of the two values of Y in the middle. Thus for a sorted list of 5 Ys, the median would be Y3; for a sorted list of 6 Ys, the median would be the arithmetic mean of Y3 and Y4. The median is given by the spreadsheet function MEDIAN(Ys).
The median is useful when dealing with highly skewed distributions. For example, if you were studying acorn dispersal, you might find that the vast majority of acorns fall within 5 meters of the tree, while a small number are carried 500 meters away by birds. The arithmetic mean of the dispersal distances would be greatly inflated by the small number of long-distance acorns. It would depend on the biological question you were interested in, but for some purposes a median dispersal distance of 3.5 meters might be a more useful statistic than a mean dispersal distance of 50 meters.
The second situation where the median is useful is when it is impractical to measure all of the values, such as when you are measuring the time until something happens. Survival time is a good example of this; in order to determine the mean survival time, you have to wait until every individual is dead, while determining the median survival time only requires waiting until half the individuals are dead.
There are statistical tests for medians, such as Mood's median test, but they are rarely used due to their lack of power and won't be discussed in this handbook. If you are working with survival times of long-lived organisms (such as people), you'll need to learn about the specialized statistics for that; Bewick et al. (2004) is one place to start.
Mode: This is the most common value in a data set. It requires that a continuous variable be grouped into a relatively small number of classes, either by making imprecise measurments or by grouping the data into classes. For example, if the heights of 25 people were measured to the nearest millimeter, there would likely be 25 different values and thus no mode. If the heights were measured to the nearest 5 centimeters, or if the original precise measurements were grouped into 5-centimeter classes, there would probably be one height that several people shared, and that would be the mode.
It is rarely useful to determine the mode of a set of observations, but it is useful to distinguish between unimodal, bimodal, etc. distributions, where it appears that the parametric frequency distribution underlying a set of observations has one peak, two peaks, etc. The mode is given by the spreadsheet function MODE(Ys).
The Maryland Biological Stream Survey used electrofishing to count the number of individuals of each fish species in randomly selected 75-m long segments of streams in Maryland. Here are the numbers of blacknose dace, Rhinichthys atratulus, in streams of the Rock Creek watershed:
Mill_Creek_1 76 Mill_Creek_2 102 North_Branch_Rock_Creek_1 12 North_Branch_Rock_Creek_2 39 Rock_Creek_1 55 Rock_Creek_2 93 Rock_Creek_3 98 Rock_Creek_4 53 Turkey_Branch 102
Here are the statistics of central tendency. In reality, you would rarely have any reason to report more than one of these:
Arithmetic mean 70.0 Geometric mean 59.8 Harmonic mean 45.1 Median 76 Mode 102
How to calculate the statistics
I have made a descriptive statistics spreadsheet that calculates the arithmetic, geometric and harmonic means, the median, and the mode, for up to 1000 observations.
This web page calculates arithmetic mean, median, and mode for up to 80 observations. It also includes most of the other common descriptive statistics: range, variance, standard deviation, coefficient of variation, and standard error of the mean.
This web page calculates arithmetic mean and median for up to 10,000 observations. It also calculates standard deviation, standard error of the mean, and confidence intervals.
This web page calculates arithmetic mean and median, along with range, variance, standard deviation, and standard error of the mean. I don't know the maximum number of observations it can handle.
There are three SAS procedures that do descriptive statistics, PROC MEANS, PROC SUMMARY, and PROC UNIVARIATE. I don't know why there are three. PROC UNIVARIATE will calculate a longer list of statistics, so you might as well use it. Here is an example, using the fish data from above.
data fish; input location $ dacenumber; cards; Mill_Creek_1 76 Mill_Creek_2 102 North_Branch_Rock_Creek_1 12 North_Branch_Rock_Creek_2 39 Rock_Creek_1 55 Rock_Creek_2 93 Rock_Creek_3 98 Rock_Creek_4 53 Turkey_Branch 102 ; proc univariate data=fish; run;
There's a lot of output from PROC UNIVARIATE, including the arithmetic mean, median, and mode:
Basic Statistical Measures Location Variability Mean 70.0000 Std Deviation 32.08582 Median 76.0000 Variance 1030 Mode 102.0000 Range 90.00000 Interquartile Range 45.00000
You can specify which variables you want the mean, median and mode of, using a VAR statement. You can also get the statistics for just those values of the measurement variable that have a particular value of a nominal variable, using a CLASS statement. This example calculates the statistics for the length of mussels, separately for each of two species, Mytilus edulis and M. trossulus.
data mussel; input species $ length width; cards; edulis 49.0 11.0 tross 51.2 9.1 tross 45.9 9.4 edulis 56.2 13.2 edulis 52.7 10.7 edulis 48.4 10.4 tross 47.6 9.5 tross 46.2 8.9 tross 37.2 7.1 ; proc univariate; var length; class species; run;
Surprisingly, none of the SAS procedures calculate harmonic or geometric mean. There are functions called HARMEAN and GEOMEAN, but they only calculate the means for a list of variables, not all the values of a single variable.
Sokal and Rohlf, pp. 39-47.
Zar, pp. 20-28.
Blacknose dace data from Maryland Biological Stream Survey.
This page was last revised September 14, 2009. Its address is http://udel.edu/~mcdonald/statcentral.html. It may be cited as pp. 95-101 in: McDonald, J.H. 2009. Handbook of Biological Statistics (2nd ed.). Sparky House Publishing, Baltimore, Maryland.
©2009 by John H. McDonald. You can probably do what you want with this content; see the permissions page for details.