## B22 Grain size distribution terminology

There are five measurements that are used by Graf (1993) when describing the grain size distribution of lunar soil samples. These are mean, median, mode, sorting, skewness, and kurtosis. Figure B.7a shows examples of mean, median, and skewness; Figure B.7b illustrates kurtosis. Table B.2 illustrates the relationship between phi and micrometer measurements.

Geologists sometimes use the symbol 0 for measuring particle size, which is a logarithmic mean of the particle diameter for each size grouping. It is obtained by calculating, for each size category:

0 = -log2 d, where d is the diameter in millimeters. Figure B.9, which is a graphical description of the particle size distribution of lunar soil "78221,8", shows a 0 scale along the bottom and its corresponding diameter scale along the top. A larger value for 0 denotes a smaller grain size.

The mean is the sum of all the individual grain sizes in the sample, divided by the number of grains. It can be thought of as the center of gravity of the size distribution.

### B.2.2.2 Median

The median is a number dividing the higher half of a sample from the lower half. The median can be found by arranging all the grain sizes from lowest value to highest value and picking the middle one.19 The median is the value where 50 percent of the measured values are larger and 50 percent are smaller. The mean is primarily used for skewed distributions, which it represents differently than the arithmetic mean. Consider the set {1, 2, 2, 2, 3, 9}. The median is 2 in this case, as is the mode, and either of these might be seen as a better indication of central tendency than the arithmetic mean of 3.166.

The mode is the most frequent value of grain size occurring in the sample. It is the midpoint of the most abundant size class, or the number that occurs most frequently in a set of numbers. For example, in Figure B.11 the mode is the range between ^ = 5 (32 ^m) and ^ = 6 (16 ^m).

Of the three, the mode is the most variable and easily-manipulated term. Changing the increment of the step size will change the value of the mode. To keep internal consistency in his data set, Graf (1993) always uses a step size of for histogram graphs and for finding the mode. This limits the allowable values of the mode to 2.5^, 3.5^, and so on.

### B.2.2.4 Sorting

Sorting means the degree of similarity of the sizes of the grains or particles in a specific sample. If there is a wide variation of sizes, the sample is said to be poorly-sorted; whereas if all the grains are nearly the same size, the sample is said to be well-sorted.

19 The difference between the median and mean is illustrated in this example. Suppose 19 paupers and 1 billionaire are in a room. Everyone removes all money from their pockets and puts it on a table. Each pauper puts $5 on the table; the billionaire puts $1 billion (that is, $109) there. The total is then $1,000,000,095. If that money is divided equally among the 20 persons, each gets $50,000,004.75. That amount is the mean (or "average") amount of money that the 20 persons brought into the room. But the median amount is $5, since one may divide the group into two groups of 10 persons each, and say that everyone in the first group brought in no more than $5, and each person in the second group brought in no less than $5. In a sense, the median is the amount that the typical person brought in. By contrast, the mean (or "average") is not at all typical, since no one present - pauper or billionaire - brought in an amount approximating $50,000,004.75 (Wikipedia).

Sorting can be thought of as the standard deviation of the size distribution. In the strictest sense, we cannot use the term "standard deviation" if the size distribution has a ^ scale (a logarithmic scale). Nevertheless, both sorting and standard deviation are measurements that describe the shape of a distribution. A nearly-perfect sorting with all grains of similar size would have a sorting parameter of less than 1^>. As the size distribution becomes more spread out, the sorting parameter becomes larger.

### B.2.2.5 Skewness

Skewness is the degree of asymmetry of a distribution. It can be described with the diagram in Figure B.7. If the median is on the coarse side of the mean, the central trend is coarse, the extended tail is fine, and the distribution is said to be positively skewed. Similarly, if the median (central trend) is finer than the mean, the distribution is said to be negatively skewed.

### B.2.2.6 Kurtosis

Kurtosis expresses the sharpness of the peak of a frequency distribution, and is usually described relative to a normal distribution. A distribution having a relatively-high peak is called "leptokurtic", while a flat-topped curve is called "platykurtic" (Figure B.7b).

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