# The Basic Statistical Numbers in Particle-Size Distribution Analysis

**Mean:**The mean is an average over the entire distribution. There are different types of means that can be calculated from the same set of data. (See the next blog for more details of various means.)

**Median: **The median is the value that divides the population into two equal halves. Just like the mean value, populations with different weightings will have different median values.

**Mode:**The mode is the most common value of the distribution, i.e., the highest point of the distribution curve. It is also commonly called the peak value. If there is more than one high-frequency region in the distribution, the distribution is called multi-modal.

**Variance (s**^{2}**): **A measure of distribution broadness defined as, for arithmetic and geometric number distribution, respectively:

(1)

(2)

**Standard Deviation (s): **The square root of the variance. Note that the geometric standard deviation is not a standard deviation in its true sense.

**Coefficient of Variation (arithmetic statistics only):**

The coefficient of variation (CV) is the standard deviation divided by the mean. It relates the breadth of the distribution to the mean in percentage.

**Skewness (g1):**

Skewness is the degree of distortion from a symmetrical distribution. If on one side of the mean has extreme values, but the other does not, the distribution is said to be skewed. If the dispersion of on either side of the mean is roughly symmetrical (i.e. one is a mirror reflection of the other), the distribution is said to be not skewed. When a distribution is perfectly symmetrical the skewness (g1) is equal to zero. For a right-skewed distribution, there is a long tail at the right side and a steep rise at the left side, and the skewness is positive. For a left-skewed distribution, the skewness is negative and the tail is on the left with the bulk of the distribution on the right (Figure 1.). The approximate equalities in Eqs. 3 and 4, which are for number distribution, are held when N is large.

**Arithmetic:**

(3)

**Geometric:**

(4)

**Kurtosis (g**_{2}**):**

Kurtosis is a measure of the weight of the tails of a distribution or the peakedness of a distribution. A normal distribution is defined as having zero kurtosis (mesokurtic). When all values of the distribution are closer to the mean, the distribution is narrower or sharper than the normal distribution and the kurtosis (g_{2}) is positive (leptokurtic). When the values tend towards the extremes, the distribution is broader than the normal distribution and the kurtosis is negative (platykurtic). These three situations are illustrated in Figure 2. The approximate equalities in Eqs. 5 and 6, which are for number distribution, are held when N is large.

**Arithmetic:**

(5)

**Geometric:**

(6)

**x**_{w%}**:**

The x value at which w% of particles has smaller x values. It is most commonly used in diameter distribution presentation, in which d_{w%} means the diameter at which w% of particles is smaller (cumulative undersize by volume.)

From the definitions of arithmetic mean, median, and mode, one can predict that for positively skewed distributions that account for most of the industrial particulate systems, mode < median < mean. The degree of spread in these three values depends on the symmetry of the distribution. For completely symmetric distributions, these three values overlap.

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