# Basic Statistics

### Mean

The mean of a set of numbers is the arithmetic average of that set of numbers, as given by the formula below:

$$\Large{\left(\frac{1}{N}\right)\sum_{i=1}^N X_i = \frac{(X_1 + X_2 + \cdots + X_N)}{N}}$$

Example for a set of eight numbers:

{6,3,12,15,4,9,4,8}

the mean value is (6 +3 +12 + 15 + 4 + 9 + 4 + 8)/8 = (61)/8 = 7 5/8

Unlike the median calculation below, it makes no difference if the number of data points is an even number or an odd number.

### Median

The median of a set of numbers is found by ordering the set in numeric order (sorting it by increasing value), and in the case an odd number of data points, choosing the center value of the ordered list. In the case of an even number of data points, the middle two data points would be averaged together to produce the median.

Example for an odd numbered set:

{6,3,12,4,9,4,8}

would produce the ordered set (3,4,4,6,8,9,12}, and the center point of that ordered set is element [3], which as the value 6. So the median value is 6.

Example for an even numbered set:

{6,3,12,15,4,9,4,8}

would produce the ordered set (3,4,4,6,8,9,12,15}, and the center point of that ordered set falls between element [3] and element [4], which are the values 6 and 8 respectively. So the median value is (6+8)/2 = 7. Note that the median value does not actually have to be in the original set of numbers.

### Standard Deviation

The standard deviation of a set of numbers is a measure of the dispersion of the data about the mean of the data. There are two similar formulas used for the calculation of the standard deviation. Which one is used depends on whether or not the set of data values is the total population of data values represented by the process being measured. The one which uses the total population of data values uses the formula:

$$\Large {S = \sqrt{\frac{\sum_{i=1}^N X_i – \bar X}{N} }}$$

But, in SPC Charts, the data being processed does not represent the total population of values. Instead the data represents a sampled subset of possible values. In this case a different formula is used, referred to as the sample standard deviation. It also commonly referred to as the unbiased standard deviation for a sample population.

$$\Large {S = \sqrt{\frac{\sum_{i=1}^N X_i – \bar X}{N-1} }}$$

Note that the difference between the two formulas is the (N) versus (N-1) in the denominator of the fraction.

Every calculation for a standard deviation in this software uses the sample standard deviation formula.