Statistics

Usually, the first topic in statistics is descriptive statistics. The mean, median and mode are used to describe statistical data. Variance and standard deviation help to understand how the data is spread out.

Mean

The mean most frequently used is the arithmetic mean, which is the same as the average, although the geometric mean is used also at times. It is the arithmetic mean that is referred to when the word mean is used by itself. Expected value is another way of saying the mean.

The lower case Greek letter μ is used to represent the mean. If the mean is from a sample of data, then x is used to represent the sample mean. Also, variables and Sigma notation is used to write the general form of the mean.

x_i = data values

= Sum of data values

n = number of data values

Median

The median is the exact middle value of a set of data values that have been sorted from the lowest value to highest. If the number of data values even, then the median is the average of the two middle values.

Examples:

Data: 1, 2, 3, 4, 5
Median: 3
Data: 1, 2, 3, 4, 5, 6
Median: 3.5

Mode

The mode is the data value that occurs at a greater frequency than the others.

Data: 1, 2, 3, 3, 3, 4, 4, 5
Mode: 3

Range

The range is the highest data value minus the lowest data values.

Data: 1, 2, 3, 4, 5, 6, 7
Highest Data Value: 7
Lowest Data Value: 1
Range: 7 - 1 = 6

Variance

Variance is used to measure how far the data is away from the mean. The distance of the data point from the mean is a deviation. The deviations are added together to get a value representing all the deviations together. However, since some deviations can be negative, the total could be zero. To account for this, the deviations are squared and then added together. When divided by the number of deviations, the result is variance. The symbol used to represent variance is the lower case Sigma squared, σ².

Standard Deviation

The standard deviation is just the square root of the variance.

  σ = √(σ²)