How to Measure Typical Value of a Probability Distribution

In statistics, we employ on different averages to measure representative or “typical” value of a set of data or distribution. In following, we will look at some of most commonly used averages.

Arithmetic Mean

The most familiar average is the arithmetic mean or simply the mean. A value obtained by dividing the sum of all the observations by total number of observations is known as arithmetic mean,

$$\bar{X} = \frac{\sum x_{i}}{N}$$

Weighted Arithmetic Mean

The relative importance of various observations in a multipliers or a set of numbers which express more or less adequately are technically called the weights. Let the weights $w_{1}, w_{2}, ..., w_{n}$ denote relative importance of the observations in a data set, when the observations are not of equal importance. The weighted mean, $\bar{X}_{w}$, of a set of n values $x_{1}, x_{2}, ..., x_{n}$ with corresponding weights $w_{1}, w_{2}, ..., w_{n}$ is then defined as

$$\bar{X}_{w}=\frac{w_{1}x_{1}, w_{2}x_{2}, ..., w_{n}x_{n}}{w_{1}+w_{2}+…..+w_{n}}$$

Mean From Grouped Data

Data is organized into a frequency distribution when the number of observations is very large, which is used to calculate the approximate values of descriptive measures as the identity of the observations is lost. The observations in each class are assumed to be identical with the class midpoint so that the product of the midpoint by the number of observations, in order to calculate the approximate value of the mean, i.e. frequency would be approximately equal to the sum of observations for each class.

Change of Origin and Scale

A change of the origin and scale can be made in order to reduce the computational labour and to save time. If X1 denotes an observed value, x and y are two constants with y is not equal to zero then the operations Xi + x, yXi and yXi + x are known respectively as the change of origin, the change of scale and both change of origin and scale.

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