Statistics

Statistics

Statistics is a branch of mathematics that involves the collection, analysis, interpretation, and presentation of data. It helps us make sense of the world by summarizing information, identifying patterns, and making predictions based on data collected from various sources. Statistics is used in many fields, including science, business, economics, and social sciences, to understand trends, make informed decisions, and solve problems.

Measure of Dispersion

A measure of dispersion is the method of finding information regarding the amount of variability or spread or deviation, or scatterness present in the data. If all the values of a data set are identical, there is no dispersion. Measures of dispersion are statistical tools that quantify the degree to which data points vary around a central value, providing insights into the spread or variability within a dataset.

Key Points:

• The amount of dispersion in different values may be small if the values are close together.
• The methods of measure of dispersion are as follows.

1. Quartile Deviation (QD)
2. Mean Deviation (MD)
3. Standard Deviation (SD)

Quartile Deviation and Its Coefficient

What is Quartile?

The three values that divide the given dataset into four parts or segments are called the quartiles. The three values are \(Q_1\), \(Q_2\) and \(Q_3\). The first quartile \(Q_1\) is called the lower quartile, which holds 25% of the given dataset. The second quartile \(Q_2\) is also called the median, which holds 50% of the given dataset. And the third quartile \(Q_3\) is called the upper quartile, which holds 75% of the given dataset. 

Quartile Deviation (QD)

The half of the difference between the third quartile/upper quartile \(Q_3\) and the first quartile/lower quartile \(Q_1\) is called the Quartile Deviation. The difference between \(Q_3\) and \(Q_1\) is called inter-quartile range. Hence, the formula is:

Coefficient of Quartile Deviation (CQD)

It is a normalized version of the quartile deviation, providing a relative measure of dispersion. The formula is:

Mean Deviation

Mean Deviation (MD) is a measure of dispersion that shows the average of the absolute deviations of each data point from a central value (usually mean, median).

It tells us how much, on average, the values in a dataset differ from the central value.

Formula for Mean Deviation

The mean deviation can be calculated from the mean and from the median. The formulas are as follows.

1. From Mean

2. From Median

where: \(\bar{x}\) = Mean, \(M_d\) = Median, \(n\) = Number of observations

Formula for Coefficient of Mean Deviation

The coefficient makes MD dimensionless, allowing comparison between datasets. The formulas for the coefficient of Mean Deviation are as follows.

1. From Mean

2. From Median