Introduction to Dispersion
* Statistics is a branch of mathematics that deals with the collection, classification and analysis of numerical data.
* Statistics that signify the extent of the spread of items around the measure of the central tendency is known as dispersion.
* The range of distribution is the difference between the greatest and the least values observed.
Mean Deviation for Ungrouped Data
* The deviation of an observation x from a fixed value k is equal to x – k.
* Mean deviation about the mean = MD (x̅) = \frac{1}{n} \sum_{i=1}^{n} | {X_i} − x̅|, where x̅ = mean.
* Mean deviation about the median = MD (M) = \frac{1}{n} \sum_{i=1}^{n} | {X_i} − M|,
Where M = median.
Mean Deviation about Mean for Grouped Data
Mean (x̅ ) = \frac{1}{N} \sum_{i=1}^{n} { f_{i} }{x_i},
Mean deviation about the mean = MD (x̅ ) = \frac{1}{N} \sum_{i=1}^{n} { f_{i} } | {x_i} - \overline{x} |
Mean Deviation about Median for Grouped Data
Mean deviation about the median of a grouped data
= MD (M) = \frac{ \sum_{i=1}^{n} f_{i} | x_{i - M } | }{N}
Variance
* The variance of a data is the mean of the squares of the deviations of the
observations from the mean. σ2= \frac{1}{n} \sum_{i=1}^{n} ( x_{i} - \overline{x} )2
σ2= \frac{1}{n} \sum_{i=1}^{n} f_{i} ( x_{i} - \overline{x} )
Xi ‘S represents the observations.
Sum of the frequencies: N = \sum_{i=1}^{n} f_{i}
σ2= \frac{1}{N} \sum_{i=1}^{n} f_{i} ( x_{i} - \overline{x} )²
Xi `s represents the mid values of the class intervals.
Sum of the frequencies: N = \sum_{i=1}^{n} f_{i}
Standard Deviation
* Standard deviation is the positive square root of the variance. It is denoted by σ.
* The standard deviation of the observations x_{1} , x_{2} , x_{3} , ....., x_{n} is
σ= \sqrt{ \frac{1}{n} \sum_{i=1}^{n}({X_i} - \overline{X} )²}
* The standard deviation of a discrete frequency distribution is
σ= \sqrt{ \frac{1}{N} \sum_{i=1}^{n} f_{i} ({X_i} - \overline{X} )²}
The standard deviation of a continuous frequency distribution is
σ= \sqrt{ \frac{1}{N} \sum_{i=1}^{n} f_{i} ({X_i} - \overline{X} )²}
Xi‘s are the mid-values of the class intervals.
Shortcut Method to Find Variance and Standard Deviation
Formula for the variance (σ2) = \frac{ h^{2} }{ N^{2} } [ N \sum f_{i} y_{i}² - ( \thinspace \sum f_{i} y_{i} )² ]
Formula for the standard deviation (σ) = \frac{h}{N} \sqrt{N \sum f_{i} y_{i}² - ( \thinspace \sum f_{i} y_{i})² }
Analysis of Frequency Distribution
* Coefficient of variation is defined as standard deviation by mean, and is expressed as a percentage.
* Coefficient of variation (CV) = \frac{ \sigma }{x} × 100, where x̅≠ 0
* The data in a series with a greater coefficient of variation (CV) is said to be more variable.
* The data in a series with a less coefficient of variation (CV) is said to be more consistent.
* The variability of the variance of the distribution.