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Measurement of Dispersion, Defining Deviation, Standard Deviation and Variance
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Defining Deviation, Measurement of Dispersion
In statistics, the average is used to measure the center of data, which forms one aspect of observations. The other set of observation is how the observations are spread across the center. The observation can either be close to the center or away from it. When the observation is close to the center, the dispersion or variance is said to be small and when they are spread away, the dispersion is called large. Measures of dispersion are used to study the variability of distribution or data. These data can be scattered around the measures of central tendency. For studying dispersion, some measures are needed which are able to show whether a dispersion is large or small. We get two types of measures of dispersion, these are
1. The absolute measure of dispersion, and
2. The relative measure of dispersion.
The absolute measure of dispersions- is used to give an idea about the amount of observations in a given set of experiments. They give the answers in units which are the same as the units used in the original observations. Hence, when the observation is in kilogram, the absolute measure is also in kilogram. In two sets of observations, the absolute measures for comparing the dispersions cannot be always obtained. The absolute measures which are commonly used are-
2. The quartile deviation
3. The mean deviation
4. Standard deviation and variance.
Range- is defined as the difference between the smallest and largest sample value. It is one of the simplest measures of variability to calculate and depend on the extreme values. It does not provide any information on how the remaining data is distributed.
Standard deviation-the standard deviation is obtained by taking the square root of a sample variance. It is defined for making inferences about the population variance. It can be calculated form the following equation:-
Standard deviation provides significant information of the distributed data around the mean values, thereby approximating normality. These are often used by climatologists for classifying abnormal climatic conditions.
Variance- it is simply the mean of all squared and summed deviations for example the average of squared deviations. The symbol for variance is s2
For finding out variance of a variable X, the equation used is
The relative measure of dispersion- we use this measure for calculating the comparisons of dispersions in two or more than two sets of observations. The units used in original data are not used in these measures. For example, if the original data is expressed in kilometers or dollars, these units do not find mention in relative measures of dispersion. These measures form a ratio of sorts and are called coefficients. These absolute measures of dispersion can be converted into the relative measures. The various relative measures of dispersion are
1. Coefficient of dispersion or coefficient of range.
2. Coefficient of quartile deviation.
3. Coefficient of mean deviation.
4. Coefficient of standard deviation
5. Coefficient of variation.
Sources of statistical dispersion- in physical sciences, the variability’s result form the random errors in measurement, for example instrument measurements etc. In biological sciences, the quantity which is being measured is very seldom unchanging and stable and the observed variable is also intrinsic to the phenomenon. In finance, economics and some other subjects, the regression analysis focuses to explain the dispersion in independent variables, which are generally measured by their variance by using one or more independent variables. All of them have positive dispersions. The fraction of variance which is explained is called the coefficient of determination.
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