Which are various measure of dispersion? Explain Variance and Standard Deviation

In many ways, measures of central tendency are less useful in statistical analysis than measures of dispersion of values around the central tendency The dispersion of values within variables is especially important in social and political research because: 

• Dispersion or "variation" in observations is what we seek to explain.
• Researchers want to know WHY some cases lie above average and others below-

Average for a given variable:

o   TURNOUT in voting: why do some states show higher rates than others?

o   CRIMES in cities: why are there differences in crime rates?

o   CIVIL STRIFE among countries: what accounts for differing amounts?

• Much of statistical explanation aims at explaining DIFFERENCES in observations -- also known as 

o    VARIATION, or the more technical term, VARIANCE

If everything were the same, we would have no need of statistics. But, people's heights, ages, etc., do vary. We often need to measure the extent to which scores in a dataset differ from each other. Such a measure is called the dispersion of a distribution Some measure of dispersion are: 

1. Range 
2. Percentile Range 
3. Inter-quartile Range 
4. Average deviation 
5. Standard deviation and variance 
6. Concentration ratio 

Variance and Standard Deviation 

Variance is the average squared difference of score from the means score of the distribution. Standard deviation is the square root of the variance. In calculating the variance of date points, we square the difference between each and the mean because if we summed the difference directly, the result would always be zero. 

For example, three friends work on campus and earn $5.50, $7.50, and $8.00 per hour, respectively. The mean of these values is $(5.5+7.50+8.00)/3 = $7.00 per hr. if we summed the differences of the mean from each wage, we would get: 

$(5.50 - 7.00) + $(7.50 - 7.00) + $(8.00 - 7.00) 

= (1.50) + (- 0.50) + (-1.00) 

= 0.00 

Instead we squared the terms to obtain the variance equal to 2.25 + 0.25 + 1.00 = 3.50. This figure is a measure of dispersion in the set of score. 

The variance is the minimum sum of squared differences of each score from any number, In other words, if we used any number other than the mean as the value from which each score is subtract, the resulting sum of squared differences would be greater. 

The standard deviation is simply the square root of the variance. In some sense, taking the square root of the variance "undoes" the squaring of the differences that we did when we calculated the variance. Variance and standard deviation of a population are designated by and, respectively. Variance and standard deviation of a sample are designated by s2 and s, respectively. 

The standard deviation (or s) and variance (or s2) are more complete measures of dispersion which take into account every score in a distribution. The other measures of dispersion we have discussed are based on considerably less information. However, because variance relies on the squared differences of scores from the mean, a single outlier has greater impact on the size of the variance than does a single score near the mean. 

Some statisticians view this property as a shortcoming of variance as a measure of dispersion, especially when there is reason to doubt the reliability of some of the extreme scores. 

The standard deviation and variance are the most commonly used measures of dispersion in the social sciences because: 

·  Both take into account the precise difference between each score and the mean. Consequently, these measures are based on a maximum amount of information. 

· The standard deviation is the baseline for defining the concept of standardized score or "z-score". 

·  Variance in a set of scores on some dependent variable is a baseline for measuring the correlation between two or more variables (the degree to which they are related).