SOCY201 Lecture Notes 2 14 2012 Measures of Dispersion Measures of Dispersion Measures of central tendency estimate the numerical center of a distribution these are measures of location Measures of dispersion estimate the spread or variability of a distribution around the center They indication the amount of variation or average differences among the scores of a frequency distribution We re less familiar with such concepts in daily life although a range of values is sometimes reported o Forecast high temp if 78 80 degrees Measures include o Variation ratio o Index of diversity o Index of qualitative variation o Range o Interquartile range o Quartile deviation o Mean deviation o Variance standard deviation Variation Ratio The variation ratio can be used with grouped data and is most useful for nominal level data V R 1 fmodal n This is the proportion of cases that lie outside of the modal category Higher V R indicate the cases are more equally spread across a variables k categories i e they are less concentrated Advantages The V R can be used with data that do not contain a lot of information i e nominal level data and is easily interpretable Disadvantages The V R may be dependent on the categorization scheme used by the researcher i e it is somewhat arbitrary and does not reflect the distribution of data in the non modal categories Index of Diversity Index of Diversity D measures whether two randomly selected observations are likely to fall into the same or different categories D 1 summation of p2 i SOCY201 Lecture Notes 2 14 2012 Measures of Dispersion Measures of Dispersion Measures of central tendency estimate the numerical center of a distribution these are measures of location Measures of dispersion estimate the spread or variability of a distribution around the center They indication the amount of variation or average differences among the scores of a frequency distribution We re less familiar with such concepts in daily life although a range of values is sometimes reported o Forecast high temp if 78 80 degrees Measures include o Variation ratio o Index of diversity o Index of qualitative variation o Range o Interquartile range o Quartile deviation o Mean deviation o Variance standard deviation Variation Ratio The variation ratio can be used with grouped data and is most useful for nominal level data V R 1 fmodal n This is the proportion of cases that lie outside of the modal category Higher V R indicate the cases are more equally spread across a variables k categories i e they are less concentrated Advantages The V R can be used with data that do not contain a lot of information i e nominal level data and is easily interpretable Disadvantages The V R may be dependent on the categorization scheme used by the researcher i e it is somewhat arbitrary and does not reflect the distribution of data in the non modal categories Index of Diversity Index of Diversity D measures whether two randomly selected observations are likely to fall into the same or different categories D 1 summation of p2 i Index of Qualitative Variation IQV k k 1 D Index of qualitative variation IQV adjusts D for the number of categories k IQV gives a bigger boost to D for a variable with fewer categories allowing comparison of its dispersion to a variable having more categories o The minimum value is 0 o The maximum value is 1 Range The difference between the values of the largest and smallest observations Range max min Advantages extremely simple measure to calculate and interpret useful looking for out of range values Disadvantages dependent on just two values the most extreme and therefore most variable sample dependent observations in a data set Interquartile Range Numerical difference of distance between the third and first quartiles in a distribution the middle 50 of the distribution Mean Deviation With data that contain more information we may calculate a measure that uses all of Calculate the deviation of each score from the mean and compute an average Note absolute values are taken as a convenience that unfortunately has poor this information deviation mathematical properties Advantages mean deviation uses all the valid observations of a variable to produce this summary statistic it is a democratic measure It may be interpreted somewhat intuitively Disadvantages absolute values are not easily algebraically manipulated There is no easy to use metric to aid the interpretation of this statistic as there is for the standard deviation Standard Deviation Most widely used measure of variability usually paired with the mean Advantages Like the mean deviation the standard deviation uses all the valid observations of a variable to produce this summary statistic it is also a democratic measure may be interpreted using Gaussian normal distribution varies from low to high with the spread of the distribution Disadvantages Squaring the differences gives greater weight to more extreme values Skewness Non symmetry in a continuous distribution Values on one side of the distribution tend to be further from the middle than values on the other side o Positive skew tail to right of md o Negative skew tail to left of md o No skew has equal tails on both sides of the md and mean o Skewness 3 Ybar md SY