3.2 Measures of Dispersion
Coefficient of variation (CV)
The ratio of the standard deviation to the mean as a percentage; allows comparison of the spreads of data from different sources, regardless of differences in units of measurement
Variance
The square of the standard deviation
Population Standard Deviation
The standard deviation of a population data set
Sample Standard Deviation
The standard deviation of a set of sample data
Empirical Rule
Used with bell shaped distributions of data to estimate the percentage of values within a few standard deviations of the mean
Empirical Rule for bell-shaped distributions (68% of the data values)
lie within 1 standard deviations of the mean
Empirical Rule for bell-shaped distributions (95%of the data values)
lie within 2 standard deviations of the mean
Empirical Rule for bell-shaped distributions (99.7% of the data values)
lie within 3 standard deviations of the mean
Properties of the Range
1) Easiest measure of dispersion to calculate 2) Only affected by the largest and smallest values in the data set, so it can be misleading
Properties of the Standard Deviation
1) Easily computed using a calculator or computer 2) Affected by every value in the data set 3) Population standard deviation and sample standard deviation formulas yield different results 4) interpreted as the average distance a data value is from the mean; thus it cannot take on negative values 5) Same units as the units of the data 6) Larger standard deviation indicates that data values are more spread out, smaller standard deviation indicates that data values lie closer together 7) If it equals 0, then all of the data values are equal to the mean 8) Equal to the square root of the variance
Properties of the variance
1) Easily computed using a calculator or computer 2) Affected by every value in the data set 3) Population variance and sample variance formulas yield different results 4) Difficult to interpret because of its unusual squared units 5) Equal to the squared of the standard deviation 6) Preferred over the standard deviation in many statistical test because of its simpler formula
Standard Deviation
A measure of how much we might expect a typical member of the data set to differ from the mean
Chebyshev's Theorem 75% of the data values lie
At least 1-1/2^2=3/4=within 2 standard deviations of the mean
Chebyshev's Theorem 88.9% of the data values
At least 1-1/3^2 =8/9= lie within 3 standard deviations of the mean
Chebyshev's Theorem
Gives a minimum estimate of the percentage of data within a few standard deviations of the mean for any distribution
Range
The difference between the largest and smallest values in the data set, given by Range= Max data value- Min data value
Chebyshev's Theorem
The proportion of data that lie within K standard deviation of the mean is at least 1-1/K^2 for K>1. When K=2: At least 1-1/2^2=3/4=75% of the data values lie within two standard deviations of the mean K=3: At least 1-1/3^2 =8/9= 88.9% of the data values lie within 3 standard deviations of the mean
