CHAPTER 8: Measures of Dispersion Flashcards
What are measures of dispersion?
Descriptive summary measure of how varied are observations from each other
What does a small measure of dispersion indicate?
Observations do not vary much and are concentrated about the center of the distribution
What does a large measure of dispersion indicate?
Observations greately vary and are spread out from the center of the distribution
What indicates an absence of variation?
Value of 0
Measure of dispersion is never negative
Differentiate measures of absolute dispersion and measures of relative dispersion
Measures of absolute dispersion:
has the same unit of the observations
Measures of relative dispersion:
has no unit, can be used to compare data sets
Give examples of measures of absolute dispersion
range, interquartile range, standard deviation
*variance is not counted because it is a squared value
Give examples of measures of relative dispersion
coefficient of variation, z score
Define the range
Distance between minimum and maximum values in a data set
Range = Maximum - Minimum
Sometimes presented by stating largest and smallest value
How is the range approximated from the FDT?
Subtracting the LCL of the lowest class interval and the HCL of the highest class interval
Range = UCLhci - LCLlci
Describe the characteristics of the range
An easy-to-use measure of dispersion, not mathematically tractable
Fails to examine the clustering of observations in the middle of the data set
Greately affected by outliers, cannot be approximated if there are open class intervals
Small values = small range, large values = large range
Define the interquartile range (IQR)
IQR = Q3-Q1
Reflects the range of the middle 50% of a data set
Seen as a data set trimmed 25% (tig top and bottom)
The value is dependent on how the quartiles were determined
What does the IQR address?
Addresses sensitivity of the range to outliers, but does not address possible variations in the outer 25% portions
Define population variance
Mean of squared deviations between each observed value and the mean
Squared deviation: (Observation - mean)^2
Define sample variance
Summation of squared deviations between each observed value and the mean, divided by the total number of observations minus 1
Divided by (n-1) to prevent underestimation of the actual variance
Define standard deviation
Positive square root of the variance
What is the purpose of computational formulas?
Removes computation errors if the mean used for it is rounded off
If each observation is transformed by addition or subtraction of a constant, what is the effect on the standard deviation?
According to Property 1, the standard deviation will not change
If each observation is transformed by multiplication or division by a constant, what is the effect on the standard deviation?
According to Property 2, the standard deviation will be multiplied or divided by the same constant
What are the characteristics of the standard deviation?
Uses every observation in the data set
Distorted by outliers
Never negative
Subject to algebraic treatment
Applies to at least interval level data
What does the Bienayme-Chebyshev Rule determine?
How values are described based on the standard deviation
Percentage of observations that are within k standard deviations below and above the mean must be at least [1 - (1/k^2)] * 100
(k > 1)
What percent of observed values are 2 standard deviations below and above the mean?
75%
[1 - (1/2^2)] * 100 = 75%
What percent of observed values are 3 standard deviations below and above the mean?
88.89%
[1 - (1/3^2)] * 100 = 88.89%
What percent of observed values are 3 standard deviations below and above the mean?
99.75%
[1 - (1/4^2)] * 100 = 99.75%
Define the coefficient of variation (CV)
The ratio of the standard deviation to the mean in percentage
Can be used to compare variability of two or more data sets even with different means and units
(Standard deviation/mean)*100
What will the coefficient of variation be when the mean is zero?
Undefined
What will the coefficient of variation be when the mean is negative?
Meaningless
Define the standard score or the Z-score
Measures by how many standard deviations an observation is above or below the mean (relative position of the observation in the data set)
(Observation-mean)/standard deviation
Can be used to compare data sets that differ in unit, mean, standard deviation, or both mean and standard deviation
Based on the z-score, when is an observation an outlier?
When the absolute value of the computed z-score is greater than or equal to 3
How is the z-score interpreted?
If positive: number of standard deviations the observation is above the mean
If negative: number of standard deviations the observation is below the mean
If zero: the observations are equal to the mean
