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Chapter 4: measures of dispersion
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Terms in this set (33)
Dispersion def
refers to the variety, diversity, or amount of variation among scores
Greater the dispersion of variable
greater the range of scores and the greater the differences between scores
*Function of larger sized samples
propensity for numbers to be closer to the mean
Range (R)
R= High score- Low Score
Meaning of range
greater the distance from high to low score, the greater the dispersion and distribution
Range can be used with what level of variable?
ordinal and interval ratio but NOT nominal
Range limitations (3)
1) Based on only 2 scores
2) distorted by atypically high or low scores
3) no info about variation between high+ low scores
What is the Index of qualitative variation IQV?
A measure of dispersion for any variable that has been organized into frequency distributions
Theoretical range for IQV
0.00 (no variation) to 1.00 (total variation)
Interquartile Range (Q)
A kind of range that considers only the middle 50% of the cases in a distribution
Limitation of interquartile range
Because the interquartile range is based on only 2 scores, it fails to yield any info from all of the other scores
How to find Q? (3)
1) Arrange the scores from highest to lowest
2) Divide the distribution into quarters (as distinct from halves when locating the median)
3) Q1 is 25% of cases, Q2 is 50%, Q3 is 75%
Interquartile range formula
Q=Q3-Q1
Difference between Q and R
Unlike range, Q avoids the problem of being based on the most extreme scores, but it also fails to yield any info about the variation of the scores other than the 2 on which it is based
A good measure of dispersion should (3)
1) Use all the scores in the distribution. The statistics should use all the info available.
2) Describe the average or typical deviation of the scores. The statistic should give us an idea about how far the scores are from each other or from the center of distribution
3) Increase in value as the scores become more diverse. This would be a very handy feature when comparing distributions because it would permit us to tell at a glance which was more variable.
The most important and widely used measure of dispersion
Standard Deviation
What level of measurement should standard deviation be used with
Interval-ratio (but sometimes ordinal)
What are deviations
distances between each score and the mean (Xi-X)
If we sum the deviations of any set scores from their mean, we would always wind up with what value?
0
Standard deviation formula and variance
look at textbook pg 97
Use N-1 in denominators when?
when we are working with random samples
5 steps to find standard D
1) subtract mean from each score
2) square deviations
3) sum the squared deviations
4) divide sum of squared deviations by N
5) find square root of the result
What are boxplot or box and whiskers plot?
A graph that provides a helpful way to visualize and analyze dispersion and gives us an opportunity to apply some of our growing array of statistical tools
What do boxplots use? (4)
1) median
2) range (R)
3) Interquartile range (Q)
4) Outliers
What does the boxplot represent?
Q Interquartile range
What do the whiskers represent?
Outliers, or any score beyond 1.5 times the height of the box in either direction
What are extreme outliers?
Scores that are beyond 3 times the height of the box in either direction
When are boxplots especially useful?
When we want to compare the distribution of a variable across different conditions or times
Reread pg 102-103 for boxplot example
good job
Highest interquartile range y indicate what?
Greatest dispersion
Meaning of standard deviation can be expressed in what 3 ways
1) normal curve
2) As an index of dispersion that increases in value as the distribution becomes more variable
3) comparing one distribution to another
What is the lowest value for standard deviation?
0 (when every single case in the sample had exactly the same score)-- Zero is the lowest value for standard deviation although there is no upper limit
Why will any distribution of income be positively skewed?
Low income values will have less effect on mean than very high values
Read pg 107-end
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