Variability Variability n Variability q n How tightly
![Variability Variability](https://slidetodoc.com/presentation_image_h/e70ee61dfd50ffb3d3dc5d591b389eac/image-1.jpg)
Variability
![Variability n Variability q n How tightly clustered or how widely dispersed the values Variability n Variability q n How tightly clustered or how widely dispersed the values](http://slidetodoc.com/presentation_image_h/e70ee61dfd50ffb3d3dc5d591b389eac/image-2.jpg)
Variability n Variability q n How tightly clustered or how widely dispersed the values are in a data set. Example q q q Data set 1: [0, 25, 50, 75, 100] Data set 2: [48, 49, 50, 51, 52] Both have a mean of 50, but data set 1 clearly has greater Variability than data set 2.
![Variability: The Range n The Range is one measure of variability q n The Variability: The Range n The Range is one measure of variability q n The](http://slidetodoc.com/presentation_image_h/e70ee61dfd50ffb3d3dc5d591b389eac/image-3.jpg)
Variability: The Range n The Range is one measure of variability q n The range is the difference between the maximum and minimum values in a set Example q q Data set 1: [1, 25, 50, 75, 100]; R: 100 -0 +1 = 100 Data set 2: [48, 49, 50, 51, 52]; R: 52 -48 + 1= 5 The range ignores how data are distributed and only takes the extreme scores into account RANGE = (Xlargest – Xsmallest) + 1
![Quartiles n Split Ordered Data into 4 Quarters 25% 25% n = first quartile Quartiles n Split Ordered Data into 4 Quarters 25% 25% n = first quartile](http://slidetodoc.com/presentation_image_h/e70ee61dfd50ffb3d3dc5d591b389eac/image-4.jpg)
Quartiles n Split Ordered Data into 4 Quarters 25% 25% n = first quartile n = second quartile= Median n = third quartile 25%
![Quartiles 25% 75% Md Q 1 Q 3 Quartiles 25% 75% Md Q 1 Q 3](http://slidetodoc.com/presentation_image_h/e70ee61dfd50ffb3d3dc5d591b389eac/image-5.jpg)
Quartiles 25% 75% Md Q 1 Q 3
![Variability: Interquartile Range n Difference between third & first quartiles q Interquartile Range = Variability: Interquartile Range n Difference between third & first quartiles q Interquartile Range =](http://slidetodoc.com/presentation_image_h/e70ee61dfd50ffb3d3dc5d591b389eac/image-6.jpg)
Variability: Interquartile Range n Difference between third & first quartiles q Interquartile Range = Q 3 - Q 1 n Spread in middle 50% n Not affected by extreme values
![Standard Deviation and Variance n How much do scores deviate from the mean? q Standard Deviation and Variance n How much do scores deviate from the mean? q](http://slidetodoc.com/presentation_image_h/e70ee61dfd50ffb3d3dc5d591b389eac/image-7.jpg)
Standard Deviation and Variance n How much do scores deviate from the mean? q deviation = X X- 1 0 6 1 =2 n Why not just add these all up and take the mean?
![Standard Deviation and Variance n Solve the problem by squaring the deviations! X- (X- Standard Deviation and Variance n Solve the problem by squaring the deviations! X- (X-](http://slidetodoc.com/presentation_image_h/e70ee61dfd50ffb3d3dc5d591b389eac/image-8.jpg)
Standard Deviation and Variance n Solve the problem by squaring the deviations! X- (X- )2 1 -1 1 0 -2 4 6 +4 16 1 -1 1 X =2 Variance =
![Standard Deviation and Variance n n n Higher value means greater variability around Critical Standard Deviation and Variance n n n Higher value means greater variability around Critical](http://slidetodoc.com/presentation_image_h/e70ee61dfd50ffb3d3dc5d591b389eac/image-9.jpg)
Standard Deviation and Variance n n n Higher value means greater variability around Critical for inferential statistics! But, not as useful as a purely descriptive statistic q n hard to interpret “squared” scores! Solution un-square the variance! Standard Deviation =
![Variability: Standard Deviation n n n The Standard Deviation tells us approximately how far Variability: Standard Deviation n n n The Standard Deviation tells us approximately how far](http://slidetodoc.com/presentation_image_h/e70ee61dfd50ffb3d3dc5d591b389eac/image-10.jpg)
Variability: Standard Deviation n n n The Standard Deviation tells us approximately how far the scores vary from the mean on average estimate of average deviation/distance from small value means scores clustered close to large value means scores spread farther from Overall, most common and important measure extremely useful as a descriptive statistic extremely useful in inferential statistics The typical deviation in a given distribution
![Variability: Standard Deviation n Standard Deviation can be calculated with the sum of squares Variability: Standard Deviation n Standard Deviation can be calculated with the sum of squares](http://slidetodoc.com/presentation_image_h/e70ee61dfd50ffb3d3dc5d591b389eac/image-11.jpg)
Variability: Standard Deviation n Standard Deviation can be calculated with the sum of squares (SS) divided by n
![Sample variance and standard deviation n Sample will tend to have less variability than Sample variance and standard deviation n Sample will tend to have less variability than](http://slidetodoc.com/presentation_image_h/e70ee61dfd50ffb3d3dc5d591b389eac/image-12.jpg)
Sample variance and standard deviation n Sample will tend to have less variability than popl’n if we use the population fomula, our sample statistic will be biased will tend to underestimate popl’n variance
![Sample variance and standard deviation n n n Correct for problem by adjusting formula Sample variance and standard deviation n n n Correct for problem by adjusting formula](http://slidetodoc.com/presentation_image_h/e70ee61dfd50ffb3d3dc5d591b389eac/image-13.jpg)
Sample variance and standard deviation n n n Correct for problem by adjusting formula Different symbol: s 2 vs. 2 Different denominator: n-1 vs. N n-1 = “degrees of freedom” Everything else is the same Interpretation is the same
![Definitional Formula: Variance: deviation Standard n squared-deviation Deviation: n ‘Sum of Squares’ = SS Definitional Formula: Variance: deviation Standard n squared-deviation Deviation: n ‘Sum of Squares’ = SS](http://slidetodoc.com/presentation_image_h/e70ee61dfd50ffb3d3dc5d591b389eac/image-14.jpg)
Definitional Formula: Variance: deviation Standard n squared-deviation Deviation: n ‘Sum of Squares’ = SS n degrees of freedom n
![Variability: Standard Deviation q q q let X = [3, 4, 5 , 6, Variability: Standard Deviation q q q let X = [3, 4, 5 , 6,](http://slidetodoc.com/presentation_image_h/e70ee61dfd50ffb3d3dc5d591b389eac/image-15.jpg)
Variability: Standard Deviation q q q let X = [3, 4, 5 , 6, 7] M=5 (X - M) = [-2, -1, 0, 1, 2] ñ subtract M from each number in X q (X - M)2 = [4, 1, 0, 1, 4] ñ squared deviations from the mean q S (X - M)2 = 10 ñ sum of squared deviations from the mean (SS) q S (X - M)2 /n-1 = 10/5 = 2. 5 ñ average squared deviation from the mean q S (X - M)2 /n-1 = 2. 5 = 1. 58 ñ square root of averaged squared deviation
![Variability: Standard Deviation q q q let X = [1, 3, 5, 7, 9] Variability: Standard Deviation q q q let X = [1, 3, 5, 7, 9]](http://slidetodoc.com/presentation_image_h/e70ee61dfd50ffb3d3dc5d591b389eac/image-16.jpg)
Variability: Standard Deviation q q q let X = [1, 3, 5, 7, 9] M=5 (X - M) = [-4, -2, 0, 2, 4 ] ñ subtract M from each number in X q (X - M)2 = [16, 4, 0, 4, 16] ñ squared deviations from the mean q S (X - M)2 = 40 ñ sum of squared deviations from the mean (SS) q S (X - M)2 /n-1 = 40/4 = 10 ñ average squared deviation from the mean q S (X - M)2 /n-1 = 10 = 3. 16 ñ square root of averaged squared deviation
![In class example n Work on handout In class example n Work on handout](http://slidetodoc.com/presentation_image_h/e70ee61dfd50ffb3d3dc5d591b389eac/image-17.jpg)
In class example n Work on handout
![Standard Deviation & Standard Scores n Z scores are expressed in the following way Standard Deviation & Standard Scores n Z scores are expressed in the following way](http://slidetodoc.com/presentation_image_h/e70ee61dfd50ffb3d3dc5d591b389eac/image-18.jpg)
Standard Deviation & Standard Scores n Z scores are expressed in the following way n Z scores express how far a particular score is from the mean in units of standard deviation
![Standard Deviation & Standard Scores n Z scores provide a common scale to express Standard Deviation & Standard Scores n Z scores provide a common scale to express](http://slidetodoc.com/presentation_image_h/e70ee61dfd50ffb3d3dc5d591b389eac/image-19.jpg)
Standard Deviation & Standard Scores n Z scores provide a common scale to express deviations from a group mean
![Standard Deviation and Standard Scores n Let’s say someone has an IQ of 145 Standard Deviation and Standard Scores n Let’s say someone has an IQ of 145](http://slidetodoc.com/presentation_image_h/e70ee61dfd50ffb3d3dc5d591b389eac/image-20.jpg)
Standard Deviation and Standard Scores n Let’s say someone has an IQ of 145 and is 52 inches tall q q n n IQ in a population has a mean of 100 and a standard deviation of 15 Height in a population has a mean of 64” with a standard deviation of 4 How many standard deviations is this person away from the average IQ? How many standard deviations is this person away from the average height?
![Homework n Chapter 4 q 8, 9, 11, 12, 16, 17 Homework n Chapter 4 q 8, 9, 11, 12, 16, 17](http://slidetodoc.com/presentation_image_h/e70ee61dfd50ffb3d3dc5d591b389eac/image-21.jpg)
Homework n Chapter 4 q 8, 9, 11, 12, 16, 17
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