Variability

Variability How tightly clustered or how widely dispersed

the values are in a data set. Example

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

Variability: The Range

The Range is one measure of variability The range is the difference between the maximum

and minimum values in a set

Example 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

Split Ordered Data into 4 Quarters

= first quartile

= second quartile= Median

= third quartile

25% 25% 25% 25%

1Q 2Q 3Q

1Q

3Q

2Q

Quartiles

MdQ1 Q3

75%25%

Variability: Interquartile Range

Difference between third & first quartiles Interquartile Range = Q3 - Q1

Spread in middle 50% Not affected by extreme values

Standard Deviation and Variance How much do scores deviate from the mean?

deviation =

Why not just add these all up and take the mean?

X

X X-1

0

6

1

= 2 )-(X

Standard Deviation and Variance Solve the problem by squaring the deviations!

X

X- (X-)2

1 -1 1

0 -2 4

6 +4 16

1 -1 1 = 2

Variance =

N

uX

22 )(

Standard Deviation and Variance Higher value means greater variability around Critical for inferential statistics! But, not as useful as a purely descriptive statistic

hard to interpret “squared” scores!

Solution un-square the variance!

Standard Deviation =N

uX

2)(

Variability: Standard Deviation

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

Standard Deviation can be calculated with the sum of squares (SS) divided by n

Variability: Standard Deviation

N

SS

N

X

2)(

Sample variance and standard deviation

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 Correct for problem by adjusting formula

Different symbol: s2 vs. 2 Different denominator: n-1 vs. N n-1 = “degrees of freedom” Everything else is the same Interpretation is the same

1

)( 22

n

MXs

Definitional Formula:

deviation squared-deviation ‘Sum of Squares’ = SS degrees of freedom

1n

ss

df

SS

1

)( 22

n

XXs

1n

ss

df

SS

1

)( 2

n

XXs

Variance:

Standard Deviation:

Variability: Standard Deviation

let X = [3, 4, 5 ,6, 7] M = 5 (X - M) = [-2, -1, 0, 1, 2]

subtract M from each number in X (X - M)2 = [4, 1, 0, 1, 4]

squared deviations from the mean (X - M)2 = 10

sum of squared deviations from the mean (SS)

(X - M)2 /n-1 = 10/5 = 2.5 average squared deviation from the mean

(X - M)2 /n-1 = 2.5 = 1.58 square root of averaged squared deviation

1

)( 2

n

XXs

Variability: Standard Deviation

let X = [1, 3, 5, 7, 9] M = 5 (X - M) = [-4, -2, 0, 2, 4 ]

subtract M from each number in X (X - M)2 = [16, 4, 0, 4, 16]

squared deviations from the mean (X - M)2 = 40

sum of squared deviations from the mean (SS) (X - M)2 /n-1 = 40/4 = 10

average squared deviation from the mean (X - M)2 /n-1 = 10 = 3.16

square root of averaged squared deviation

1

)( 2

n

XXs

In class example

Work on handout

Standard Deviation & Standard Scores Z scores are expressed in the following way

Z scores express how far a particular score is from the mean in units of standard deviation

X

Z

Standard Deviation & Standard Scores Z scores provide a common scale to express

deviations from a group mean

ZX

X

Z

Let’s say someone has an IQ of 145 and is 52 inches tall 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?

Standard Deviation and Standard Scores

Homework

Chapter 4 8, 9, 11, 12, 16, 17