Measures of Variability

• A single summary figure that describes the spread of observations within a distribution.

Measures of Variability

• Range– Difference between the smallest and largest

observations.

• Variance– Mean of all squared deviations from the mean.

• Standard Deviation– Rough measure of the average amount by which

observations deviate from the mean.– The square root of the variance.

Variability Example: Range• Las Vegas Hotel Rates

52, 76, 100, 136, 186, 196, 205, 150, 257, 264, 264, 280, 282, 283, 303, 313, 317, 317, 325, 373, 384, 384, 400, 402, 417, 422, 472, 480, 643, 693, 732, 749, 750, 791, 891

• Range: 891-52 = 839• Pros

– Very easy to compute.– Scores exist in the data set.

• Cons– Value depends only on two scores.– Very sensitive to outliers.

Variance

• The average amount that a score deviates from the typical score.– Score – Mean = Difference

Score– Average Mean Difference

Score

n

XX

Σ=

35

15

5

54321==

++++=X 0)( =Σ deviations

Score Score-Mean

Difference

1 1-3 -2

2 2-3 -1

3 3-3 0

4 4-3 1

5 5-3 2

Variance

– In order to make this number not 0, square the difference scores (no negatives to cancel out the positives).

10)( 2 =Σ deviations

Score Score-Mean

Answer Answer2

1 1-3 -2 4

2 2-3 -1 1

3 3-3 0 0

4 4-3 1 1

5 5-3 2 4

25

10)( 2

===∑N

deviationsAverage

Variance: Definitional Formula

• Population • Sample

N

X∑ −=

22 )( μ

σ1

)( 22

−−

=∑n

XXS

“sigma” *Note the “n-1” in the sample formula!

Variance

• Use the definitional formula to calculate the variance.

-1

Variance: Computational Formula

• Population • Sample

2

222

)(

N

XXN∑ ∑−=σ

N

X∑ −=

22 )( μ

σ 1

)( 22

−−

=∑n

XXS

)1(

)( 22

2

−

−=∑ ∑

nnX

XS

Variance

• Use the computational formula to calculate the variance.

X X2

3 94 164 164 166 367 497 498 648 649 81

Sum: 60 Sum: 400)1(

)( 22

2

−

−=∑ ∑

nnX

XS

44.4

9

360400910)60(

400

2

2

2

2

=

−=

−=

S

S

S

Variability Example: Variance

• Las Vegas Hotel Rates

37.60774

34

45119571.31668620213535

)13386()6686202(

2

2

2

2

=

−=

−

−=

S

S

S

X X2

472 222784303 91809280 78400282 79524417 173889400 160000254 64516205 42025384 147456264 69696317 10048976 5776

643 413449480 230400136 18496250 62500100 10000732 535824317 100489264 69696384 147456750 562500402 161604422 178084373 139129325 105625313 97969749 561001791 625681196 38416891 793881283 8008952 2704

186 34596693 480249

Sum: 13386 Sum: 6686202

)1(

)( 22

2

−

−=∑ ∑

nnX

XS

Standard Deviation

• Population • Sample

-Rough measure of the average amount by which observations deviate on either side of the mean.

-The square root of the variance.

-Returns squared units to original units (more meaningful)

σ = σ 2 s= s2

σ =(X −μ)∑N

2 2

1

)(

−−

= ∑n

XXS

σ =N X2 −( X∑ )∑

N2

2

)1(

)( 22

−

−=

∑ ∑

nnX

XS

Variability Example: Standard Deviation

)1(

)( 22

2

−

−=

∑ ∑

nnX

XS

Mean: 6

Standard Deviation: 2.11

11.29

40

110

)69()68()68()67()67()66()64()64()64()63(

)(

2222222222

2

==

−−+−+−+−+−+−+−+−+−+−

=

−= ∑

S

S

nXX

S

11.2

44.4

9

360400

910

)60(400

2

==

−=

−=

SS

S

S

Pros and Cons of Standard Deviation

• Pros– Lends itself to computation of other stable measures

(and is a prerequisite for many of them).– Average of deviations around the mean.– Majority of data within one standard deviation above or

below the mean.– Combined with mean:

• Efficiently describes a distribution with just two numbers• Allows comparisons between distributions with different scales

• Cons– Influenced by extreme scores.

Ch. 4 homework16 8 23 7 31 18 12 19 15 20 28 27 9 18 49

11 14 5 18 17 3 6 25 1• 1. Find the range, variance, and standard deviation for the

sample data (# candy bars eaten) above.• 2. How would your answers in #1 change if this were a

population? Why?• 3. Repeat #1 if the sample data included the 100 candy bar

eater. How do these results compare to before?• 4. If everyone in the sample ate 10 more candy bars the

following month, what would the range, variance, and standard deviation be?