Measures of Variability
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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?