Measures of Variability

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

DESCRIBING VARIABILITY

The amount by which scores are dispersed/spread/scattered in a distribution

0

2

4

6

Freq

uenc

y

FEAR

#1

0

2

4

6

Freq

uenc

y

FEAR

#2

0

2

4

6

Freq

uenc

y

FEAR

#3

0

2

4

6

Freq

uenc

y

FEAR

#4

Range

¨  Difference between the smallest and largest observations ¨  Pros and Cons

¤ Easy! J ¤ Values exist in the data set J ¤ Value depends on only two scores L ¤ Very sensitive to outliers L

¨  Examples: ¤ Fear scores: 1, 1, 5, 7, 9, 3, 1 ¤ Height

Deviations

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

¤  Average Mean Difference Score

nXX Σ

=

3515

554321

==++++

=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

*deviations always sum to zero To fix this, square each one…

Variance

¨  Mean of all squared deviation scores ¨  Steps

¤ 1. Calculate sample mean: ¤ 2. Calculate difference scores: score - mean ¤ 3. Square the difference scores (aka the Sum of Squares [SS]) ¤ 4. Add them up:

¤ 5. Take the average

Fear Score

Score -Mean

Difference Difference2

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

3515

554321

==++++

=X

10)( 2 =Σ deviations

2510)( 2

===∑N

deviationsAverage

2σ“sigma”

Variance: Definitional Formula

¨  Population ¨  Sample

NX∑ −

=2

2 )( µσ

1)( 2

2

−

−=∑

nXX

Syour old friend “sigma” …but lower case!

*Note the “n-1” in the sample formula! ** Degrees of freedom (df)

Symbol for sample variance

“mu”

Variance

¨  Use the definitional formula to calculate the variance.

44.4940

110)69()68()68()67()67()66()64()64()64()63(

)(

2

22222222222

22

==

−

−+−+−+−+−+−+−+−+−+−=

−=∑

S

S

nXX

S -1

Variance: Computational Formula

¨  Population ¨  Sample

2

222 )(

N

XXN∑ ∑−=σ

NX∑ −

=2

2 )( µσ 1

)( 22

−

−=∑

nXX

S

)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

44.49360400910)60(400

2

2

2

2

=

−=

−=

S

S

S

Standard Deviation

¨  Rough measure of the average amount by which scores deviate on either side of the mean

¨  Steps: ¤ 1. Calculate variance (we just did this) ¤ 2. Take the square root

σ = σ 2

σ =(X − µ)∑N

2

s = s22

1)(

−

−= ∑

nXX

S

¨  Population ¨  Sample

Variability Example: Standard Deviation

)1(

)( 22

2

−

−=∑ ∑

nnX

XS

Mean: 6

Standard Deviation: 2.11

11.2940

110)69()68()68()67()67()66()64()64()64()63(

)(

2222222222

2

==

−

−+−+−+−+−+−+−+−+−+−=

−= ∑

S

S

nXX

S

11.244.49360400910)60(4002

=

=

−=

−=

SS

S

S

Practice!

¨  Calculate the range, variance, and standard deviation for the following set of “fear” scores

¨  Do this for the population AND the sample formulas

10, 8, 5, 0, 3, 4

Practice!

10, 8, 5, 0, 3, 4 ¨  Mean = 5 ¨  10-5 = 5 à 25 ¨  8-5 = 3 à 9 ¨  5-5= 0 à 0 ¨  0-5= -5 à 25 ¨  3-5= -2 à 4 ¨  4-5= -1 à 1 ¨  Sum of Squares = 64

¨  Population -  Range: 10-0 = 10 -  Variance: 64/6 = 10.67 -  SD = 3.27 ¨  Sample ¨  Range: 10 ¨  Variance: 64/5 = 12.8 ¨  SD = 3.58 ¨  What is the ONLY

difference between the two formulas? (N vs. n-1)

Standard Deviation

¨  A majority (68% for a normal distribution) of all scores are within one standard deviation on either side of the mean

¨  Only a small minority (5% for a normal distribution) is more than two standard deviations on either side of the mean

Pros and Cons of Standard Deviation

¨  Pros ¤ Used in calculating many other measures. ¤ Average of deviations around the mean. ¤ Majority of data within 1 s.d. above or below the mean. ¤ Combined with mean:

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

¨  Cons

¤  Influenced by extreme scores.