Statistics

Variability

Variability

What does it mean?

• Variability:

Degrees of Variability

Variability

Variability

Traits of good measures of variability

Variability

1.)

2.)

3.)

4.)

4 types:

Measures of variability

1.) Range

2.) Interquartile Range

3.) Standard Deviation*

4.) Variance*

The range

Formula

Range =

Measures of variability

ex. Calculate the range for the following ages:

2, 4, 28, 34, 35, 59, 61, 70, 73, 83

The Interquartile Range & Semi-Interquartile Range

• Quartile sounds like what?

Distribution slicing

Measures of variability

• Why do we use? How do we find the quartiles?

First Quartile (Q1):

Second Quartile (Q2):

Third Quartile (Q3):

The Interquartile Range & Semi-Interquartile Range

Measures of variability

Interquartile range:

Semi-Interquartile Range:

Measures of variability

Measures of variability

Figure 4.2, p. 108

Q1=

24, 24, 25, 26, 26, 26, 27, 27, 30, 33, 33, 35, 35, 36, 43

• SIQR=

• IQR=

Find the Interquartile & Semi-Interquartile Ranges

Q3=Median=

Standard Deviation & Variance

• The mean is the reference point for calculating these

measures

• Goal:

Measures of variability

The mean and standard deviation: Population

Standard Deviation & Variance

Step 1: find the deviation of

each score from the mean

• Formulas

Step 2a: Calculate the mean

of the deviation scores

Deviation score = X - µ

X

6

10

4

4

6

7

11

7

3

11

6

12

X - µ

-1.25

2.75

-3.25

-3.25

-1.25

-0.25

3.75

-0.25

-4.25

3.75

-1.25

4.75

Step 2b: To get rid of the

negative signs square

each deviation score first

(X - µ)²

1.56

7.56

10.56

10.56

1.56

0.06

14.06

0.06

18.06

14.06

1.56

22.56

Measures of variability

Measures of variability

Standard Deviation & Variance

Step 3: Compute the mean

squared deviation “Variance”

• Formulas

X

6

10

4

4

6

7

11

7

3

11

6

12

X - µ

-1.25

2.75

-3.25

-3.25

-1.25

-0.25

3.75

-0.25

-4.25

3.75

-1.25

4.75

(X - µ)²

1.56

7.56

10.56

10.56

1.56

0.06

14.06

0.06

18.06

14.06

1.56

22.56

Mean squared deviation =∑ (X - µ)²

OR

N

Mean squared deviation =

Variance (²) =

SS

N

=

Measures of variability

Standard Deviation & Variance

• Formulas

Step 4: “Unsquare” to correct for

the squaring of all the

individual distances (i.e., take

the square root).

X

6

10

4

4

6

7

11

7

3

11

6

12

X - µ

-1.25

2.75

-3.25

-3.25

-1.25

-0.25

3.75

-0.25

-4.25

3.75

-1.25

4.75

(X - µ)²

1.56

7.56

10.56

10.56

1.56

0.06

14.06

0.06

18.06

14.06

1.56

22.56Standard Deviation ( ) = Variance =

Measures of variability

Standard Deviation ( ) =

Standard Deviation & Variance

• Quick Steps for computing

Step 1: Find the distance from

Step 2: Square

Step 3: Find the Sum

Step 4: Find the mean

Step 5: Take the square root

Measures of variability

Population vs. Sample Variability

Figure 4.6, p. 117

Standard Deviation & Variance

• Steps for computing for a sample

Step 1: Find the distance from the mean for each individual

Step 2: Square each distance

Step 3: Sum the Squared distances (SS)

X - X

(X – X)²

SS = ∑(X – X)² Definitional formula

Measures of variability

Standard Deviation & Variance

• Steps for computing for a sample

Step 4: Find the mean of the squared distances (Sample Variance)

Step 5: Take the square root of the sample variance (Std. Deviation)

• Must correct for bias in sample variability

Sample variance =

Sample standard deviation = S = S² =

Measures of variability

In class exercise (Part 1)

1.) Calculate the mean (by hand)

2.) Calculate the variance (by hand)

3.) Calculate the standard deviation (by hand)

Degrees of freedom (df)

n - 1df=

Measures of variability

Biased vs. unbiased statistics

• Biased statistic

• Unbiased statistic

Measures of variability

Importance of Variance and Std. Deviation

• Provides information

• Large variance:

• Small variance:

Measures of variability

Factors that Affect Variability