Variability

Variability refers to the Spread or Dispersion of the Distribution

Variability of the Distribution

Range Max - min

Variance Average Squared Distance from Mean

Standard Deviation Average Distance from Mean

(Common Statistics)

(Verbal definitions of Variance and Standard Deviation are not exactly right, but close enough to right and easy to remember.)

Range

Maximum – minimum. Quick, easy. 2, 3, 4, 5, 5, 5, 6, 7. Range is 7 – 2 = 5 2, 3, 4, 5, 5, 5, 6, 19. Range is 19 – 2 = 17

What is the range for the distribution shown in the boxplot?

18N =

VAR00001

9

8

7

6

5

4

3

2

1

Variance

Population Variance: Where means population variance, means population mean, and the other terms have their usual meaning. The variance is equal to the average squared deviation from the mean. To compute, take each score and subtract the mean. Square the result. Find the

average over scores. Ta da! The variance. Think of this as the average squared distance from the mean. The farther scores are

from the mean, the bigger the variance.

N

X

22 )(

2

Computing the Variance(N=5)

5 15 -10 100

10 15 -5 25

15 15 0 0

20 15 5 25

25 15 10 100

Total: 75 0 250

Mean: Variance Is 50

N

XX

22 )(

X X XX 2)( XX NX

X

Standard Deviation

Variance is average squared deviation from the mean.

To return to original, unsquared units, we just take the square root of the variance. This is the standard deviation.

Population formula:

N

X

2)(

Standard Deviation

Sometimes called the root-mean-square deviation from the mean. This name says how to compute it from the inside out.

Find the deviation (difference between the score and the mean).

Find the deviations squared. Find their mean. Take the square root.

Computing the Standard Deviation(N=5)

5 15 -10 100

10 15 -5 25

15 15 0 0

20 15 5 25

25 15 10 100

Total: 75 0 250

Mean: Variance Is 50

Sqrt SD Is

X X XX 2)( XX

07.750

N

XX

2)(

Two Population Distributions

Both distributions have a mean of zero (mu). One has a standard deviation of 1.6, the other has a standard deviation of 10. The SD can be considered the average distance from the mean.

Example: Age Distribution

5040302010

age

16

12

8

4

0

Fre

qu

en

cy

5040302010

age

Distribution of Age

Mean=25.73

5040302010

age

SD = 6.47

Average Distrance from Mean

5040302010

age

Central Tendency, Variability, and Shape

Median = 23

Mode = 21

What is the variance of this distribution (approximately)?

Heiman’s notation for Variance and Standard DeviationSample Variance Sample Standard

Deviation

Estimate of Population Variance

Estimate of Population Standard Deviation

N

XXSX

22 )(

N

XXSX

2)(

1

)( 22

N

XXsX 1

)( 2

N

XXsX

Stats builds. You have to understand and remember these.

Uses

Range is used when a simple description is desired or when the variance or standard deviation cannot be computed.

Standard Deviation is used to communicate the variability of the distribution. Has applications we will cover later.

Variance is used in many statistical calculations for significance tests.

Review

What is variability? Define terms:

Range Variance Standard Deviation

Examples and Computation

Age from demographics Grades from last semester

Computation

How do we compute the range statistic? 1 99th percentile – 1st percentile 2 75th quartile – 25th quartile 3 maximum – minimum 4 minimum – maximum

Definition

The standard deviation is an index of a distribution’s __________.

1 central tendency 2 kurtosis 3 skew 4 variability

Graphs

Which distribution appears to have the largest variance?

3 41 2

9N =

VAR00001

7

6

5

4

3

2

19N =

VAR00001

7

6

5

4

3

2

1

9

9N =

VAR00001

7.0

6.5

6.0

5.5

5.0

4.5

4.0

3.5

3.0

2.5

2.0

1.51.0

9N =

VAR00001

7.0

6.5

6.0

5.5

5.0

4.5

4.0

3.5

3.0

2.5

2.0

1.51.0

Graphs

Which graph shows a higher mean?a. Youngb. Oldc. Samed. Cannot tell

Which graph shows a higher standard deviation?

a. Youngb. Oldc. Samed. Cannot tell

Variability

Documents

Transcript of Variability