Post on 01-Jan-2016
description
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
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