Transcript Lecture 4 Slides (Variability)

Wednesday, October 3
Variability
nominal
ordinal
interval
nominal
ordinal
Range
Interquartile Range
interval
Variance
Standard Deviation
610
600
590
580
570
560
DIS2
550
540
.4
T
.6
.8
1.0
1.2
1.4
1.6
610
600
590
580
Range
570
560
DIS2
550
540
.4
T
.6
.8
1.0
1.2
1.4
1.6
610
610
600
600
590
590
580
580
Interquartile
Range
DIS2
Range
570
570
560
560
550
550
540
540
.4
T
.6
.8
1.0
N =1.2
23
1.4
DIS2
1.6
610
600
590
_
X
580
570
560
DIS2
550
540
.4
T
.6
.8
1.0
1.2
1.4
1.6
Population
µ
Sample
_
X
_
The population mean is µ. The sample mean is X.
Population
µ

Sample
_
X
s
_
The population mean is µ. The sample mean is X.
The population standard deviation is , the sample sd is s.
Variance of a population, 2 (sigma squared).
It is the sum of squares divided (SS) by N
2
SS
=
N
Variance of a population, 2 (sigma squared).
It is the sum of squares divided (SS) by N
2
2
SS
=
N
 (X –  )
The Standard Deviation of a population, 
It is the square root of the variance.

SS
=
N
This enables the variability to be expressed in the same unit of measurement
as the individual scores and the mean.
Population
µ
Sample
_
X
_
The population mean is µ. The sample mean is X.
Population
µ

Sample
_
X
s
_
The population mean is µ. The sample mean is X.
The population standard deviation is , the sample sd is s.
Sample
_ C
XC
Sample
_ D
XD
Population
Sample
_ B
µ
Sample
_ E
XE
XB
Sample
_ A
XA
In reality, the sample mean is just one of many possible sample
means drawn from the population, and is rarely equal to µ.
Sample
_ C
XC sc
Sample
_ D
XD sd
Population
Sample
_ B
µ 
Sample
_ E
XE se
XB sb
Sample
_ A
XA sa
In reality, the sample mean is just one of many possible sample
means drawn from the population, and is rarely equal to µ.
Sampling error = Statistic - Parameter
_
Sampling error for the mean = X - µ
Sampling error for the standard deviation = s - 
Unbiased and Biased Estimates
An unbiased estimate is one for which the mean sampling
error is 0. An unbiased statistic tends to be neither larger
nor smaller, on the average, than the parameter it estimates.
_
The mean X is an unbiased estimate of µ.
The estimates for the variance s2 and standard deviation s
have denominators of N-1 (rather than N) in order to be
unbiased.
2
SS
=
N
SS
s2
=
(N - 1)
_2
 (X – X )
SS
s2
=
(N - 1)
s
SS
=
(N - 1)
Conceptual formula
VS
Computational formula
What is a measure of variability
good for?