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Wednesday, October 3 Variability
| Date post: | 21-Dec-2015 |
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Wednesday, October 3
Variability
nominal
ordinal
interval
nominal
ordinal
interval
RangeInterquartile Range
VarianceStandard Deviation
T
1.61.41.21.0.8.6.4
DIS
2
610
600
590
580
570
560
550
540
T
1.61.41.21.0.8.6.4
DIS
2
610
600
590
580
570
560
550
540
Range
T
1.61.41.21.0.8.6.4
DIS
2
610
600
590
580
570
560
550
540
Range
23N =
DIS2
610
600
590
580
570
560
550
540
InterquartileRange
T
1.61.41.21.0.8.6.4
DIS
2
610
600
590
580
570
560
550
540
X_
Population
Sample
X
µ
_
The population mean is µ. The sample mean is X.
_
Population
Sample
X
µ
_
The population mean is µ. The sample mean is X. The population standard deviation is , the sample sd is s.
_
s
SS
Variance of a population, 2 (sigma squared).
It is the sum of squares divided (SS) by N
N2 =
SS
Variance of a population, 2 (sigma squared).
It is the sum of squares divided (SS) by N
N2 =
(X – )2
SS
The Standard Deviation of a population, It is the square root of the variance.
N =
This enables the variability to be expressed in the same unit of measurement as the individual scores and the mean.
Population
Sample
X
µ
_
The population mean is µ. The sample mean is X. _
Population
Sample
X
µ
_
The population mean is µ. The sample mean is X. The population standard deviation is , the sample sd is s.
_
s
Population
SampleA XA
µ
_
SampleB XB
SampleE XE
SampleD XD
SampleC XC
_
_
_
_
In reality, the sample mean is just one of many possible samplemeans drawn from the population, and is rarely equal to µ.
Population
SampleA XA
µ
_
SampleB XB
SampleE XE
SampleD XD
SampleC XC
_
_
_
_
In reality, the sample mean is just one of many possible samplemeans drawn from the population, and is rarely equal to µ.
sa
sb
sc
sd
se
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 samplingerror is 0. An unbiased statistic tends to be neither largernor 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 shave denominators of N-1 (rather than N) in order to beunbiased.
_
SS
N2 =
SS
(N - 1)s2 =
SS
(N - 1)s2 =
(X – X )2
_
SS
(N - 1)s =
Conceptual formulaVS
Computational formula
What is a measure of variability good for?
