“measures of central tendency” - wordpress.com · 7/6/2016 · measures of central tendency....
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Session VI
“Measures of Central Tendency”
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Measures of Central Tendency
The three measures in common use are:
Average
Median
Mode
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Average
There are three different techniques available for
calculating the average
Ungrouped data
Grouped data
Weighted average
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Average-Ungrouped Data
1
ni
i
XX
n
Xi=readings n=number of observations
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Average-Grouped Data
1
1 1 2 2
1 2
... .
...
hi i
i
h h
h
f XX
n
f X f X f X
f f f
h = number of cells fi=frequencyXi=midpoint n=number of observations
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Average-Weighted Average
1
1
n
i iiw n
i
i
w XX
w
Used when a number of averages are combined with different frequencies
Xi=readings n=number of observationsW=weight of average
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Median-Grouped Data
2m
d m
m
ncf
M L if
lm = lower boundary of the cell with the median
n = total number of observations
cfm= cumulative frequency of all cells below m
fm = frequency of median cell
i =cell interval
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Example
Boundaries Midpoint Frequency Computation
23.6-26.5 25.0 4 100
26.6-29.5 28.0 36 1008
29.6-32.5 31.0 51 1581
32.6-35.5 34.0 63 2142
35.6-38.5 37.0 58 2146
38.6-41.5 40.0 52 2080
41.6-44.5 43.0 34 1462
44.6-47.5 46.0 16 736
47.6-50.5 49.0 6 294
Total 320 11549
Frequency Distribution of the Life of 320 tires in 1000 km
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Median-Grouped Data
2m
d m
m
ncf
M L if
320154
235.6 3 35.958
Md
Using data from example table
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Mode
The Mode is the value that occurs with the
greatest frequency.
It is possible to have no modes in a series or
numbers or to have more than one mode.
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Relationship Among theMeasures of Central Tendency
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Measures of Dispersion
Range
Standard Deviation
Variance
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Range
The range is the simplest and easiest to
calculate of the measures of dispersion.
Range = R = Xh – Xl
Largest value - Smallest value in data set
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Standard Deviation
Sample Standard Deviation:
2
1( )
1
n
iXi X
Sn
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Standard Deviation
Ungrouped Technique
2 2
1 1( )
( 1)
n n
i in Xi Xi
Sn n
2
2
11
/
1
nn
ii
Xi Xi n
Sn
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Standard Deviation
2 2
11
( ) ( )
( 1)
hh
i i i ii
i
n f X f X
sn n
Grouped Technique
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Variance
2
2
11
/
1
nn
ii
Xi Xi n
Sn
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Comments on the Measures of Dispersion
As n increases, accuracy of R decreases
Use R when there is small amount of data or data is
too scattered
If n > 10 use standard deviation
A smaller standard deviation means better quality
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Comparison of two distributions with equal average and range
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Other Measures
There are three other measures that are
frequently used to analyze a collection of data:
Skewness
Kurtosis
Coefficient of Variation
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Skewness is the lack of symmetry of the data.
For grouped data:
Skewness
3
13 3
( ) /h
i ii
f X X na
s
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Skewness
Left (negative) and right (positive) skewness distributions
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Kurtosis provides information regarding the shape of the
population distribution (the peakedness or heaviness of
the tails of a distribution).
For grouped data:
Kurtosis
4
14 4
( ) /h
i ii
f X X na
s
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Kurtosis
Leptokurtic and Platykurtic distributions
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Correlation variation (CV) is a measure of how much
variation exists in relation to the mean.
(100%)sCV
X
Coefficient of Variation
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Population
Set of all items that possess a characteristic of
interest
Sample
Subset of a population
Population and Sample
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Parameter is a characteristic of a population, it describes a
population
Example: average weight of the population, e.g. 50,000 cans
made in a month.
Statistic is a characteristic of a sample, used to make
inferences on the population parameters that are
typically unknown, called an estimator
Example: average weight of a sample of 500 cans from that
month’s output, an estimate of the average weight of
the 50,000 cans.
Parameter and Statistic
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Comparison of Sample and Population
Statistic Parameter
X average μ(Xo) mean
s sample standard deviation σ(so) standard deviation