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Descriptive Statistics
Objectives: (Chapter 3, Decoursey)
- To understand the definition of
, median,
variance, standard deviation, mean absolute
deviation and coefficient variation and calculate
these quantities.
- To calculate some of these quantities using the
statistical functions of
.
Descriptive Statistics
Arithmetic Mean:
1
x or  
N
x
i
x
xi
is the mean of a sample.

is the mean of a population.
is the ith data point in the sample or population.
N is the total number of data points.
Descriptive Statistics
Median: If all the items with which we are concerned
are sorted in order of increasing
(size),
from the smallest to the largest, then the median is
the middle item.
If the number of the items is odd, the median is the
middle item.
e.g. 12, 13, 21, 27, 31
The median is .
If the number of the items are even, the median is given
by the arithmetic mean of the two middle items.
e.g. 12, 13, 21, 27, 33, 37
The median is (21+27)/2=24.
Descriptive Statistics
Variance: it is the mean of the squares of the deviations of
each measurement from the mean of the population.
N
 
2
 (x
i 1
i
 )
2
N
σ2 stands for the variance of the
.
Standard Deviation: A representative of the deviations
from the mean.
N

 (x
i 1
i
 )
N
2
Descriptive Statistics
Estimation of variance from a sample by Bessel’s
N
correction: N/(N-1).
2
( xi  x )
(Sample Variance)
s 
2

i 1
N 1
s2 stands for the variance of the sample.
Sample Standard Deviation:
N
s
 (x
i 1
i
 x)
N 1
2
Descriptive Statistics
shows the variability of the
data by simple calculation though it is not related to the
parameters of theoretical distributions.
N
x
i 1
i
 x /N
Descriptive Statistics
Coefficient of Variation is the ratio between the standard
deviation and the mean for the same set of data,
expressed as a percentage.
/ or s/ x