Descriptive statistics formulae Part I

Descriptive Statistics Formulae
Part I -
Measures of Central Tendency
Prepared By:
Medha Kulkarni
Email: medha.dataanalyst@gmail.com
Linkedin: http://in.linkedin.com/in/statsanalyst
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Follow @letstalkstatistics
Blog: letstalkstatistics.wordpress.com
Blog: letstalkstatistics.wordpress.comInstagram: @letstalkstatistics
Let’s Talk
Statistics
∑ fi, N n xi
r R2 α
β……

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ARITHMETIC MEAN
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Statistics
Arithmetic
Mean
 Add the values of all the observations.
 Divide the above sum by the total number of
observations.
 This calculated value is Mean
 If x1, x2, x3…..xn are n observations, then
mean = (x1+x2+x3+…..xn)/ n
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WEIGHTED ARITHMETIC MEAN
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Statistics
Weighted
Mean
The Weighted Arithmetic Mean of a set of
numbers x1, x2, ... xn, with corresponding
weights w1, w2, ...,wn, is,
n
nn
w
wwww
xwxwxwxw
x



....
.....
321
332211
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MEAN FOR UNGROUPED FREQUENCY
TABLE Follow @letstalkstatistics
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Statistics
Mean for
ungrouped
data
Mean for an ungrouped frequency distribution
is calculated using the formula:
where fi is the frequency of the variable xi
Mean =∑fi xi
∑fi
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MEAN FOR GROUPED FREQUENCY TABLE
Let’s Talk
Statistics
Mean for a grouped frequency distribution is
calculated using the formula:
where fi is the frequency of the variable xi
xi is the class mark of the corresponding class
Mean =∑fi xi
∑fi
Mean for
grouped
data
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MEDIAN
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Statistics
Median
@letstalkstatistics
 Middle value in the ordered data set.
 Arrange the data in ascending or descending
order.
 If number of observations are odd then
[(n+1)/2]
item is Median.
 If number of observations are even then
average of {(n/2)th item + [(n/2)+1]th item} will
be Median.

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MEDIAN FOR UNGROUPED FREQUENCY
TABLE
 Find N/2 where N= ∑fi
 Check cumulative frequency just greater
than N/2
 Corresponding x value is Median
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Let’s Talk
Statistics
Median for
Ungrouped
Table

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MEDIAN FOR GROUPED FREQUENCY TABLE
 Median = L + h ( N - c )
f 2
Median class = class corresponding to cumulative
frequency (cf) just greater than N/2
L = lower limit of median class
f = frequency of median class
h = width of median class
c = cf of the class preceding the median class
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Let’s Talk
Statistics
Median for
Grouped
Table

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MODE FOR UNGROUPED FREQUENCY
TABLE Follow @letstalkstatistics
@letstalkstatistics
Let’s Talk
Statistics
Mode for
Ungrouped
Table
 Mode for ungrouped frequency table is the value
that has the maximum frequency.
 Data can have more that one mode.
 Data with one mode value is called uni-modal, with
two modes is called bi-modal & with more than
two modes is called multi-modal.

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MODE FOR GROUPED FREQUENCY TABLE
 Mode = l + {h(f1-f0) /(f1-f0)-(f2-f1)}
OR
 Mode = l +{h( f1 – f0 )/( 2f1 – f0 – f2 )}
Modal class = the class interval with the largest
frequency.
l = lower limit
h = width
f1 = frequency of modal class
f0 = frequency of class preceding modal class
f2 = frequency of class succeeding modal class
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Let’s Talk
Statistics
Mode for
Grouped
Table

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EMPIRICAL RELATION BETWEEN MEAN,
MODE & MEDIAN Follow @letstalkstatistics
@letstalkstatistics
Let’s Talk
Statistics
Empirical
Rule
 (Mean –Mode) = 3*(Mean-Median)
 Mode = 3Median – 2Mean

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GEOMETRIC MEAN
Let’s Talk
Statistics
Geometric
Mean
The Geometric Mean (GM) of n observations is
the nth root
of the product of the n observations.
GM x x x xnn ( )*( )*( ). .. *( )1 2 3
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GEOMETRIC MEAN FOR GROUPED TABLE
Let’s Talk
Statistics
Geometric
Mean for
grouped data
The Geometric Mean (GM) of n non-zero &
positive observations is given by
where values have
frequencies.
  Nn
i
f
i
Nf
n
ff in
xxxxGM
1
1
1
21
21






 

nxxx ,........,, 21
nfff ,.......,, 21
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WEIGHTED GEOMETRIC MEAN
Let’s Talk
Statistics
Weighted
Geometric
Mean
The Weighted Geometric Mean of a set of
G.Mw =
where N = i.e. sum of weights
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HARMONIC MEAN
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Statistics
Harmonic
Mean
 The harmonic mean (HM) is the number of
variables divided by the sum of the reciprocals
of the variables.
where n = total number of observations

 n
i ix
n
HM
1
1
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HARMONIC MEAN FOR GROUPED DATA
Let’s Talk
Statistics
 The harmonic mean (HM) for grouped data is
given by
where fi is the frequency of the variable xiHarmonic
Mean for
Grouped
Data

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WEIGHTED HARMONIC MEAN
Let’s Talk
Statistics
Weighted
Harmonic
Mean
The Weighted Harmonic Mean of a set of
where n = sum of weights
nn
w
xwxwxw
n
HM
/....// 2211 

@letstalkstatistics

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RELATION BETWEEN ARITHMETIC MEAN,
GEOMETRIC MEAN & HARMONIC MEAN
@letstalkstatistics
Let’s Talk
Statistics
AM, GM,
HM
 For positive numbers

THE END
THANK YOU!
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Let’s Talk
Statistics
∑ fi, N n xi
r R2 α
β……

Descriptive statistics formulae Part I

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Descriptive statistics formulae Part I