Descriptive statistics

Descriptive
Statistics
Rajesh Gunesh
http://pages.intnet.mu/cueboy

ConsulDesign

DESCRIPTIVE (SUMMARY) STATISTICS

Dispersion Skewness

B C

Location A D Kurtosis
Characteristics
of a distribution

© Rajesh Gunesh – Jul 2008

MOMENTS OF A DISTRIBUTION

Fourth moment
KURTOSIS

Third moment
SKEWNESS

Second moment
DISPERSION

First moment
LOCATION


IN PLAIN AND SIMPLE ENGLISH

Location = Central tendency

Dispersion = Spread

Skewness = Symmetry

Kurtosis = Peakedness


LOCATION

Location

ThemeGallery Location
is a Design Digital A measure of location,
Content & Contents otherwise known as
mall developed by central tendency, is a
Guild Design Inc. point in a distribution
that corresponds to a
typical, representative
or middle score in that
distribution.


MEASURES OF LOCATION

 A measure of location, otherwise
known as central tendency, is a point
in a distribution that corresponds to a
typical, representative or middle score
in that distribution.

 The most common measures of location
are the mean (arithmetic), median and
mode.



Mode

Location
Median

Mean



1

The
mean


LOCATION – THE MEAN

 The arithmetic mean is the most
common form of average. For a
given set of data, it is defined as
the sum of the values of all the
observations divided by the total
number of observations. The
mean is denoted by x for a
sample and by µ for a population.
Other existing means are the
geometric, harmonic and
weighted means.


LOCATION – THE MEAN

Arithmetic

Geometric

Mean
Harmonic

Weighted


THE ARITHMETIC MEAN

Mean

Definition Formula

The sum of all

∑
n
observations 1
divided by the x= xi
total number of n
Arithmetic i =1
observations

Text


THE GEOMETRIC MEAN

Mean

Definition Formula

The nth root of n
the product of n
observations
Geometric
GM = n
∏
i =1
xi

Text


THE HARMONIC MEAN

Mean

Definition Formula

The reciprocal of 1
HM =
the mean of the
∑
n

reciprocals of all
1 1
observations n xi
Harmonic i =1

Text


THE WEIGHTED MEAN

Mean

Definition Formula

∑
n
The mean of a
set of numbers wi xi
that have been
weighted Weighted x= i =1

∑
n
(multiplied by
their relative
wi
importance or × i =1
of occurrence).

Text



2

The
median


LOCATION – THE MEDIAN

 The median is the middle observation
of a distribution and can only be
determined after arranging numerical
data in ascending (or descending)
order. If n is the total number of
observations, then the rank of the
median is given by (n+1)/2. For
ungrouped data, if n is odd, the
median is simply the middle
observation but, if n is even, then the
median is the mean of the two middle
observations.


THE MEDIAN (UNGROUPED DATA)

Median

Definition Formula

The median is
the middle 1
observation of a Q2 = (n + 1)
distribution of 2
Ungrouped
arranged
numerical data

Text


THE MEDIAN (GROUPED DATA)

Median

Definition Formula

The median is
the middle  n +1
− cf 
observation of a Q2 = LCB +  2
÷(c)
 f 
distribution of Grouped
arranged
numerical data



Median

Definition Formula
When n is odd, the
median is the 1
middle observation. Q2 = (n + 1)
2
When n is even,
Ungrouped
the median is the
average or midpoint
of the two middle
observations.

Text



Median (n odd)

Example 1:
Ungrouped
Find the median of 27 13 62 5 44 29 16

1 Solution:
Q2 = (n + 1)
2 First re-arrange the numbers in ascending order:
5 13 16 27 29 44 62 (n = 7)

Rank of median = (7 + 1)/2 = 4
Median = 27


Median (n even)

Example 2:
Ungrouped
Find the median of 5 13 16 27 29 44

Solution:
First re-arrange the numbers in ascending order:
5 13 16 27 29 44 (n = 6)
Rank of median = (6 + 1)/2 = 3.5
1
Median = (16 + 27) = 21.5
2


3

The
mode


LOCATION – THE MODE

 The mode is the observation which
occurs the most or with the highest
frequency. For ungrouped data, it may
easily be detected by inspection. If
there is more than one observation
with the same highest frequency, then
we either say that there is no mode or
that the distribution is multimodal.


THE MODE (GROUPED DATA)

Mode

Definition Formula

The mode is the
observation  f1 
which occurs the x = LCB + 
ˆ ÷(c)
most or with the  f1 + f 2 
Grouped
highest
frequency.

Text


DISPERSION

Dispersion

ThemeGallery Dispersion
is a Design Digital A measure of
Content & Contents dispersion shows the
mall developed by amount of variation or
Guild Design Inc. spread in the scores
(values of
observations) of a
variable.


MEASURES OF DISPERSION

 A measure of dispersion shows the
amount of variation or spread in the
scores (values of observations) of a
variable. When the dispersion is large,
the values are widely scattered
whereas, when it is small, they are
tightly clustered.

 The two most well-known measures of
dispersion are the range and standard
deviation.



Standard deviation

B

Mean absolute
Range A C deviation

Dispersion

Coefficient of variation E D Quartile deviation



1

The
range


DISPERSION – THE RANGE

 The range is the difference between
the values of the maximum and
minimum observations of a
distribution. It can only measure the
extent to which the distribution
spreads between its endpoints.


THE RANGE

Dispersion

Definition Formula

The difference
between the
values of the R = xmax − xmin
maximum and Range
minimum
observations of a
distribution

Text



2

The
standard
deviation


DISPERSION – STANDARD DEVIATION

 The standard deviation is defined as
the positive square root of variance or
the square root of the average of the
squared distances of the observations
of a distribution from its mean. We
also use the term standard error in
the case of an estimate.


THE STANDARD DEVIATION

Dispersion

Definition Formula

The square root
of the average of
the squared 1 n
distances of the Standard
s= ∑ (x − x )
2

observations deviation n i =1
from the mean.

Text



3

The
mean
absolute
deviation


DISPERSION – MEAN ABSOLUTE DEVIATION


 The mean absolute deviation (MAD) is
defined as the average of the
distances of the observations of a
distribution from its mean.


THE MEAN ABSOLUTE DEVIATION

Dispersion

Definition Formula

The average of
the distances of
the observations 1 n
of a distribution
MAD = ∑ xi − x
from its mean. MAD n i =1

Text



4

The
quartile
deviation


DISPERSION – QUARTILE DEVIATION


 The quartile deviation is equal to half
the difference between the lower and
upper quartiles and is sometimes
called the semi inter-quartile range.


THE QUARTILE DEVIATION

Dispersion

Definition Formula

It is half the
difference
between the Q3 − Q1
lower and upper Quartile
QD =
quartiles deviation 2

Text



5

The
coefficient
of
variation


DISPERSION – COEFFICIENT OF VARIATION


 The coefficient of variation (CV) is
mainly used to compare the dispersion
of two distributions that have different
means and standard deviations; it is
thus considered to be a relative
measure of dispersion.


COEFFICIENT OF VARIATION

Dispersion

Definition Formula

It is used to
compare two
distributions s
when they have Coefficient of CV = × 100
the same mean variation x
but different
standard
deviations

Text


SKEWNESS

Skewness

ThemeGallery Skewness
is a Design Digital Skewness is a
Content & Contents measure of symmetry
mall developed by – it determines whether
Guild Design Inc. there is a
concentration of
observations
somewhere in
particular in a
distribution.


MEASURES OF SKEWNESS

Skewness
Quartile
coefficient

Pearson’s
coefficient


SKEWNESS

 Skewness is a measure of symmetry –
it determines whether there is a
concentration of observations
somewhere in particular in a
distribution. If most observations lie
at the lower end of the distribution,
the distribution is said to be positively
skewed. If the concentration of
observations is towards the upper end
of the distribution, then it is said to
display negative skewness. A
symmetrical distribution is said to
have zero skewness.

SKEWNESS

Positively skewed Symmetrical Negatively skewed

 The vertical bars on each diagram indicate the
respective positions of the mean (bold), median
(dashed) and mode (normal). In the case of a
symmetrical distribution, the mean, median and
mode are all equal in values (for example, the
normal distribution).



1

Pearson’s
coefficient


SKEWNESS – PEARSON’S COEFFICIENT

 This is the most accurate measure of
skewness since its formula contains
two of the most reliable statistics, the
mean and standard deviation.


PEARSON’S COEFFICIENT

Skewness

Definition Formula
The most accurate
measure of
3( x − Q2 )
skewness since its
formula contains α=
the two most Pearson’s s
reliable statistics:
the mean and
standard deviation

Text



2

Quartile
coefficient


SKEWNESS – QUARTILE COEFFICIENT

 A less accurate but relatively quicker
way of estimating skewness is by the
use of quartiles of a distribution


QUARTILE COEFFICIENT

Skewness

Definition Formula

A less accurate
but relatively
Q1 + Q3 − 2Q2
quicker way of α=
estimating Quartile Q3 − Q1
skewness is by
the use of
quartiles of a
distribution

Text


KURTOSIS

Kurtosis

ThemeGallery Kurtosis
is a Design Digital Kurtosis may be
Content & Contents considered as a
mall developed by measure of the relative
Guild Design Inc. concentration of
observations in the
centre, upper and
lower ends and the
shoulders of a
distribution.


KURTOSIS

 Kurtosis indicates the degree of
peakedness of a unimodal frequency
distribution. It may be also considered
as a measure of the relative
concentration of observations in the
centre, upper and lower ends and the
shoulders of a distribution. Kurtosis
usually indicates to which extent a
curve (distribution) departs from the
bell-shaped or normal curve.


KURTOSIS

 It is customary to subtract 3 from the
coefficient of kurtosis for the sake of
reference to the normal distribution. A
negative value would indicate a
platykurtic curve whereas a positive
coefficient of kurtosis indicates a
leptokurtic distribution. A value close
to 0 means that the distribution is
mesokurtic, that is, close to the
normal.


KURTOSIS

Platykurtic Mesokurtic Leptokurtic


A MEASURE OF KURTOSIS

Kurtosis

Coefficient
of kurtosis


MEASURES OF KURTOSIS

1

Coefficient
of kurtosis


COEFFICIENT OF KURTOSIS

Kurtosis

Definition Formula

It is customary to

∑
subtract 3 from
for the sake of ( x − x )4
reference to the
normal
Coefficient β=
distribution ns 4

Text


SUMMARY OF DESCRIPTIVE STATISTICS

Distribution
Distribution

Characteristics

Location Dispersion Skewness Kurtosis


Descriptive statistics

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Descriptive statistics