MBA Super Notes on Descriptive Statistics

MBA Super Notes© M S Ahluwalia Sirf Business
Version 1.0
Descriptive measures

MBA SUPER NOTES
Statistics

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Numerical Descriptive measures
4
Measures of central tendency/
location
Mean and
its types
Measures
of location
Mode
Measures of dispersion/ variation
Range and
IQR
Mean and
standard
deviation
Coefficient
of variation
Measures of shape/
symmetry
Skewness Kurtosis
Numerical
descriptive
measures
• Large data sets can often be adequately described by just a few numbers:
• Populations are described by parameters. The symbols are notated by
Greek symbols, or upper case English symbols
• Samples are described by statistics. The symbols are notated by lower
case English symbols
• Populations and parameters are seldom encountered in the real world,
therefore, it would be worthwhile to focus attention on samples and
statistics
Types of descriptive measures

Interpreting histograms (1/2)
5
Interpreting
histograms
• The main reason for drawing a histogram is to graphically summarize the
data.
• We can also use the histogram to understand the data. Following are some
things to look for in a histogram:
• Patterns
• Is there an overall pattern, and any striking deviations from that
pattern
• Overall pattern of a distribution
• Look for the center of the distribution
• Look for the spread of the distribution
• Does the distribution have a simple shape that you can describe in
a few words
• Outliers
• any individual observation that lies outside the overall pattern of
the graph

Spread Shape
Interpreting histograms (2/2)
6
0
5
10
15
20
0-4
>4-8
>8-12
>12-16
>16-20
>20-24
>24-28
>28-32
>32-36
>36-40
Peaked distribution
0
5
10
15
20
0-4
>4-8
>8-12
>12-16
>16-20
>20-24
>24-28
>28-32
>32-36
>36-40
Skewed to the left (-ve skew)
0
5
10
15
20
1
3
5
7
9
11
13
15
17
19
Bimodal distribution
0
5
10
15
0-4
>4-8
>8-12
>12-16
>16-20
>20-24
>24-28
>28-32
>32-36
>36-40
Skewed to the right (+ve skew)
0
5
10
15
20
0-4
>4-8
>8-12
>12-16
>16-20
>20-24
>24-28
>28-32
>32-36
Flat distribution

Measures of central tendency/
location
1
7

Measures of Central tendency
8
1
Mean
Arithmetic
mean
Harmonic
mean
Geometric
mean
Measures of location
Median Quartiles Deciles Percentiles
Mode
Measures of
central
tendency
• Indicates a number, which all the observations tend to have, or a value where
all the observations can be assumed to be located or concentrated (center of
a distribution)
Use • Gives a single value that is representative of the distribution i.e. gives us
some idea of what the ‘average’ or ‘middle’ or ‘most occurring’ number in
the data set is
• Facilitates comparison of:
• One sample at different points of time
• More than one sample at a point of time
• It is analogous to the concept of center of gravity
• It is the most common measure used to describe data sets
Types

Arithmetic mean (1/2)
9
1
Arithmetic
mean
• For ungrouped data mean is defined as the sum of all the values, divided by
the number of values.
• Commonly referred to as mean
• in a sample is called x bar, symbol 𝑥
• in a population is called mu (pronounced Mew), symbol μ
• Sample mean is often used as an estimate of the population mean
• Sum of deviation of all observations from arithmetic mean is 0
• Because the mean is calculated by summing every observation, it is greatly affected by any
extreme values, and can as such present a distorted representation of the data.
Formulae
𝑀𝑒𝑎𝑛 𝑜𝑓 𝑢𝑛𝑔𝑟𝑜𝑢𝑝𝑒𝑑 𝑑𝑎𝑡𝑎 =
𝑆𝑢𝑚 𝑜𝑓 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛𝑠
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛𝑠
=
𝑥 𝑖
𝑛
𝑖=1
𝑛
𝑀𝑒𝑎𝑛 𝑜𝑓 𝑑𝑎𝑡𝑎 𝑔𝑟𝑜𝑢𝑝𝑒𝑑 𝑖𝑛𝑡𝑜 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑡𝑎𝑏𝑙𝑒𝑠 =
𝑓𝑖 𝑥𝑖
𝑘
𝑖=1
𝑓𝑖
𝑘
𝑖=1

Arithmetic mean (2/2)
10
1
Change of
origin
• When a number, say M, is subtracted from each observation then it shifts the
origin from 0 to point M
• Mean of original data = Mean of new observations + M (origin)
Change of
scale
• When the original observations are scaled or divided by a number to reduce
the value of original observations
• Mean of original data = Mean of new observations x Factor by which
observations were divided
Change of
both origin
and scale
• Origin of variable shifted to M and observations are scaled by a factor, say N
• Mean of original data = M + (N x Mean of new observations)

Median
11
1
Usage • When the data is not unimodal and symmetrical (i.e. skewed) the median is
preferred
• It is a positional measure
• It is the middle value when the data is arranged in order - there are an equal
number of observations above and below the median
Calculation • Odd number of values: median is the value of the middle observation
• Even number of values: it is somewhere between the two middle values, and
generally calculated as the average of these two numbers:
• arrange the data in order (decreasing or increasing)
• locate the middle value using the formula:
• 𝑚𝑖𝑑𝑑𝑙𝑒 𝑣𝑎𝑙𝑢𝑒 =
𝑛+1
2
• The median is only affected by the number of observations, not the value of the
observations. Hence, extreme values do not influence the median
• Ex: 3, 3, 5, 7, 8, 12, 13 and 3, 3, 5, 7, 8, 12, 95 have the same median

Mode
12
1
Mode • The mode is the value(s) in the distribution with the maximum frequency –
the most common observation in the series
• Useful on nominal scale data, where it is not possible to calculate the mean
or median
• A distribution can have more than one mode (e.g. two modes = bimodal)
• Does not necessarily indicate the centre of a distribution – mode may even
be a class interval rather than a data value
Examples
0
5
10
15
20
1
3
5
7
9
11
13
15
17
19
Bimodal distribution
0
5
10
15
0-4
>4-8
>8-12
>12-16
>16-20
>20-24
>24-28
>28-32
>32-36
>36-40
Skewed to the right
modes = 5 and 16 mode = >8 - 12

Comparison of mean, median & mode
13
1
If data is uni-modal and
symmetrical, the three
measures of central
tendency will be of
similar value
If data is skewed, mean
and median will not be
equal. The mean will be
‘pulled towards the
skew’.
The mode acts similarly,
but not always.
Skewed to the right or
+ve skew: Mean >
Median > Mode
• Skewed to the left or
-ve skew: Mean <
Median < Mode
*Right and left refers to the side of the long tail

Measures of Dispersion
2
14

Measures of Dispersion
15
2
Range Variance Standard deviation
Coefficient of
variation
Measures of
dispersion
• Indicate the extent to which the observations differ from each other
Major types of measures of dispersion

Range
16
2
Range • It is the difference between the maximum and minimum observations in a
data set:
𝑅𝑎𝑛𝑔𝑒 = 𝑥 𝑚𝑎𝑥 − 𝑥 𝑚𝑖𝑛
• Usually the actual values are given. For example; “Chocolate prices ranged
between $1 and $1.5 per bar during 2014”
• It gives no indication of the dispersion of values between these two extreme
values, i.e., there may be a lot of values clumped at either end of the
distribution
Q1 Q3MLowest
number
Highest
number
Range
Inter Quartile Range

Variance
17
2
Variance • Variance and standard deviations are the two commonly used measures
which take into account all data values
• A data set that is more variable will have a larger variance than a data set
that is relatively homogeneous
• Variance is the sum of the squared deviations divided by the number of
observations
Calculation • Calculate deviation - distance of each observation from the mean = 𝑥𝑖 − 𝜇
• Square the deviations = (𝑥𝑖−𝜇)2
• Sum the squared deviations and divide by number of observations to get the
variance
• The variance is hence the average squared deviation of the data
Formulae • For a population, variance is notated by 𝜎2
𝜎2 =
(𝑥𝑖−𝜇)2𝑁
𝑖=1
𝑁
• For a sample, variance is notated by 𝑠2:
𝑠2 =
(𝑥𝑖−𝑥)2𝑛
𝑖=1
𝑛 − 1

Standard deviation
18
2
Standard
deviation
• Absolute measure of the deviation of the observation from its arithmetic
mean
• Also known as the Root Mean Square (RMS) value
Calculation • The standard deviation is simple the +ve square root of the variance. Hence
for a population the standard deviation is s and σ for a sample, i.e., the
standard deviation is in the same units as the mean.
Formulae Population:
𝜎 = 𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒
Sample:
𝑠 =
𝑥𝑖 − 𝑥 2𝑛
𝑖=1
𝑛 − 1

Coefficient of variance
19
2
Coefficient
of variance
(CV)
• It is a relative measure of variability which has no units
• It expresses standard deviation as a proportion of arithmetic mean
• It is used for comparing data that are not measured using the same units, or
when comparing data with significantly different means
• The CV can only be calculated on data collected at the ratio level
Formulae Population:
𝐶𝑉 =
𝜎
𝜇
Sample:
𝐶𝑉 =
𝑠
𝑥

Quartiles
20
2
Quartiles • Quartile is a positional measure like median (calculation is also similar)
• There are three quartiles that divide the distribution into four equal parts:
• The first quartile lies one quarter of the way through the data, i.e., 25%
of the observations are less than the first quartile
• The second quartile (median) is the middle value of the data set, i.e.,
the value that 50% of observations are greater than and 50% of
observations are less than the second quartile
• The third quartile lies three quarters of the way through the data, i.e.,
three quarters of the data values are less than the third quartile
Inter
quartile
range
• The difference between the 1st and 3rd quartiles is called the Inter Quartile
Range
• It signifies the central 50% of the observation
0
5
10
15
20
25
30
35
375
425
475
525
575
625
675
725
775
825
875
925
975
1025
Frequency
Salary midpoint ($)
Company weekly salaries

Percentiles
21
2
Percentiles • Positional measure like median and quartiles
• There are 99 percentiles in a distribution. They divide the data into 100 equal
parts

Five Number Summary
22
2
Five number
summary
• Quartiles, median and range can be used collectively to determine the five-
number Summary
• It offers a reasonably complete description of the centre and the spread of
the data around the centre

Boxplots
23
2
Boxplots • The five number Summary lends itself nicely to a new type of graph, the
boxplot
• With boxplots it is imperative that the plot is drawn OFF the axis, that the
axis is drawn to scale and clearly labelled with units.
• The plot itself should have a clear title attached.
• They can be drawn either vertically or horizontally
• Different computer programs use different methods for generating boxplots:
• Most programs like to identify outliers in the data - usually any
observation(s) that are more than 1.5 Inter Quartile Ranges from 1st or
3rd quartile
Use • Boxplots allow the viewer to easily assess the range, spread and centre of a
distribution
• They are useful for comparing more than one distribution (better than
histograms or stem leaf displays)
Q1 Q3Mmin max

Approximate statistics for grouped data
24
2
Statistics for
grouped
data
• When data is given in a frequency distribution table, we cannot calculate the
exact mean and standard deviation
• However, it is possible to calculate the approximate mean and variance
Formulae • Mean
𝑥 ≅
𝑓𝑖 𝑥𝑖
𝑘
𝑖=1
𝑛
• Variance
𝑠2 ≅
1
𝑛 − 1
𝑓𝑖 𝑥𝑖
2 − 𝑛𝑥2
𝑘
𝑖=1

The Empirical rule
25
2
The
empirical
rule
• The great benefit of standard deviation is that in certain circumstances it
allows us to calculate the number of observations lying within particular
intervals of the distribution
• The Empirical rule evolved from studies involving ‘mound’ shaped
distributions like the following:
• If a sample (or population) of measurements has a mound shaped
distribution:
• approximately 68 % of observations lie within one standard deviation of
the mean
• approximately 95 % of observations lie within two standard deviations
of the mean
• approximately 99.7 % of observations lie within three standard
deviations of the mean
99.7 %
68 %
95 %
  


Measures of Shape
3

Skewness
27
3
Definition
• Measure of asymmetry of
a frequency distribution
• It measures the deviation
of the distribution from a
symmetrical uniform bell
shaped curve
Types
• Following are the 3
possibilities:
• Skewed to left (negatively
skewed)
• Mean < Median < Mode
• Symmetric or unskewed
• Mean = Median = Mode
• Skewed to right (positively
skewed)
• Mean > Median > Mode
Formulae
• Bowley’s coefficient of
skewness to measure
extent of skewness
• Varies from -1 to +1
• Positively skewed >0
• Negatively skewed <0
• Pearson’s measure of
skewness0
2
4
6
1 2 3 4 5 6
(Q3 − Med ) – ( Med – Q1 )
Sk = −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
2 Q2 or ( Q3 − Q1)
0
2
4
6
1 2 3 4 5 6
0
2
4
6
1 2 3 4 5 6
=
Mean − Mode
𝜎
=
3 (Mean − Median)
𝜎

Kurtosis
28
3
Definition
• Measure of flatness or ‘peaked-ness of a
frequency distribution
• Also known as ‘Convexity of the curve’
Types
• Platykurtic (relatively flat distribution)
• Mesokurtic (not too flat, nor too peaked
distribution)
• Leptokurtic (relatively peaked
distribution)
0
2
4
6
8
1 2 3 4 5 6 7
0
2
4
6
8
1 2 3 4 5 6 7
0
2
4
6
8
1 2 3 4 5 6 7

Standardised variable
4

Standardised variable
30
4
Calculation Standardised variable,
Where:
x = variable
m = mean
σ = standard deviation
Definition • A variable whose origin is shifted to its arithmetic mean and which is then
scaled by its standard deviation
• Standardized variable has Mean = 0 and Standard deviation = 1
• It is also known as ‘Standardised Score’
z =
x − m
σ

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Send an email to
super.msahluwalia@yahoo.com
Thank You!
31

M S Ahluwalia, is a top B-School graduate (MBA, Finance), CAIIB & JAIIB (both with ‘First class with
Distinction’) and ex-Banker from India.
He’s also a visual artist, blogger, designer and photographer. To know more please visit Estudiante De La
Vida or follow on Twitter or Facebook:
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MBA Super Notes on Descriptive Statistics

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