Statistics.pdf

• Measures of Central Tendency
• Measures of Dispersion
• Measures of Shape
• Basics of Probability
• Marginal Probability
• Bayes Theorem
• Probability Distributions
• Binomial
• Poisson
• Normal

▪ Raw Data
• Frequency Distribution - Histograms
• Cumulative Frequency Distribution
• Measures of Central Tendency
• Mean, Median, Mode
• Measures of Dispersion
• Range, IQR, Standard Deviation, coefficient of variation
• Normal distribution, Chebyshev Rule.
• Five number summary, boxplots, QQ plots, Quantile plot, scatter
plot.
• Visualization: scatter plot matrix.
• Correlation analysis

Data versus Information
• When analysts are bewildered by plethora of data, which do not
make any sense on the surface of it, they are looking for methods to
classify data that would convey meaning. The idea here is to
help them draw the right conclusion. Data needs to be arranged
into information.

Raw Data
• Raw Data represent numbers and facts in the original format in
which the data have been collected. We need to convert the raw data
into information for decision making.

Frequency Distribution - Histograms
• In simple terms, frequency distribution is a summarized
table in which raw data are arranged into classes and frequencies.
• Frequency distribution focuses on classifying raw data into
information. It is a widely used data reduction technique in
descriptive statistics.

Histogram (also known as frequency
histogram) is a snap shot of the frequency
distribution.
Histogram is a graphical representation of the
frequency distribution in which the X-axis
represents the classes and the Y-axis
represents the frequencies in bars
Histogram depicts the pattern of the
distribution emerging from the characteristic
being measured.

Histogram- Example
The inspection records of a hose assembly operation revealed a high
level of rejection. An analysis of the records showed that the "leaks" were
a major contributing factor to the problem. It was decided to
investigate the hose clamping operation. The hose clamping force
(torque) was measured on twenty five assemblies. (Figures in foot-
pounds). The data are given below: Draw the frequency histogram and
comment.
8 13 15 10 16
11 14 11 14 20
15 16 12 15 13
12 13 16 17 17
14 14 14 18 15

Histogram Example Solution
2
7
12
3
1
0
5
10
15
8-11 11-14 14-17
Classes
17-20 20-23
Fre
quency

Cumulative Frequency Distribution
A type of frequency distribution that shows how many observations are
above or below the lower boundaries of the classes. You can formulate
the following from the previous example of hose clamping force(torque)
Class Frequency Relative
Frequency
Cumulative
Frequency
Cumulative
Relative
Frequency
8-11
11-14
14-17
17-20
20-23
2
7
12
3
1
0.08
0.28
0.48
0.12
0.04
2
9
21
24
25
0.08
0.36
0.84
0.96
1.00
Total 25 1.00

What is Central Tendency?
• Whenever you measure things of the same kind, a fairly large number of such measurements
will tend to cluster around the middle value. Such a value is called a measure of "Central
Tendency". The other terms that are used synonymously are "Measures of Location", or
"Statistical Averages".

Arithmetic Mean
• Arithmetic Mean (called mean) is defined as the sum of all observations in a data set divided by
the total number of observations. For example, consider a data set containing the
following observations:
• In symbolic form mean is given by
X = 𝛴𝑥
𝑛
𝑛 = Total number of observations(Sample Size)
σ 𝑥 = Indicates sum all X values in the data set
X = Arithmetic Mean

Arithmetic Mean -Example
The inner diameter of a particular grade of tire based on
5 sample measurements are as follows: (figures in
millimeters)
565, 570, 572, 568, 585
Applying the formula
We get mean = (565+570+572+568+585)/5 =572
Caution: Arithmetic Mean is affected by extreme values or
fluctuations in sampling. It is not the best average to use
when the data set contains extreme values (Very high or
very low values).
X = 𝛴𝑥
𝑛

Median
• Median is the middle most observation when you arrange data in ascending order of magnitude.
Median is such that 50% of the observations are above the median and 50% of the observations are below
the median.
• Median is a very useful measure for ranked data in the context of consumer preferences and rating. It is
not affected by extreme values (greater resistance to outliers)
th value of ranked data
n = Number of observations in the sample
𝑛 + 1
2
Median =

Median - Example
Marks obtained by 7 students in Computer Science
Exam are given below: Compute the median.
45 40 60 80 90 65 55
Arranging the data after ranking gives
90 80 65 60 55 45 40
Median = (n+1)/2 th value in this set = (7+1)/2 th
observation= 4th observation=60
Hence Median = 60 for this problem.

Mode
Mode is that value which occurs most often. It has the maximum frequency
of occurrence. Mode also has resistance to outliers.
Mode is a very useful measure when you want to keep in the inventory, the most
popular shirt in terms of collar size during festive season.

Mode -Example
The life in number of hours of 10 flashlight batteries are as
follows: Find the mode.
340 occurs five times. Hence, mode=340.
340 350 340 340 320 340 330 330
340 350

Comparison of
Mean, Median, Mode Cont.
Mean Median Mode
Affected by extreme
values.
Can be treated
algebraically. That is,
Means of several groups
can be combined.
Not affected by
extreme values.
Cannot be treated
Medians of several
groups cannot be
combined.
Not affected by
extreme values.
Cannot be treated
Modes of several
groups cannot be
combined.

Measures of Dispersion
• In simple terms, measures of dispersion indicate how large the
spread of the distribution is around the central tendency. It answers
unambiguously the question " What is the magnitude of
departure from the average value for different groups having
identical averages?".

Range
• Range is the simplest of all measures of dispersion. It is
calculated as the difference between maximum and
minimum value in the data set.
Range = XMaximum
− XMinimum

Inter-Quartile Range(IQR)
IQR= Range computed on middle 50% of the observations after
eliminating the highest and lowest 25% of
observations in a data set that is arranged in ascending
order. IQR is less affected by outliers.
IQR =Q3-Q1

Interquartile Range-Example
The following data represent the annual percentage
returns of 9 mutual funds.
Data Set: 12, 14, 11, 18, 10.5, 12, 14, 11, 9
Arranging in ascending order, the data set becomes
9, 10.5, 11, 11, 12, 12, 14, 14, 18
IQR=Q3-Q1=14-10.75=3.25

Standard Deviation
To define standard deviation, you need to define another term
called variance. In simple terms, standard deviation is the square
root of variance.

Example for Standard Deviation
The following data represent the percentage return on investment
for 10 mutual funds per annum. Calculate the sample standard
deviation.
12, 14, 11, 18, 10.5, 11.3, 12, 14, 11, 9

Coefficient of Variation
(Relative Dispersion)
CoefficientvVariation (CV) is defined as the ratio of
Standard Deviation to Mean.
In symbolic form
CV = S
for the sample data and = for the population
μ
σ
X

Coefficient of Variation
Example
Consider two SalesPersons working in the same territory
The sales performance of these two in the context of
selling PCs are given below. Comment on the results.
Sales Person 1
Mean Sales (One year
average)
50 units
Sales Person 2
Mean Sales (One year
average)
75 units
Standard Deviation
5 units
Standard deviation
25 units

Interpretation for the Example
The CV is 5/50 =0.10 or 10% for the Sales Person1
and 25/75=0.33 or 33% for sales Person2.
The moral of the story is "don't get carried away by by
averages. Consider variation (“risk”).

• The empirical rule approximates the variation of data in a bell-
shaped distribution
• Approximately 68% of the data in a bell shaped distribution
is within 1 standard deviation of the mean or
The Empirical Rule
μ ± 1σ
68%
μ
μ ± 1σ

• Approximately 95% of the data in a bell-shaped distribution
lies within two standard deviations of the mean, or µ ± 2σ
• Approximately 99.7% of the data in a bell-shaped distribution
lies within three standard deviations of the mean, or µ ± 3σ
The Empirical Rule
μ ± 3σ
99.7%
95%
μ ± 2σ

• Regardless of how the data are distributed, at least (1 -
1/k2) x 100% of the values will fall within k standard
deviations of the mean (for k > 1)
• For Example, when k=2, at least 75% of the values of any data
set will be within μ ± 2σ
Chebyshev Rule

The Five Number Summary
The five numbers that help describe the center, spread and shape
of data are:
▪ Xsmallest
▪ First Quartile (Q1)
▪ Median (Q2)
▪ Third Quartile (Q3)
▪ Xlargest

Distribution Shape
Right-Skewed
Left-Skewed Symmetric
Q1 Q2 Q3 Q1 Q2 Q3
Q1 Q2 Q3

Relationships among the five-number
summary and distribution shape
Left-Skewed Symmetric Right-Skewed
Median – Xsmallest
>
Xlargest – Median
≈
Xlargest – Median
<
Xlargest – Median
Q1
– Xsmallest
>
Xlargest
– Q3
Q1
– Xsmallest
≈
Xlargest
– Q3
Q1
– Xsmallest
<
Xlargest
– Q3
Median – Q1
>
Q3 – Median
Median – Q1
≈
Q3 – Median
Median – Q1
<
Q3 – Median

Five Number Summary and The
Boxplot
• The Boxplot: A Graphical display of the data based on the five-
number summary:
Example:
40
Xsmallest Q1 Median Q3
Xlargest
25% of data 25%
of data
25%
of data
25% of data

Five Number Summary:
Shape of Boxplots
• If data are symmetric around the median then the box
and central line are centered between the endpoints
• A Boxplot can be shown in either a vertical or
horizontal orientation
Xsmallest Q1 Q3
Median Xlargest

Distribution Shape and
The Boxplot
Right-Skewed
Left-Skewed Symmetric
Q1 Q2 Q3 Q1 Q2 Q3
Q1 Q2 Q3

Boxplot Example
5
The data are right skewed in the following plot
2277
00 22 33 55

10 15 20 25 30
0 5
Box plot example showing an outlier
• The boxplot below of the same data shows the
• outlier value of 27 plotted separately
• A value is considered an outlier if it is more than 1.5 times the interquartile
range below Q1 or above Q3

Graphic Displays of Basic Statistical Descriptions
•Boxplot: graphic display of five-number summary
•Histogram: x-axis are values, y-axis repres. frequencies
•Quantile plot: each value xi is paired with fi indicating that approximately 100 fi % of
data are ≤ xi
•Quantile-quantile (q-q) plot: graphs the quantiles of one univariant distribution
against the corresponding quantiles of another
•Scatter plot: each pair of values is a pair of coordinates and plotted as points in the
plane

Histograms Often Tell More than Boxplots
■ The two histograms
shown in the left may
have the same boxplot
representation
■ The same values for:
min, Q1, median, Q3,
max
■ But they have rather
different data
distributions

Quantile Plot
•Displays all of the data (allowing
the user to assess both the overall
behavior and unusual occurrences)
•Plots quantile information
•For a data xi data sorted in
increasing order, fi indicates that
approximately 100 fi% of the
data are below or equal to the
value xi

Quantile-Quantile (Q-Q) Plot
•Graphs the quantiles of one univariate distribution against the corresponding quantiles of
another
•View: Is there is a shift in going from one distribution to another?
•Example shows unit price of items sold at Branch 1 vs. Branch 2 for each quantile. Unit
prices of items sold at Branch 1 tend to be lower than those at Branch 2.

Scatter plot
Provides a first look at bivariate data to see clusters of points,
outliers, etc
Each pair of values is treated as a pair of coordinates and plotted as
points in the plane

Positively and Negatively Correlated Data
The left half fragment is positively
correlated
The right half is negative correlated

Visually Evaluating Correlation
Scatter plots
showing the
similarity from
–1 to 1.

Summary
• Histograms
• Measures of central tendency: mean, mode, median
• Measures of dispersion: range, IQR, variance, std deviation, coefficient of
variation.
• Normal distribution, Chebyshev Rule.
• Five number summary, boxplots, QQ plots, Quantile plot, scatter plot.
• Visualization: scatter plot matrix, parallel coordinates.
• Correlation analysis.

Bayes’ Theorem
• Bayes’ Theorem is used to revise previously calculated probabilities
based on new information.
• Developed by Thomas Bayes in the 18th Century.
• It is an extension of conditional probability.
Bi = ith event of k mutually exclusive and collectively exhaustive events A =
new event that might impact P(Bi )

In precise terms, a probability distribution is a total listing of the various
values the random variable can take along with the corresponding probability
of each value. A real life example could be the pattern of distribution of the
machine breakdowns in a manufacturing unit.
• The random variable in this example would be the various values the machine
breakdowns could assume.
• The probability corresponding to each value of the breakdown is the relative
frequency of occurrence of the breakdown.
• The probability distribution for this example is constructed by the actual
breakdown pattern observed over a period of time. Statisticians use the term
“observed distribution” of breakdowns.

P(x) is the probability of getting x successes in n trials

• Poisson Distribution is another discrete distribution which also plays a major role
in quality control in the context of reducing the number of defects per standard
unit.
• Examples include number of defects per item, number of defects per
transformer produced, number of defects per 100 m2 of cloth, etc.
• Other real life examples would include 1) The number of cars arriving at a
highway check post per hour; 2) The number of customers visiting a bank per
hour during peak business period

𝑃 𝑥 =
ⅇ−𝜆𝜆𝑥
𝑥!
• P(x) = Probability of x events in an interval
given an idea of λ
• λ = Average number of events per unit
• e = 2.71828(based on natural logarithm) x =
events per unit which can take values 0, 1,
2, 3,…………..∞
• λ is the Parameter of the Poisson
Distribution.

If on an average, 6 customers arrive every two minutes at a
bank during the busy hours of working,
a) what is the probability that exactly four customers arrive
in a given minute?
b) What is the probability that more than three customers
will arrive in a given minute?
Sol: 6 customers arrive every two minutes.
Therefore , 3 customers arrive every minute.
That implies my lambda=3
P(X=4)=?
P(X>3)=?
Implies 1-P(X< =3)? In the problem mean value is given as an input for a time
interval. This is one of the indication that Poisson distribution
has to be applied

The Normal Distribution is the most widely used continuous
distribution
The inferential statistics is based on the normal distribution.
When the sample size is reasonably large, almost every dataset
achieves normal distribution
• The normal distribution is a continuous distribution looking
like a bell.
• Statisticians use the expression “Bell Shaped Distribution”.
• Mean, the median, and the mode are all equal to one
another.
• It is symmetrical about its mean.
• If the tails of the normal distribution are extended, they will
run parallel to the horizontal axis without actually touching
it.
• • The normal distribution has two parameters namely the
mean µ and the standard deviation σ

• Mutually exclusive Vs Independent Events.
• Conditional Probability.
• Bayes Theorem.
• Applying Probability Concepts.
• Applying Distribution Concepts.

Irrespective of the shape of the
distribution of the original population,
the sampling distribution of the mean
will approach a normal distribution as
the size of the sample increases and
becomes large

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