SBAOC
Statistical Methods &
their Applications I
Unit : I - V
SBAOC – Statistical Methods & their Applications -I
Unit I - Syllabus
Statistics definition
Characteristics and functions of Statistics
Classification & tabulation
Diagrams & graphs
Lorenz Curve
2
Statistics Definition
“By statistics we mean aggregates of facts affected to a marked
extent by multiplicity of causes numerically expressed,
enumerated or estimated according to reasonable standards of
accuracy, collected in a systematic manner for a pre-
determined purpose and placed in relation to each other.”
- PROF.HORACE SECRIST
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CHARACTERISTICS/FEATURES
Statistics are aggregate of facts
Statistics are affected to a marked extent of multiplicity of causes.
Statistics must be Numerically Expressed
Statistics must be enumerated or estimated according to
reasonable standard of accuracy.
Statistics should be collected in a systematic manner.
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FUNCTIONS OF
STATISTICS1. Statistics presents the fact in definite form
2. It simplifies mass of data.
3. It facilitates for comparison
4. It enlarges individual experience
5. It facilitates in formulating policies
Statistics laws are true on average. Statistics are aggregates
of facts. So single observation is not a statistics, it deals with
groups and aggregates only.
Statistical methods are best applicable on quantitative data.
Limitations of Statistics
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A Taxonomy of Statistics
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Classification
The process of grouping a large number of individual facts or
observations on the basis of similarity among the items, is called
classification.
OBJECTIVES OF CLASSIFICATION
To condense the mass of data, and to present the facts in a
simple form.
To facilitate comparison between data.
To eliminate unnecessary details so that main features of the
collected data are easily understandable.
To facilitate further statistical treatment of the data
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TYPES OF CLASSIFICATION
CLASSIFICATION TYPES
GEOGRAPHICAL
CHRONOLOGICAL
QUANTITATIVE
QUALITATIVE
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TABULATION:
The mass of data collected has to be arranged in some kind of concise
and logical order.
Tabulation summarizes the raw data and displays data in form of some
statistical tables.
Tabulation is an orderly arrangement of data in rows and columns.
OBJECTIVE OF TABULATION:
1. Conserves space & minimizes explanation and descriptive statements.
2. Facilitates process of comparison and summarization.
3. Facilitates detection of errors and omissions.
4. Establish the basis of various statistical computations.
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TYPES OF TABLE
ON THE BASIS OF
COVERAGE
ON THE BASIS OF
OBJECTIVE
ON THE BASIS OF
ORGINALITY
SIMPLE AND COMPLEX TWO – WAY TABLE
MANIFOLD OR
HIGHER ORDER TABLETHREE WAY TABLE
GENERAL PURPOSE SPECIAL PURPOSE
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A diagram is a visual form for presentation of statistical data,
highlighting their basic facts and relationship. If we draw diagrams
on the basis of the data collected they will easily be understood and
appreciated by all. It is readily intelligible and save a
considerable amount of time and energy.
A graph is a visual form of presentation of statistical data. A
graph is more attractive than a table of figure. Even a common
man can understand the message of data from the graph.
Comparisons can be made between two or more phenomena very
easily with the help of a graph.
Diagrams
Graphs
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TYPES OF DIAGRAMS
ONE DIMENSIONAL OR
BAR DIAGRAMS
TWO DIMENSIONAL OR
AREA DIAGRAMS
THREE DIMENSIONAL OR
VOLUME DIAGRAMS
PICTOGRAMS AND
CARTOGRAMS
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DEVIATION
BAR
PERCENTAGE
BAR
SUBDIVIDED
BAR
MULTIPLE BAR
SIMPLE BAR
TYPES OF
ONE
DIMENSIONAL
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TYPES OF GRAPHS
GRAPHS FOR FREQUENCY DISTRIBUTION ARE AS FOLLOWS:
1. Histogram
2. Frequency polygon
3. Frequency curve
4. Ogives or cumulative frequency curve
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LORENZ CURVE
LORENZ CURVE is a graphical method of studying dispersion.
It is used to study the variability in the distribution of profits, wages,
revenue etc.,
To study the degree of inequality in the distribution of income and
wealth between countries or different periods.
SBAOC – Statistical Methods & their Applications -I 15
SBAOC – Statistical Methods & their Applications -I
Unit II - Syllabus
Sampling Definition
Simple random sampling
Systematic random sampling
Stratified random sampling
Sampling & non-sampling errors
Methods of reducing sampling errors
16
Definition of Sampling
Sampling is the process of selecting a small number of
elements from a larger defined target group of elements such
that the information gathered from the small group will allow
judgments to be made about the larger groups.
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• Sampling have various benefits to us. Some of the
advantages are listed below:
• Sampling saves time to a great extent by reducing the volume
of data. You do not go through each of the individual items.
• Sampling Avoids monotony in works. You do not have to
repeat the query again and again to all the individual data.
• When you have limited time, survey without using
sampling becomes impossible. It allows us to get near-
accurate results in much lesser time
SBAOC – Statistical Methods & their Applications -I
Advantages of Sampling
18
• Every coin has two sides. Sampling also have some demerits.
Some of the disadvantages are:
• Since choice of sampling method is a judgmental task, there
exist chances of biasness as per the mindset of the person
who chooses it.
• Improper selection of sampling techniques may cause the
whole process to defunct.
• Selection of proper size of samples is a difficult job.
SBAOC – Statistical Methods & their Applications -I
Disadvantages of Sampling
19
Sample size
• The number of cases used for the research analysis
• Sample size may be determined by using:
– Subjective methods (less sophisticated methods)
• The rule of thumb approach: eg. 5% of population
• Conventional approach: eg. Average of sample sizes of
similar other studies;
• Cost basis approach: The number that can be studied with
the available funds;
– Statistical formulae (more sophisticated methods)
• Confidence interval approach.
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Simple random sampling
A sample of size n draw from a finite population of size N is said
to be a random sample if it is chosen in such a way that each of the
possible sample has the same probability of being selected.
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Systematic random sampling
Systematic random sampling is a method of probability sampling in
which the defined target population is ordered and the sample is
selected according to position using a skip interval
K = N
n
K = Sampling Interval
N = Universe Size
n= Sample Size
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Stratified random sampling
• This method is used when the population distribution of items is
skewed.
• It allows us to draw a more representative sample. Hence if there
are more of certain type of item in the population the sample has
more of this type and if there are fewer of another type, there are
fewer in the sample.
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Sampling & Non-Sampling Errors
• In statistics, the word „Error‟ is used to denote the difference
between the true value and the estimated value.
• The term error should be distinguished from mistake or inaccuracies
which may be committed in the course of making observations,
counting calculations etc.,
• The error, mainly arise at the stage of ascertainment and
processing of data in complete enumeration and sample
surveys.
• Errors are classified into sampling errors and non-sampling
errors.
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Methods of Reducing Sampling Errors
Once the absence of bias has been ensured, attention should be
given to the random sampling errors. Such errors may be reduced to
the minimum to attain the desired accuracy.
Apart from reducing errors of bias, the simplest way of increasing
the accuracy of a sample is to increase its size.
Slide number / Total slidesSBAOC – Statistical Methods & their Applications -I 29
SBAOC – Statistical Methods & their Applications -I
Unit III - Syllabus
Central tendency
Dispersion
Coefficient of variation
Skewness
Kurtosis
30
Measures of Central Tendency
The value or the figure which represents the whole series is neither the
lowest value in the series nor the highest it lies somewhere between
these two extremes.
1. The average represents all the measurements made on a group,
and gives a concise description of the group as a whole.
2. When two are more groups are measured, the central tendency
provides the basis of comparison between them.
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KINDS OF AVERAGES
CENTRAL
TENDENCY
(AVERAGES)
AVERAGES OF
LOCATION
(POSITION)
COMMERCIAL
AVERAGE
(BUSINESS)
MATHEMATICAL
AVERAGES
3. COMPOSITE
AVG
2. PROGRESSIVE
AVG
1.MOVING AVG
2. MODE
1. MEDIAN
4. QUADRATIC
MEAN
3. HARMONIC
MEAN
2. GEOMETRIC
MEAN
1.ARITHMETIC
MEAN
2. WEIGHTED
1. SIMPLE
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Mean
The mean (arithmetic mean or average) of a set of data is found
by adding up all the items and then dividing by the sum of the
number of items.
x
The mean of a sample is denoted by (read “x bar”).
The mean of a complete population is denoted by (the lower
case Greek letter mu).
The mean of n data items x1, x2,…, xn, is given by the formula
or
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Example:
Ten students were polled as to the number of siblings in their
individual families.
The raw data is the following set: {3, 2, 2, 1, 3, 6, 3, 3, 4, 2}.
Find the mean number of siblings for the ten students.
siblings
SBAOC – Statistical Methods & their Applications -I 34
Median
Another measure of central tendency, is the median.
The median is not as sensitive to extreme values as the mean.
This measure divides a group of numbers into two parts, with half
the numbers below the median and half above it.
To find the median of a group of items:
Rank the items.
If the number of items is odd, the median is the middle item
in the list.
If the number of items is even, the median is the mean of the
two middle numbers.
SBAOC – Statistical Methods & their Applications -I 35
Example:
Ten students in a math class were polled as to the number of
siblings in their individual families and the results were:
3, 2, 2, 1, 1, 6, 3, 3, 4, 2.
Find the median number of siblings for the ten students.
Data in order: 1, 1, 2, 2, 2, 3, 3, 3, 4, 6
Median
Median
= (2+3)/2 = 2.5
Position of the median: 10/2 = 5
Between the 5th and 6th values
siblings
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The mode of a data set is the value that occurs the most often.
If a distribution has two modes, then it is called bimodal.
Mode
In a large distribution, this term is commonly applied even
when the two modes do not have exactly the same frequency
Ten students in a math class were polled as to the number of siblings
in their individual families and the results were: 3, 2, 2, 1, 3, 6, 3, 3, 4,
2. Find the mode for the number of siblings.
Example:
3, 2, 2, 1, 3, 6, 3, 3, 4, 2
The mode for the number of siblings is 3.
SBAOC – Statistical Methods & their Applications -I 37
Geometric mean:
Geometric mean is the nth root when you multiply n numbers
together.
Harmonic mean:
• Harmonic mean as the reciprocal of the arithmetic mean of
the reciprocals.
• So far we have looked at ways of summarising data by
showing some sort of average (central tendency).
• But it is often useful to show how much these figures differ
from the average. This measure is called dispersion.
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METHODS OF MEASURING DISPERSION
• Range
• Quartile Deviation
• Mean Deviation
• Standard Deviation
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Range:
The range is the difference between the maximum and minimum
values.
Quartile deviation (or) inter-quartile range :
The inter-quartile range is the range of the middle half of the values.
Mean deviation:
The mean of the absolute values of the numerical differences between
the numbers of a set (such as statistical data) and their mean or
median.
Standard deviation:
A statistic used as a measure of the dispersion or variation in a
distribution, equal to the square root of the arithmetic mean of the
squares of the deviations from the arithmetic mean.
SBAOC – Statistical Methods & their Applications -I 41
Video Link: Quartile Deviation
https://www.youtube.com/watch?v=N_YnzL8mEqw
Video Link: Quartile Deviation
https://www.youtube.com/watch?v=K3wsOqIqA6k
Video Link: Mean Deviation
https://www.youtube.com/watch?v=z9AJk7TvdpQ
SBAOC – Statistical Methods & their Applications -I 42
Coefficient of Variation (C.V.):
The variance and the standard deviation are useful as
measures of variation of the values of a single variable for a
single population (or sample).
If we want to compare the variation of two variables we
cannot use the variance or the standard deviation because:
1. The variables might have different units.
2. The variables might have different means.
SBAOC – Statistical Methods & their Applications -I 43
We need a measure of the relative variation that will not
depend on either the units or on how large the values are. This
measure is the coefficient of variation (C.V.) which is defined by:
%100*.x
VC
(free of unit or unit less)
Mean S.D C.V.
%100.1
11
xVC
%100.2
22
xVC
11x
2x 2
1st data
set
2nd data
set
The relative variability in the 1st data set is larger than the relative
variability in the 2nd data set if C.V1> C.V2 (and vice versa).
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Skewness
• Skewness is the measure of the shape of a nonsymmetrical
distribution
• Two sets of data can have the same mean & SD but different
skewness
Two types of skewness:
– Positive skewness
– Negative skewness
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Kurtosis
A measure of whether the curve of a distribution is:
Bell-shaped -- Mesokurtic
Peaked -- Leptokurtic
Flat -- Platykurtic
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SBAOC – Statistical Methods & their Applications -I
Unit IV - Syllabus
Probability
Additive Rule
Independent events
Conditional probability
Multiplicative Rule for Intersections
Bayes' Rule
47
TM
Slide number / Total slides
Probability:
Definition: The numerical chance that a specific outcome will
occur (i.e.)., it is a mathematical measure of “Measuring the
certainty or uncertainty of an event”.
Sometimes you can measure a probability with a number: "10%
chance of rain", or you can use words such as impossible,
unlikely, possible, even chance, likely and certain.
Example: "It is unlikely to rain tomorrow".
SBAOC – Statistical Methods & their Applications -I 48
TM
Approaches to Probability
1. Relative frequency event probability = x/n, where x= no.of
occurrences of event of interest, n = total no. of observations
2. Subjective probability
Individual assigns prob. based on personal experience, anecdotal
evidence, etc
3.Classical approach: Every possible outcome has equal probability
(more later)
The probability for the occurrence of an event A is defined as the
radio between the number of favourable outcomes for the occurrence
of the event and the total number of possible outcomes,
Probability of an event =
Number of favourable outcomes / Total number of outcomes
P(E) = m/n
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TM
Basic Definitions
Experiment: act or process that leads to a single outcome that cannot
be predicted with certainty
Sample space: all possible outcomes of an experiment. Denoted by S
Event: any subset of the sample space S;typically denoted A, B, C, etc.
Simple event: event with only 1 outcome
Certain event: S
Laws of Probability
1. 0 P(A) 1 for any event A
2. P() = 0, P(S) = 1 and P(A‟) = 1 – P(A)
4. If A and B are disjoint events, then P(A B) = P(A) + P(B)
5. If A and B are independent events, then P(A B) = P(A) × P(B)
6. For any two events A and B, P(A B) = P(A) + P(B) – P(A B)
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TM
Examples
1. Toss a fair coin. P(Head) = ½
2. Toss a fair coin twice. What is the probability of observing at least
one head
P(at least 1 head) = P(E1) + P(E2) + P(E3) = 1/4 + 1/4 + 1/4 = ¾
3. Select a student from the classroom and record his/her hair color
and gender.
A: student has brown hair
B: student is female
C: student is male Mutually exclusive; B = CC
What is the relationship between events B and C?
•AC: Student does not have brown hair
•BC: Student is both male and female =
•BC:Student is either male and female = all students = S
SBAOC – Statistical Methods & their Applications -I 51
TM
The Additive Rule for Unions:
For any two events, A and B, the probability of their union, P(A B), is
When two events A and B are mutually exclusive, P(AB) = 0
and P(AB) = P(A) + P(B).
)()()()( BAPBPAPBAP AUB BA
SBAOC – Statistical Methods & their Applications -I 52
TM
Example: Additive Rule
Example: Suppose that there were 120 students in the classroom,
and that they could be classified as follows:
A: Statistics Student
P(A) = 50/120
B: Female
P(B) = 60/120
Allied Statistics Non
Statistics
Male 20 40
Female 30 30
P(selecting a statistics female student) = P(AB) = P(A) + P(B) – P(AB)= 50/120 + 60/120 - 30/120 = 80/120 = 2/3
SBAOC – Statistical Methods & their Applications -I 53
TM
Independent Event & Conditional Probabilities
1.Two events, A and B, are said to be independent if the occurrence or
nonoccurrence of one of the events does not change the probability of the
occurrence of the other event. In terms of conditional, Two events A and B
are independent if and only if P(A|B) = P(A) or P(B|A) = P(B)
Otherwise, they are dependent.
2. The probability that A occurs, given that event B has occurred is called
the conditional probability of A given B and is defined as
0)( if )(
)()|(
BP
BP
BAPBAP
“given”
SBAOC – Statistical Methods & their Applications -I 54
TM
Example 1
1. Toss a fair coin twice. Define
A: head on second toss
B: head on first toss
P(A|B) = ½
P(A|not B) = ½
P(A) does not change, whether B
happens or not…
A and B are
independent!
2.Toss a pair of fair dice. Define
A: red die show 1
B: green die show 1
P(A|B) = P(A and B)/P(B)
=1/36 / 1/6=1/6=P(A)
SBAOC – Statistical Methods & their Applications -I 55
TM
The Multiplicative Rule for
IntersectionsFor any two events, A and B, the probability that both A and B occur is
= P(A) P(B given that A occurred)
If the events A and B are independent, then the probability that both A and
B occur P(A B)=P(A)P(B/A) if A & B are independent,
Example:
In a certain population, 10% of the people can be classified as being high
risk for a heart attack. Three people are randomly selected from this
population. What is the probability that exactly one of the three are high
risk? Define H: high risk N: not high risk
P(exactly one high risk) = P(HNN) + P(NHN) + P(NNH)
= P(H)P(N)P(N) + P(N)P(H)P(N) + P(N)P(N)P(H)
= (.1)(.9)(.9) + (.9)(.1)(.9) + (.9)(.9)(.1)= 3(.1)(.9)2 = .243
( ) ( ). ( )P A B P A P B
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TM
The Law of Total Probability
Let S1 , S2 , S3 ,..., Sk be mutually exclusive and exhaustive events
(that is, one and only one must happen). Then the probability of
any event A can be written as
P(A) = P(A S1) + P(A S2) + … + P(A Sk)
= P(S1)P(A|S1) + P(S2)P(A|S2) + … + P(Sk)P(A|Sk)
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TM
Bayes’ Rule
,...k, i SAPSP
SAPSPASP
ii
iii 21for
)|()(
)|()()|(
)|()(
)|()(
)(
)()|(
)|()()()(
)()|(
Proof
ii
iiii
iii
i
ii
SAPSP
SAPSP
AP
ASPASP
SAPSPASPSP
ASPSAP
SBAOC – Statistical Methods & their Applications -I 58
TM
ExampleSuppose a rare disease infects one out of every 1000 people in a
population. And suppose that there is a good, but not perfect, test for this
disease: if a person has the disease, the test comes back positive 99% of
the time. On the other hand, the test also produces some false positives:
2% of uninfected people are also test positive. And someone just tested
positive. What are his chances of having this disease?
Solution:
Define A: has the disease B: test positive
We know:
P(A) = .001 P(Ac) =.999
P(B|A) = .99 P(B|Ac) =.02
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TM
Example:
( ) ( | )( | )
( ) ( | ) ( ) ( | )
.001 .99.0472
.001 .99 .999 .02
P A P B AP A B
c cP A P B A P A P B A
SBAOC – Statistical Methods & their Applications -I 60
SBAOC – Statistical Methods & their Applications -I
Unit V - Syllabus
Correlation
Methods of studying Correlation
Scatter diagram
Regression
Least squares method
Partial Correlation & Multiple Correlation
61
TM
Correlation coefficient is a statistical measure of the degree to
which changes to the value of one variable predict change to
the value of another.
When the fluctuation of one variable reliably predicts a similar
fluctuation in another variable, there‟s often a tendency to think
that means that the change in one causes the change in the
other.
CORRELATION
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TM
Coefficient of Correlation lies between -1 and +1.
Coefficients of Correlation are independent of Change of
Origin.
Coefficients of Correlation possess the property of symmetry.
Coefficient of Correlation is independent of Change of Scale.
Co-efficient of correlation measures only linear correlation
between X and Y.
Properties of Correlation Coefficient
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TM
Types of correlation
Correlation are of three types:
Positive Correlation
Negative Correlation
No correlation
In correlation, when values of one variable increase with the increase
in another variable, it is supposed to be a positive correlation.
On the other hand, if the values of one variable decrease with the
decrease in another variable, then it would be a negative correlation.
There might be the case when there is no change in a variable with
any change in another variable. In this case, it is defined as no
correlation between the two.
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TM
Scatter diagram.
Karl pearson's coefficient of correlation.
Spearman's Rank correlation coefficient.
Methods of studying Correlation
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TM
The scatter diagram graphs pairs of numerical data, with one
variable on each axis, to look for a relationship between them. If the
variables are correlated, the points will fall along a line or curve. The
better the correlation, the tighter the points will hug the line.
The following video explains the scatter diagram with an example
https://www.youtube.com/watch?v=NcgRa0uotXs
Scatter Diagram
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TM
Karl pearson's coefficient of correlation
The formula to calculate Karl pearson‟s coefficient correlation
is as follows:
The following video explains the correlation problem.
2222 )()()()(
))((
yynxxn
yxxynr
https://www.youtube.com/watch?v=pOKWxNTterg
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TM
A rank correlation coefficient measures the degree of similarity
between two rankings, and can be used to assess the significance of
the relation between them.
The following video explains the rank correlation problem
Spearman's Rank correlation coefficient.
https://www.youtube.com/watch?v=KK-wuL7WN20
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TM
In statistical modeling, regression analysis is a set of statistical
processes for estimating the relationships among variables.
It includes many techniques for modeling and analysing several
variables, when the focus is on the relationship between a
dependent variable and one or more independent variables (or
'predictors').
Regression
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TM
Anyone who wants to make predictions or inferences based on a
particular data set. Companies,
Example, Companies, use regression to determine their profit,
revenue, and cost functions and scientists use regression to
analyze and predict all sorts of time series phenomenon including
the population of certain bacteria, crop data, planetary orbits, and
projectile tracking. These days, regression is the most used method
for data analysis.
Uses of Regression
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TM
If Y depends on X then the regression line is Y on X. Y is
dependent variable and X is independent variable.
The regression equation Y on X is Y = a + bx, is used to estimate
value of Y when X is known.
If X depends on Y, then regression line is X on Y and X is
dependent variable and Y is independent variable.
The regression equation X on Y is X = c + dy is used to estimate
value of X when Y is given
Here a, b, c and d are constant.
Regression Lines
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TM
Formula to calculate Regression lines
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TM
Least-squares Method
If a straight line is fitted to the data it will serve as a satisfactory
trend, perhaps the most accurate method of fitting is that of least
squares.
The formula for a straight-line trend can most simply be expressed
as
Yc = a + bX
where X represents time variable, Yc is the dependent variable for
which trend values are to be calculated and a and b are the
constants of the straight tine to be found by the method of least
squares.
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TM
https://www.youtube.com/watch?v=T7LauV6AnRg
Regression
The following link explains the regression problem in detail .
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Correlation between two statistical variables under the condition
that all other relevant variables are fixed.
Partial Correlation:
Correlation involving two or more independent mathematical
variables.
Multiple Correlation:
SBAOC – Statistical Methods & their Applications -ISBAOC – Statistical Methods & their Applications -I
Partial correlation & Multiple Correlation
75
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