1. Applied Statistics in Business & Economics
Applied Statistics in Business & Economics
Discrete Data
List of observation eg 14,20, 23, 25, 28
Grouped Data
List of observations & frequency e.g. Weekly Wages & No Of Labours
Continuous Series
In series from 1 value to another eg.
Marks 0-10 10-20 20-30 30-40 40-50
# Of Student 6 11 14 20 15
Measures of Central Tendencies
Arithmetic Mean
Discrete Data
Mean =
X
N
Grouped Data
Mean =
fX
f
Continuous Series
Mean =
fX
* X here is the Mid Value of the data in series eg. If Marks is 0-10, 10-20 then the
f
Mid Values will be 5, 15 respectively.
Median
Median is the middle most observation if the data is sorted.
If the number of observations (N) is odd then the centre most value is the
observation.
Eg. In Observations: 14, 20, 23, 25, 28
Median is 23
If the number of observations (N) is even then the N/2-1th and N/2+1th
observations are summed and divided by 2.
Eg. In Observations: 14, 20, 23, 25, 28, 32
Median is 24 i.e. (23+25 / 2)
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2. Applied Statistics in Business & Economics
Mode
Mode is the observation with highest frequency/occurrence.
E.g. In Observations: 10, 20, 30, 30, 40
Mode is 30 because it occurs twice which is highest in the set.
If there are more than one observation values which have the highest occurrence
then there is NO Mode to the data. Eg. 10, 20, 20, 30, 30, 40, Here 20 and 30
both have Frequency of 2 which is the highest, in this case there is No Mode to
the data.
Midrange
Midrange is the average of just the minimum value and maximum value.
Xmin + Xmax
Midrange =
2
E.g. In Observations: 10, 20, 30, 30, 40
Midrange is (10+40) / 2 i.e. 25
Geometric Mean
-
Trimmed Mean
Similar to Arithmetic Mean, but a few extreme values are excluded.
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3. Applied Statistics in Business & Economics
Measures of Dispersion
Dispersion, also known as scatter spread or variation, measures the extent to
which the items vary from some central value and they measure only the degree
but not the direction of variation.
Significance of Measuring Dispersion
To determine the reliability of an average.
To facilitate comparison.
To facilitate control.
To facilitate the use of other statistical measures.
Range
Difference between minimum and maximum value
Range = XMax – Xmin
Mean Deviation
Mean Deviation is the arithmetic mean of the absolute deviations of all items of
the distribution from a measure of central tendency.
If nothing is specified, ‘Mean Deviation’ means ‘Mean Deviation’ about the
Arithmetic Mean.
Steps to Compute Mean Deviation
1. Calculate the Arithmetic Mean
2. Take the absolute deviations of each observation from the Mean ( Say |D| )
3. Calculate the sum of all these deviations i.e. | D |
4. Calculate the Mean Deviation by dividing this sum by total number of
observation.
|D|
Mean Deviation =
N
Coefficient of Mean Deviation
Mean Deviation
Coefficient of Mean Deviation =
Mean
Standard Deviation ( )
Standard Deviation is the Square Root of the Arithmetic Mean of the Squares of
deviations of all items of the distribution from the Arithmetic Mean.
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4. Applied Statistics in Business & Economics
Properties of Standard Deviation
The Sum of the Squares of the Deviations of Items from Arithmetic Mean
is minimal.
If all the observations are added by the same constant C, then the
standard deviation remains unchanged.
If all the observations are multiplied by the same constant C, then the
standard deviation will be | C | times the standard deviation.
Standard Deviation is the square root of Variance.
= x2
N
Where x = X - X
Variance
Variance is the arithmetic mean of the squares of deviations of all items of
distributions from the arithmetic mean in other words variance is the square of
standard deviation.
2
V=
Smaller the value of variance, lesser is the variability in population or greater the
uniformity in population and vice-versa.
Coefficient of Variation
CV = X 100%
X
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5. Applied Statistics in Business & Economics
Regression Analysis
Regression is the measure of average relationship between 2 or more variables
in terms of the original unit of the data.
Regression analysis is a statistical tool to study the nature & extent of functional
relationship between 2 or more variables and to estimate the unknown values of
dependent variables from known values of independent variable.
The terms dependence & independence does not necessarily indicate a cause-
effect relationship between the variables.
Regression Analysis is a valuable tool in business economics & business
research.
Linear Regression
Incase if a Linear Regression model involving 2 variables there are 2 regression
lines possible. Regression of X on Y and Regression of Y on X.
Regression of X on Y
X = a + bY
Where,
X = Dependent variable
Y = Independent variable
a = X Intercept. (Value of dependent variable when value of independent
variable is zero).
b = Slope of the Line. (The amount of change in the amount of dependent
variable per unit change in independent variable).
The value of constants a & b for the given data of X & Y can be calculated by
solving the following two algebraic equations (simultaneous equations) called
NORMAL EQUATIONS.
X = aN + bY
XY = aY + bY2
Where N is the number of pairs of X & Y variables and denotes the respective
summation.
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6. Applied Statistics in Business & Economics
Regression of Y on X
Y = a + bX
Where,
X = Independent variable
Y = Dependent variable
a = Y Intercept. (Value of dependent variable when value of independent
variable is zero).
b = Slope of the Line. (The amount of change in the amount of dependent
variable per unit change in independent variable).
The value of constants a & b for the given data of X & Y can be calculated by
solving the following two algebraic equations (simultaneous equations) called
NORMAL EQUATIONS.
Y = aN + bX
XY = aX + bX2
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7. Applied Statistics in Business & Economics
Correlation
Correlation is the relationship that exists between 2 or more variable.
Correlation Analysis is a statistical technique to measure the degree and
direction of relation between the variables.
If both the variables incase in the same direction then the correlation is
said to be positive. Whereas if both the variables vary in opposite
direction, the correlation is said to be negative.
When only 2 variables are considered, it is called simple correlation where
as if 3 or more variables are considered then it is multiple correlations.
Covariance
Given a set of N pairs , our observation relating to 2 variables X & Y, the
covariance pf X & Y are denoted by COV(X,Y) and is given by the formula:
(X – X ) ( Y- Y )
COV(X,Y) =
N
Covariance may be +ve, –ve or zero and take any value from - to +
Coefficient Of Correlation (Karl Pearson’s)
Given a set of ‘N’ pairs of observations relating to 2 variables X & Y, the
coefficient of Correlation between X & Y is denoted by the symbol ‘ r ‘
And is given by the formula
COV(X,Y)
r=
x . y
Where x and y are the standard deviations of variables X & Y
respectively.
Spearman’s Rank Correlation
Spearman’s Rank Correlation uses ranks rather than actual observations and
makes no assumptions about the population from which actual observations are
drawn.
The correlation coefficient between 2 series of ranks is called rank correlation
coefficient.
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8. Applied Statistics in Business & Economics
It is denoted by ‘ R ‘ and is given by the formula
6 D2
r= 1-
N3 - N
Where D is the difference of ranks between paired items in 2 series and
N is the number of pairs of ranks.
‘R’ lies between -1 and +1 ( -1 <= R >= +1 ), it can be interpreted in the same
fashion as Karl Pearson’s Coefficient of Correlation.
The Sum of difference of ranks will always be Zero. i.e. D = 0
Spearman’s Rank Correlation Coefficient is very useful when dealing with
Qualitative data.
Incase of tied ranks, average rank is allotted to each of these items and the
factor (m3 – m) / 12 is added to D2 for each instance of such tie.
Coefficient of Determination
The coefficient of determination is defined as the ratio of explained variance to
the total variance.
The coefficient of determination is calculated by squaring the coefficient of
correlation.
Coefficient of Determination = r2
For illustration , if r = 0.8 then r2 = 0.64 which means that 64% of the variation in
the dependent variable has been explained by the independent variable.
r2 takes the values between 0 and 1. 0 <= r2 <= 1
Coefficient of Non Determination
It is defined as the ratio between unexplained variance & total variance. It is
denoted by K2 and its value is calculated by subtracting r2 from 1.
K2 = 1 = r2
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9. Applied Statistics in Business & Economics
Exercise
For the following set of data
X 10 20 30 40 50 60 70 80 90
Y 20 22 30 45 50 65 67 78 85
Central Tendencies
1. Find Mean of X & Y
2. Find Median of Y
3. Find Mode of Y
4. Find Midrange of Y
Dispersion
5. Find Mean Deviation of Y
6. Find Coefficient of Mean Deviation of Y
7. Find Standard Deviation of X & Y
8. Find Variance of Y
9. Find Coefficient of Variation of Y
Regression Analysis
10. Find Both Regression Equations ( X on Y and Y on X)
11. Find the value of Y when X is 100, 150 and 200
12. Find the value of X when Y is 100, 150 and 200
Correlation
13. Find Covariance
14. Find Coefficient of Correlation (Karl Pearson’s)
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