Chapter: Organisation of Data
1. Width of Class Interval
2. Mid-Point or Mid-Value
Chapter: Diagrammatic Presentation of Data
1. Conversion of percentage into degrees in Pie Diagram
2. Adjustment Factor for any Class (Histogram)
OR
Chapter: Measures of Central Tendency: Arithmetic Mean
1. Arithmetic Mean
i) Individual Series:
Where,
N = Total Number of Items
Where,
A = Assumed Mean
d = X – A (deviations of variables from assumed mean)
∑d = ∑(X – A) (sum of deviations of variables from assumed mean)
N = Total Number of Items
Where,
A = Assumed Mean
d = X – A (deviations of variables from assumed mean)
C = Common Factor
N = Total Number of Items
ii) Discrete Series:
Where,
∑fX = Sum of the product of variables with the respective frequencies
∑f = Total Number of Items
Where,
A = Assumed Mean
d = X – A (deviations of variables from assumed mean)
∑fd = Sum of the product of deviations (d) with the respective frequencies
∑f = Total Number of Items
Where,
A = Assumed Mean
d = X – A (deviations of variables from assumed mean)
C = Common Factor
∑f = Total Number of Items
iii) Continuous Series:
Where,
∑fm = Sum of the product of mid-points with the respective frequencies
∑f = Total Number of Items
Where,
A = Assumed Mean
d = m – A (deviations of mid-points from assumed mean)
∑fd = Sum of the product of deviations (d) with the respective frequencies
∑f = Total Number of Items
Where,
A = Assumed Mean
d = m – A (deviations of mid-points from assumed mean)
C = Common Factor
∑f = Total Number of Items
2. Charlier’s Accuracy Check
∑f(d + 1) = ∑fd + ∑f
- For Step Deviation Method
3. Missing Value
i) Individual Series:
Where,
X1, X2, ………………… Xn-1 = Given Values
Xn = Missing Value
ii) Discrete Series:
iii) Continuous Series:
4. Combined Mean
Where,
N1 = Number of Items of first distribution
N2 = Number of Items of second distribution
5. Corrected Mean
6. Weighted Arithmetic Mean
Where,
∑WX = Sum of product of items and respective weights
∑W = Sum of the weights
Chapter: Measures of Central Tendency: Median and Mode
1. Median
i) Individual Series:
Where,
N = Number of Items
- If the Number of Items is Even
ii) Discrete Series:
Where,
N = Total of Frequency
Find out the value of
iii) Continuous Series:
Where,
l1 = lower limit of the median class
c.f. = cumulative frequency of the class preceding the median class
f = simple frequency of the median class
i = class size of the median group or class
2. Quartiles
i) Individual Series:
Where,
N = Number of Items
ii) Discrete Series:
Where,
N = Cumulative Frequency
iii) Continuous Series:
3. Mode
i) Individual Series:
Mode is the value which occurs the largest number of times.
ii) Discrete Series:
In the case of regular and homogeneous frequencies, and single maximum frequency, Mode is the value corresponding to the highest frequency. Otherwise, the grouping method is used.
iii) Continuous Series:
Where,
Z = Value of Mode
f_0 = frequency of pre-modal class
i = size of the modal group
4. Relationship between Mean, Median, and Mode
Mode = 3 Median – 2 Mean
Chapter: Measures of Dispersion
1. Range
Range(R) = Largest Item(L) – Smallest Item(S)
2. Coefficient of Range
In Individual Series, the largest and smallest item is taken from the given observations.
In Discrete Series, the largest and smallest item is taken from the given frequencies.
In Continuous Series, the first method to calculate coefficient of range is to take the difference between the upper and lower limit of the highest and lowest class interval respectively. The second method is to take the difference between the mid-points of the highest class limit and lowest class limit.
3. Quartile Deviation
Where,
Q3 = Upper Quartile (Size of
Q1 = Lower Quartile (Size of
4. Coefficient of Quartile Deviation
Where,
Q3 = Upper Quartile (Size of
Q1 = Lower Quartile (Size of
5. Mean Deviation
i) Individual Series:
- Mean Deviation from Median
Where,
∑|D| = Sum of absolute deviations from Assumed Mean
A = Assumed Mean
∑fB = Number of Values from actual mean
∑fA = Number of values below actual mean including actual mean
N = Number of Observations
ii) Discrete Series:
- Mean Deviation from Median
Where,
M = Median
iii) Continuous Series:
- Mean Deviation from Median
Where,
M = Median
6. Coefficient of Mean Deviation
- Coefficient of Mean Deviation from Mean
- Coefficient of Mean Deviation from Median
7. Standard Deviation
i) Individual Series:
Where,
σ = Standard Deviation
∑x2 = Sum total of the squares of deviations from the actual mean
N = Number of pairs of observations
Or
Where,
σ = Standard Deviation
∑X2 = Sum total of the squares of observations
N = Number of Observations
Where,
σ = Standard Deviation
∑d = Sum total of deviations from assumed mean
∑d2 = Sum total of squares of deviations
N = Number of pairs of observations
ii) Discrete Series:
Where,
σ = Standard Deviation
∑fx2 = Sum total of the squared deviations multiplied by frequency
N = Number of pairs of observations
Or
Where,
σ = Standard Deviation
∑fx2 = Sum total of the squared deviations multiplied by frequency
N = Number of Observations
Or
Where,
σ = Standard Deviation
∑fd = Sum total of deviations multiplied by frequencies
∑d2 = Sum total of the squared deviations multiplied by frequencies
N = Number of pairs of observations
Where,
σ = Standard Deviation
N = Number of pairs of observations
iii) Continuous Series:
σ =
OR
Where,
σ = Standard Deviation
∑fx2 = Sum total of the deviations of every mid-value of the class intervals multiplied by frequency
N = Number of pair of observations
σ =
σ = Standard Deviation
∑fd2 = Sum total of the squared deviations multiplied by frequency
∑fd = Sum total of deviations multiplied by frequency
N = Number of pair of observations
σ = Standard Deviation
C = Common Factor
N = Number of pair of observations
8. Coefficient of Standard Deviation
9. Combined Standard Deviation
Where,
10. Variance
Variance = σ2
11. Coefficient of Variation
Where,
C.V. = Coefficient of Variation
σ = Standard Deviation
Chapter: Correlation
1. Degree of Correlation
2. Karl Pearson’s Coefficient of Correlation
Or,
Or,
Or,
Or,
Where,
N = Number of Pair of Observations
x = Deviation of X series from Mean
y = Deviation of Y series from Mean
r = Coefficient of Correlation
Where,
N = Number of pair of observations
∑dx = Sum of deviations of X values from assumed mean
∑dy = Sum of deviations of Y values from assumed mean
∑dx2 = Sum of squared deviations of X values from assumed mean
∑dy2 = Sum of squared deviations of Y values from assumed mean
∑dxdy = Sum of the products of deviations dx and dy
Where,
N = Number of pair of observations
3. Karl Pearson’s Coefficient of Correlation and Covariance
With Covariance formula, the formula for r (coefficient of correlation) can be written as:
Or,
Or,
Or,
4. Spearman’s Rank Correlation Coefficient
Where,
rk = Coefficient of rank correlation
D = Rank differences
N = Number of variables
Here,
m1, m2, ……. are the number of times a value has repeated in the given X, Y, …….. series respectively.
Chapter: Index Number
1. Unweighted or Simple Index Numbers
Where,
P01 = Index Number of the Current Year
∑p1 = Total of the current year’s price of all commodities
∑p0 = Total of the base year’s price of all commodities
2. Weighted Index Numbers
i) Weighted Aggregative Method
Here,
P01 = Price Index of the current year
p0 = Price of goods at base year
q0 = Quantity of goods at base year
p1 = Price of goods at the current year
Here,
P01 = Price Index of the current year
p0 = Price of goods in the base year
q1 = Quantity of goods in the base year
p1 = Price of goods in the current year
Here,
P01 = Price Index of the current year
p0 = Price of goods in the base year
q1 = Quantity of goods in the base year
p1 = Price of goods in the current year
Fisher’s Method is considered the Ideal Method for Constructing Index Numbers.
ii) Weighted Average of Price Relatives Method
3. Methods of Constructing Consumer Price Index
- Aggregate Expenditure Method
4. Purchasing Power
5. Real Wages
Chapter: Organisation of Data
1. Width of Class Interval
2. Mid-Point or Mid-Value
Chapter: Diagrammatic Presentation of Data
1. Conversion of percentage into degrees in Pie Diagram
2. Adjustment Factor for any Class (Histogram)
OR
Chapter: Measures of Central Tendency: Arithmetic Mean
1. Arithmetic Mean
i) Individual Series:
Where,
N = Total Number of Items
Where,
A = Assumed Mean
d = X – A (deviations of variables from assumed mean)
∑d = ∑(X – A) (sum of deviations of variables from assumed mean)
N = Total Number of Items
Where,
A = Assumed Mean
d = X – A (deviations of variables from assumed mean)
C = Common Factor
N = Total Number of Items
ii) Discrete Series:
Where,
∑fX = Sum of the product of variables with the respective frequencies
∑f = Total Number of Items
Where,
A = Assumed Mean
d = X – A (deviations of variables from assumed mean)
∑fd = Sum of the product of deviations (d) with the respective frequencies
∑f = Total Number of Items
Where,
A = Assumed Mean
d = X – A (deviations of variables from assumed mean)
C = Common Factor
∑f = Total Number of Items
iii) Continuous Series:
Where,
∑fm = Sum of the product of mid-points with the respective frequencies
∑f = Total Number of Items
Where,
A = Assumed Mean
d = m – A (deviations of mid-points from assumed mean)
∑fd = Sum of the product of deviations (d) with the respective frequencies
∑f = Total Number of Items
Where,
A = Assumed Mean
d = m – A (deviations of mid-points from assumed mean)
C = Common Factor
∑f = Total Number of Items
2. Charlier’s Accuracy Check
∑f(d + 1) = ∑fd + ∑f
- For Step Deviation Method
3. Missing Value
i) Individual Series:
Where,
X1, X2, ………………… Xn-1 = Given Values
Xn = Missing Value
ii) Discrete Series:
iii) Continuous Series:
4. Combined Mean
Where,
N1 = Number of Items of first distribution
N2 = Number of Items of second distribution
5. Corrected Mean
6. Weighted Arithmetic Mean
Where,
∑WX = Sum of product of items and respective weights
∑W = Sum of the weights
Chapter: Measures of Central Tendency: Median and Mode
1. Median
i) Individual Series:
Where,
N = Number of Items
- If the Number of Items is Even
ii) Discrete Series:
Where,
N = Total of Frequency
Find out the value of
iii) Continuous Series:
Where,
l1 = lower limit of the median class
c.f. = cumulative frequency of the class preceding the median class
f = simple frequency of the median class
i = class size of the median group or class
2. Quartiles
i) Individual Series:
Where,
N = Number of Items
ii) Discrete Series:
Where,
N = Cumulative Frequency
iii) Continuous Series:
3. Mode
i) Individual Series:
Mode is the value which occurs the largest number of times.
ii) Discrete Series:
In the case of regular and homogeneous frequencies, and single maximum frequency, Mode is the value corresponding to the highest frequency. Otherwise, the grouping method is used.
iii) Continuous Series:
Where,
Z = Value of Mode
f_0 = frequency of pre-modal class
i = size of the modal group
4. Relationship between Mean, Median, and Mode
Mode = 3 Median – 2 Mean
Chapter: Measures of Dispersion
1. Range
Range(R) = Largest Item(L) – Smallest Item(S)
2. Coefficient of Range
In Individual Series, the largest and smallest item is taken from the given observations.
In Discrete Series, the largest and smallest item is taken from the given frequencies.
In Continuous Series, the first method to calculate coefficient of range is to take the difference between the upper and lower limit of the highest and lowest class interval respectively. The second method is to take the difference between the mid-points of the highest class limit and lowest class limit.
3. Quartile Deviation
Where,
Q3 = Upper Quartile (Size of
Q1 = Lower Quartile (Size of
4. Coefficient of Quartile Deviation
Where,
Q3 = Upper Quartile (Size of
Q1 = Lower Quartile (Size of
5. Mean Deviation
i) Individual Series:
- Mean Deviation from Median
Where,
∑|D| = Sum of absolute deviations from Assumed Mean
A = Assumed Mean
∑fB = Number of Values from actual mean
∑fA = Number of values below actual mean including actual mean
N = Number of Observations
ii) Discrete Series:
- Mean Deviation from Median
Where,
M = Median
iii) Continuous Series:
- Mean Deviation from Median
Where,
M = Median
6. Coefficient of Mean Deviation
- Coefficient of Mean Deviation from Mean
- Coefficient of Mean Deviation from Median
7. Standard Deviation
i) Individual Series:
Where,
σ = Standard Deviation
∑x2 = Sum total of the squares of deviations from the actual mean
N = Number of pairs of observations
Or
Where,
σ = Standard Deviation
∑X2 = Sum total of the squares of observations
N = Number of Observations
Where,
σ = Standard Deviation
∑d = Sum total of deviations from assumed mean
∑d2 = Sum total of squares of deviations
N = Number of pairs of observations
ii) Discrete Series:
Where,
σ = Standard Deviation
∑fx2 = Sum total of the squared deviations multiplied by frequency
N = Number of pairs of observations
Or
Where,
σ = Standard Deviation
∑fx2 = Sum total of the squared deviations multiplied by frequency
N = Number of Observations
Or
Where,
σ = Standard Deviation
∑fd = Sum total of deviations multiplied by frequencies
∑d2 = Sum total of the squared deviations multiplied by frequencies
N = Number of pairs of observations
Where,
σ = Standard Deviation
N = Number of pairs of observations
iii) Continuous Series:
σ =
OR
Where,
σ = Standard Deviation
∑fx2 = Sum total of the deviations of every mid-value of the class intervals multiplied by frequency
N = Number of pair of observations
σ =
σ = Standard Deviation
∑fd2 = Sum total of the squared deviations multiplied by frequency
∑fd = Sum total of deviations multiplied by frequency
N = Number of pair of observations
σ = Standard Deviation
C = Common Factor
N = Number of pair of observations
8. Coefficient of Standard Deviation
9. Combined Standard Deviation
Where,
10. Variance
Variance = σ2
11. Coefficient of Variation
Where,
C.V. = Coefficient of Variation
σ = Standard Deviation
Chapter: Correlation
1. Degree of Correlation
2. Karl Pearson’s Coefficient of Correlation
Or,
Or,
Or,
Or,
Where,
N = Number of Pair of Observations
x = Deviation of X series from Mean
y = Deviation of Y series from Mean
r = Coefficient of Correlation
Where,
N = Number of pair of observations
∑dx = Sum of deviations of X values from assumed mean
∑dy = Sum of deviations of Y values from assumed mean
∑dx2 = Sum of squared deviations of X values from assumed mean
∑dy2 = Sum of squared deviations of Y values from assumed mean
∑dxdy = Sum of the products of deviations dx and dy
Where,
N = Number of pair of observations
3. Karl Pearson’s Coefficient of Correlation and Covariance
With Covariance formula, the formula for r (coefficient of correlation) can be written as:
Or,
Or,
Or,
4. Spearman’s Rank Correlation Coefficient
Where,
rk = Coefficient of rank correlation
D = Rank differences
N = Number of variables
Here,
m1, m2, ……. are the number of times a value has repeated in the given X, Y, …….. series respectively.
Chapter: Index Number
1. Unweighted or Simple Index Numbers
Where,
P01 = Index Number of the Current Year
∑p1 = Total of the current year’s price of all commodities
∑p0 = Total of the base year’s price of all commodities
2. Weighted Index Numbers
i) Weighted Aggregative Method
Here,
P01 = Price Index of the current year
p0 = Price of goods at base year
q0 = Quantity of goods at base year
p1 = Price of goods at the current year
Here,
P01 = Price Index of the current year
p0 = Price of goods in the base year
q1 = Quantity of goods in the base year
p1 = Price of goods in the current year
Here,
P01 = Price Index of the current year
p0 = Price of goods in the base year
q1 = Quantity of goods in the base year
p1 = Price of goods in the current year
Fisher’s Method is considered the Ideal Method for Constructing Index Numbers.
ii) Weighted Average of Price Relatives Method
3. Methods of Constructing Consumer Price Index
- Aggregate Expenditure Method