Statistical Tools And Interpretation Class 11 Notes Economics Chapter 3 - CBSE
Chapter : 3
What Are Statistical Tools And Interpretation ?
Statistical Tools
- Statistical tools are methods or techniques which are used to collect, organise, analyse and interpret the data.
Statistical Tools And Interpretation
- Measures of Central Tendency
- Airthmetic Mean
- Median
- Mode
- Correlation
- Index Numbers
Measure Of Central Tendency
- It refers to all those methods of statistical analysis which helps in determining the averages of statistical series.
Averages
- It refers to the ‘mid’ value or any value which repersents the given series.
Types of Statistical Averages
Mathematical Averages
- Airthmetic Mean
- Geometric Mean
- Harmonic Mean
Positional Averages
- Median
- Partition Value
- Mode
Arithmetic Mean
- The sum of a series of numbers divided by the count of that series of numbers is the simple average or called as Arithmetic Mean.
- Simple Arithmetic Mean
- Weighted Arithmetic Mean
Methods/Calculations of Simple Arithmetic Mean
For Ungrouped Data
- By Direct Method
- $$\bar{\text{X}} =\frac{\Sigma \text{X}}{\text{N}}\\\text{where,}\\\Sigma \text{X} =\text{Total value of Items}$$
- N = Total number of items
- By Assumed Mean Method
- $$\bar{\text{X}} = \text{A +}\frac{\Sigma d}{\text{N}}$$
- where, d = X – A
A = Assumed Mean
N = No. of items
∑d = Sum of deviations
- By Step Deviation Method
- $$\bar{\text{X}} = \text{A +}\frac{\Sigma \text{d'}}{\text{N}}×\text{C}$$
- where, A = Assumed Mean
- $$\text{d'} =\frac{\text{X-A}}{\text{C}}$$
- C = Common factor
- N = Number of observations
For Grouped Data
- By Direct Method
- $$\bar{\text{X}} = \frac{\Sigma fX}{\Sigma f}$$
- where, ∑fX = Sum of product of frequency and X
- ∑f = Sum of frequencies
- By Assumed Mean Method
- $$\bar{\text{X}} = \text{A} + \frac{\Sigma fd}{\Sigma f}$$
- where, A = Assumed Mean
- d = X – A
- ∑fd = Sum of product of frequency and deviation
- ∑f = Sum of frequencies
- By Step Devation Method
- $$\bar{\text{X}} =\text{A} +\frac{\Sigma fd'}{\Sigma f}×\text{C}$$
- where,
A = Assumed Mean - $$\text{d' =}\space\frac{\text{X - A}}{\text{C}}$$
- C = Class interval
- ∑f = Sum of frequencies
Method/Calculation of Weighted Arithmetic Mean
$$\bar{\text{X}}_{w} =\frac{\Sigma\text{WX}}{\Sigma\text{W}}$$
where,
∑W = Total of weights
∑WX = Total of Product of weights & items
Median
It refers to the ‘mid’ value of group given.
Methods Of Median
For Ungrouped Data
- $$\text{M = Size of}\space\bigg(\frac{\text{N+1}}{2}\bigg)\space\text{th}$$
- N = Total no. of items
- $$\text{If}\bigg(\frac{\text{N+1}}{2}\bigg)\space\text{th comes in,}$$then fraction median will be average of the two middle value of series.
For Grouped Data
In Discrete Series
- $$\text{M = Size of\space}\bigg(\frac{\text{N+1}}{2}\bigg)\text{th}$$
- N = Sum of frequencies
In Continuous Series
$$\text{M =}\frac{\text{I}_{1} +\frac{\text{N}}{2} - c.f.}{f}×\text{C}$$
l1 = lower limit of median class
c.f. = Cumulative frequency of class preceding the median class
i = size of median class
f = frequency of median class
Partition Value
- The quantity that splits the series into more than two pieces known as partition value.
Partition values are estimated as Quartiles
Individual and Discrete Series
- Q1
- $$\text{Q}_{1} =\text{Size of}\bigg(\frac{\text{N+1}}{4}\bigg)\text{th}\\\text{item of series}$$
- Q3
- $$\text{Q}_{3} =\text{Size of 3}\bigg(\frac{\text{N+1}}{4}\bigg)\text{th}\space\\\text{item of series}$$
Continuous Series
- Q1
- $$\text{Q}_{1} =\frac{\text{I}_{1}+\bigg(\frac{\text{N}}{4} -\text{c.f.}\bigg)}{f}×i$$
- Q3
- $$\text{Q}_{3} =\frac{\text{I}_{1} +\bigg[^{3}\bigg(\frac{\text{N}}{4 }\bigg)-\text{c.f.}\bigg]}{f}×i$$
Partition values are estimated as Deciles
Individual and Discrete Series
- D1
- $$\text{D}_{1} =\text{Size of\space}\bigg(\frac{\text{N+1}}{10}\bigg)\space\text{th}\space\text{item}$$
- D4
- $$\text{D}_{4} =\text{Size of}\\4\bigg(\frac{\text{N+1}}{10}\bigg)\text{th}\space\text{item}$$
- D9
- $$\text{D}_{9} =\text{Size of}\\9\bigg(\frac{\text{N+1}}{10}\bigg)\text{th}\space\text{item}$$
Continuous Series
$$\cdotp\text{D}_{1} =\text{I}_{1} +\bigg[\frac{\frac{\text{N}}{10} -\text{c.f.}}{f}\bigg]×i\\\cdotp\text{D}_{4} =\text{I}_{1} +\bigg[\frac{\frac{4\text{N}}{10}-\text{c.f.}}{f}\bigg]×i\\\cdotp\text{D}_{9} =\text{I}_{1}+\bigg[\frac{\frac{\text{9N}}{10} - c.f.}{f}\bigg]×i$$
Partition values are estimated as Percentile
Individual and Discrete Series
- P1
- $$\text{P}_{1} =\text{Size of}\space\bigg(\frac{\text{N+1}}{10}\bigg)\text{th}\\\text{item}$$
- P4
- $$\text{P}_{4} =\text{Size of}\\4\bigg(\frac{\text{N+1}}{10}\bigg)\text{th}\space\text{item}$$
- P99
- $$\text{P}_{99} =\text{Size of}\\99\bigg(\frac{\text{N+1}}{10}\bigg)\text{th}\space\text{item}$$
Continuous Series
$$\cdotp\text{P}_{1} =\text{I}_{1} + \bigg[\frac{\frac{\text{N}}{100} - c.f.}{f}\bigg]×i\\\cdotp\text{P}_{4} =\text{I}_{1} +\bigg[\frac{4\frac{\text{N}}{100} - c.f.}{f}\bigg]×i\\\cdotp\text{P}_{99} =\text{I}_{1} +\bigg[\frac{99\frac{N}{100}-c.f.}{f}\bigg]×i$$
Note: Using similar formula, D1 , D2 , D3 , D4 , D5, D6 , D7 , D8 , D9, D10 can be determined.
Mode
The variable with the highest frequency in a distribution is called mode.
Calculation Of Mode For Different Types Of Statistical Series
Individual or Discrete Series
- By Inspection Method
- By Grouping Method
Frequency Distribution
- By Inspection Method
- By Grouping Method
Note: Mode = 3 Median − 2 Mean
Correlation
Technique of studying the degree of relationship between two or more groups, classes, or series or data is called correlation.
Types of Correlation
- Positive Correlation
- When two variable moves in the same direction, i.e., when one increases other also increases.
- Negative Correlation
- When two variable move in opposite direction, i.e., when one increases other decreases.
- Linear Correlation
- When two variables change in a constant proportion.
- Non-linear Correlation
- When two variables do not change in any constant proportion.
- Simple Correlation
- Study of relationship between two variables only.
- Multiple Correlation
- Relationship among three or more than three variables.
Methods of Estimating Correlation
- Scattered Diagram Method
- It refers to the graphic representation
of direction and degree of correlation.
- It refers to the graphic representation
- Karl Pearson’s Coefficient of Correlation
- $$r =\frac{\Sigma xy}{\text{N}\sigma x\sigma y}$$
- where, r = Coefficient of correlation
- $$x =\text{x -}\bar x\\ y = y-\bar y$$
- σx = Standard deviation of x series
- σy = Standard deviation of y series
- N = Number of observations
- Spearman’s Rank Correlation Coefficient
- $$\text{r}_{\text{k}} = 1 -\frac{6\Sigma D^{2}}{\text{N}^{3}-\text{N}}$$
- where, rk = Coefficient of rank
correlation
D = Rank differences
N = Number of pairs
Note: Value of correlation coefficient lies between +1 and −1
Karl Pearson’s Coefficient of Correlation
According to Karl Pearson’s Method, the coefficient of correlation is measured as:
Basic Version
$$r= \frac{\Sigma xy}{\text{N}\sigma x\sigma y}\\\text{Here,}\space\text{x = X -}\bar{X}\\ y = Y-\bar{\text{Y}}\\\sigma x =\text{deviation of x series}\\\sigma y =\text{deviation of x series}$$
Modified Version
$$r =\frac{\Sigma xy}{\sqrt{\Sigma x^{2}×\Sigma y^{2}}}\\\text{Here,}\\ x =\text{X -}\bar{\text{X}}\\y =\text{Y -}\bar{\text{Y}}$$
Methods of Calculating Correlation
- Shortcut Method
- $$r =\frac{\Sigma dxdy-\frac{\Sigma dx×\Sigma dy}{\text{N}}}{\sqrt{\Sigma}dx^{2} -\frac{(\Sigma dx)^{2}}{\text{N}}×\sqrt{\Sigma dy^{2}} -\frac{(\Sigma dy)^{2}}{\text{N}}}$$ Here, dx = x − Ax
dy = y − Ay
∑dxdy = Sum of multiple of dx and dy
∑dx2 = Sum of square of dx
∑dy2 = Sum of square of dy
∑dx = Sum of deviation of x series
∑dy = Sum of deviation of y series
N = Total number of items
- Step Deviation Method
- $$r =\\\frac{\Sigma dx'dy' -\frac{\Sigma dx'×\Sigma dy'}{\text{N}}}{\sqrt{(\Sigma dx')^{2}} -\frac{(\Sigma dx')^{2}}{\text{N}}×\sqrt{(\Sigma dy')^{2}}-\frac{(\Sigma dy')^{2}}{\text{N}}}$$
- where,
- $$dx' =\frac{dx}{c_{1}}\\\lbrack\text{c}_{1}\space\text{is common factor of x}\rbrack\\dy' =\frac{dy}{c_{2}}\\\lbrack\text{c}_{2}\space\text{ is common factor of y}\rbrack$$
- where,
- $$r =\\\frac{\Sigma dx'dy' -\frac{\Sigma dx'×\Sigma dy'}{\text{N}}}{\sqrt{(\Sigma dx')^{2}} -\frac{(\Sigma dx')^{2}}{\text{N}}×\sqrt{(\Sigma dy')^{2}}-\frac{(\Sigma dy')^{2}}{\text{N}}}$$
Note: There are some variables, whose quantitative measurement is not possible. Hence, they are known as qualitative variables. To calculate coefficient of correlation of these qualitative variables, spearman developed a formula.
Spearman’s Rank Correlation Coefficient
When Ranks are not equal
$$r_{k} = 1-\frac{6\Sigma \text{D}^{2}}{\text{N}^{3}-\text{N}}$$
Where,
rk = Coefficient of rank correlation
d = Rank differences
N = Number of pairs
When Ranks are equal
$$r_{k} = 1 -\\\frac{6\bigg[\Sigma D^{2} +\frac{1}{12}(m_{1}^{3}-m) + \frac{1}{12}(m_{2}^{3} -m_{2}) +...\bigg]}{\text{N}^{3} -\text{N}}$$
Where, rk = Coefficient of rank correlation
d = Rank differences
N = Number of pairs
m = Number of items of equal ranks
Index Numbers
“Index Numbers are devices for measuring difference in the magnitude of a group of related variables.” – Croxton and cowder
Features of Index Numbers
- It measures relative change in variables over time
- Quantitative expression of change
- It shows changes in terms of averages
Advantages Of Index Numbers
- It measures the change in price level during different periods of time
- It helps to ascertain the living standard of people
- It serves as useful guide to businesses for planning and decision making
- Helps in ascertaining real economic competition of the country
Limitations of Index Numbers
- Not fully true
- International comparisions not possible
- With passage of time, comparisons of index numbers become difficult
- Prepared with certain specific objective. Thus have limited use
Methods of Constructing Index Numbers
Construction of Simple Index Numbers
Simple Aggregative Method
- $$\text{P}_{01} =\frac{\Sigma \text{P}_{1}}{\Sigma \text{P}_{0}}×100$$
- Here, P01 = Price index of current year
∑P1 = Sum of Prices of commodities in current year
∑P0 = Sum of Prices of commodities in base year
Simple Average of Price Relative Method
$$\text{P}_{01}=\frac{\Sigma\bigg(\frac{\text{P}_{1}}{\text{P}_{0}}×100\bigg)}{\text{N}}$$
Here,
P01 = Price index of current year
$$\frac{\text{P}_{1}}{\text{P}_{0}}×100 =\text{Price relatives}$$
N = Number of goods
P1 = Current year’s value
P0 = Base year’s value
Methods of Constructing Index Numbers
Construction of Weighted Index Numbers
Weighted Aggregative Method
Laspeyres’ Method
- $$\text{P}_{01}=\frac{\Sigma \text{P}_{1}\text{q}_{0}}{\Sigma \text{P}_{0}\text{q}_{0}}×100$$
Paasche’s Method
- $$\text{P}_{01}=\frac{\Sigma \text{P}_{1}\text{q}_{1}}{\Sigma \text{P}_{0}\text{q}_{1}}×100$$
Fishers’ Method
- $$\text{P}_{01} =\sqrt{\frac{\Sigma \text{P}_{1}\text{q}_{0}}{\Sigma \text{P}_{0}\text{q}_{0}}×\frac{\Sigma \text{P}_{1}\text{q}_{1}}{\Sigma \text{P}_{0}\text{q}_{1}}}×100$$
Here,
P0 = Base year price
P1 = Current year price
q0 = Base year quantity
q1 = Current year quantity
Weighted Average of Price Relative Method
$$\text{P}_{01} =\frac{\Sigma\text{RW}}{\Sigma\text{W}}$$
Here,
P0 = Index number for current year in relation to the base year
W = Weight
R = Price Relative or
$$\text{R} =\frac{\text{P}_{1}}{\text{P}_{0}}×100$$
Consumer Price Index Or Cost Of Living Index Number
It measures the average change in prices paid by specific class of consumers for goods and services consumed by them in current year compared to base year.
Methods of calculating Price Index
Aggregative Expenditure Method
$$\text{CPI} =\frac{\Sigma \text{P}_{2}\text{q}_{0}}{\Sigma \text{P}_{0}\text{q}_{0}}×100$$
Family Budget Method
$$\text{CPI} =\frac{\Sigma\text{RW}}{\Sigma\text{W}}$$
where,
R = Current years’ price relative of various items
w = Weights of various items
Wholesale Price Index
It measures the relative changes in the prices of commodities traded in wholesale markets.
Index Number Of Industrial Production
It measures the relative increase on decrease in the level of industrial output in a country in comparison with the level of of output in base year. It is calculated as:
Formula:
$$\text{Index for industrial production =}\\\frac{\Sigma\bigg(\frac{q_1}{q_0}\bigg)\text{W}}{\Sigma\text{W}}×100$$
Here,
q1 = Current year output
q0 = Base year output
W = Weight of industrial output
