Statistical Tools And Interpretation Class 11 Notes Economics Chapter 3 - CBSE

Chapter : 3

What Are Statistical Tools And Interpretation ?

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    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.
    • 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,
          • $$dx' =\frac{dx}{c_{1}}\\\lbrack\text{c}_{1}\space\text{is common factor of x}\rbrack\\dy' =\frac{dy}{c_{2}}\\\lbrack\text{c}_{2}\space\text{ is common factor of y}\rbrack$$

    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