Important Formulas in Statistics for Economics | Class 11

Last Updated : 12 Oct, 2023

Statistics for Economics is a field that helps in the study, collection, analysis, interpretation, and organization of data for different ultimate objectives. Statistics help a user in gathering and analyzing huge numerical data easily and efficiently. For this, it provides various statistical tools and formulas that can be used to collect, interpret, and analyse the given set of data. Following are some of the important formulas (chapter-wise) used in Class 11th Statistics for Economics.

1. Width of Class Interval

Where,

Largest Observation is the largest value of the given data set

Smallest Observation is the smallest value of thegiven data set

Number of Classes Desired is the number of class intervals required

2. Mid-Point or Mid-Value

Where,

Lower Class Limit is the lower limit  of a class interval of the given frequency distribution

Upper Class Limit is the upper limit of the same class interval of the given frequency distribution

OR

5. Arithmetic Mean

i) Individual Series:

• Direct Method

Where,

N = Total Number of Items

• Short-cut Method

Where,

A = Assumed Mean

d = X – A (deviations of variables from assumed mean)

âˆ‘d = âˆ‘(X – A) (sum of deviations of variables from assumed mean)

N = Total Number of Items

• Step Deviation Method

Where,

A = Assumed Mean

d = X – A (deviations of variables from assumed mean)

(Step Deviations; i.e., deviations divided by common factor)

C = Common Factor

N = Total Number of Items

ii) Discrete Series:

• Direct Method

Where,

âˆ‘fX = Sum of the product of variables with the respective frequencies

âˆ‘f = Total Number of Items

• Short-cut Method

Where,

A = Assumed Mean

d = X – A (deviations of variables from assumed mean)

âˆ‘fd = Sum of the product of deviations (d) with the respective frequencies

âˆ‘f = Total Number of Items

• Step Deviation Method

Where,

A = Assumed Mean

d = X – A (deviations of variables from assumed mean)

(Step Deviations; i.e., deviations divided by common factor)

C = Common Factor

âˆ‘f = Total Number of Items

iii) Continuous Series:

• Direct Method

Where,

âˆ‘fm = Sum of the product of mid-points with the respective frequencies

âˆ‘f = Total Number of Items

• Short-cut Method

Where,

A = Assumed Mean

d = m – A (deviations of mid-points from assumed mean)

âˆ‘fd = Sum of the product of deviations (d) with the respective frequencies

âˆ‘f = Total Number of Items

• Step Deviation Method

Where,

A = Assumed Mean

d = m – A (deviations of mid-points from assumed mean)

(Step Deviations; i.e., deviations divided by common factor)

C = Common Factor

âˆ‘f = Total Number of Items

6. Charlier’s Accuracy Check

• For Short-cut Method

âˆ‘f(d + 1) = âˆ‘fd + âˆ‘f

Where,

f   = Number  of Observations

d = m – A (deviations of mid-points from assumed mean)

âˆ‘fd = Sum of the product of deviations (d) with the respective frequencies

âˆ‘f = Total Number of Items

• For Step Deviation Method

Where,

f   = Number  of Observations

(Step Deviations; i.e., deviations divided by common factor)

C = Common Factor

A = Assumed Mean

âˆ‘f = Total Number of Items

7. Formula to calculate Missing Value in Individual, Discrete, and Continuous Series

i) Individual Series:

Where,

X1, X2, ………………… Xn-1 = Given Values

Xn = Missing Value

ii) Discrete Series:

Where,

âˆ‘fX = Sum of the product of variables with the respective frequencies

âˆ‘f = Total Number of Items

iii) Continuous Series:

Where,

âˆ‘fm = Sum of the product of mid-points with the respective frequencies

âˆ‘f = Total Number of Items

8. Combined Mean

Where,

N1 = Number of Items of first distribution

N2 = Number of Items of second distribution

10. Weighted Arithmetic Mean

Where,

âˆ‘WX = Sum of product of items and respective weights

âˆ‘W = Sum of the weights

11. Median

i) Individual Series:

Where,

N = Number of Items

• If the Number of Items is Even

Where,

N = Number of Items

ii) Discrete Series:

Where,

N = Total of Frequency

Find out the value of  Locate the cumulative frequency which is equal to higher than  and then find the value corresponding to this cf. This value will be the Median value of the series.

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

12. Quartiles

i) Individual Series:

Where,

N = Number of Items

ii) Discrete Series:

Where,

N = Cumulative Frequency

13. Deciles

i) Individual Series:

……….

Where,

n is the total number of observations, D1 is First Decile, D2 is Second Decile,……….D9 is Ninth Decile.

ii) Discrete Series:

……….

Where,

n is the total number of observations (âˆ‘f), D1 is First Decile, D2 is Second Decile,……….D9 is Ninth Decile.

iii) Continuous Series:

……….

Where,

n is the total number of observations (âˆ‘f), D1 is First Decile, D2 is Second Decile,……….D9 is Ninth Decile.

14. Percentiles

i) Individual Series:

……….

Where,

n is the total number of observations (âˆ‘f), P1 is First Percentile, P2 is Second Percentile, P3 is Third Percentile, ……….P99 is Ninety Ninth Percentile.

ii) Discrete Series:

……….

Where,

n is the total number of observations (âˆ‘f), P1 is First Percentile, P2 is Second Percentile, P3 is Third Percentile, ……….P99 is Ninety Ninth Percentile.

iii) Continuous Series:

……….

Where,

n is the total number of observations (âˆ‘f), P1 is First Percentile, P2 is Second Percentile, P3 is Third Percentile, ……….P99 is Ninety Ninth Percentile.

15. Mode

i) Individual Series:

Mode is the value that 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

l1 = lower limit of the modal class

f1  = frequency of modal class

f0  = frequency of pre-modal class

f2  = frequency of the next higher class or post-modal class

i = size of the modal group

16. Relationship between Mean, Median, and Mode

Mode = 3 Median – 2 Mean

17. Range

Range(R) = Largest Item(L) – Smallest Item(S)

18. 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.

19. Quartile Deviation

Where,

Q3 = Upper Quartile (Size of  item)

Q1 = Lower Quartile (Size of  item)

20. Coefficient of Quartile Deviation

Where,

Q3 = Upper Quartile (Size of  item)

Q1 = Lower Quartile (Size of  item)

21. Mean Deviation

i) Individual Series:

• Mean Deviation from Mean

• Mean Deviation from Median

• Alternate Method

Where,

âˆ‘|D| = Sum of absolute deviations from Assumed Mean

A = Assumed Mean

âˆ‘fB = Number of Values from actual mean

âˆ‘fA = Number of values below actual mean including actual mean

N = Number of Observations

ii) Discrete Series:

• Mean Deviation from Mean

• Mean Deviation from Median

Where,

âˆ‘f|D| = Sum of product of frequency and absolute deviations from Assumed Mean

M = Median

N = Number of Observations

iii) Continuous Series:

• Mean Deviation from Mean

• Mean Deviation from Median

Where,

âˆ‘f|D| = Sum of product of frequency and absolute deviations from Assumed Mean

m = Mid-value

M = Median

N = Number of Observations

22. Coefficient of Mean Deviation

• Coefficient of Mean Deviation from Mean

Where,

= Mean Deviation from Mean

• Coefficient of Mean Deviation from Median

Where,

MDM = Mean Deviation from Median

M = Median

23. Standard Deviation

i) Individual Series:

Where,

Ïƒ = Standard Deviation

âˆ‘x2 = Sum total of the squares of deviations from the actual mean

N = Number of pairs of observations

Or

Where,

Ïƒ = Standard Deviation

âˆ‘X2 = Sum total of the squares of observations

= Actual Mean

N = Number of Observations

Where,

Ïƒ = Standard Deviation

âˆ‘d = Sum total of deviations from assumed mean

âˆ‘d2 =  Sum total of squares of deviations

N = Number of pairs of observations

ii) Discrete Series:

Where,

Ïƒ = Standard Deviation

âˆ‘fx2 = Sum total of the squared deviations multiplied by frequency

N = Number of pairs of observations

Or

Where,

Ïƒ = Standard Deviation

âˆ‘fx2 = Sum total of the squared deviations multiplied by frequency

= Actual Mean

N = Number of Observations

Or

Where,

Ïƒ = Standard Deviation

âˆ‘fd = Sum total of deviations multiplied by frequencies

âˆ‘d2 =  Sum total of the squared deviations multiplied by frequencies

N = Number of pairs of observations

Where,

Ïƒ = Standard Deviation

= Sum total of the squared step deviations multiplied by frequencies

=  Sum total of step deviations multiplied by frequencies

N = Number of pairs of observations

iii) Continuous Series:

Ïƒ =

OR

Where,

Ïƒ = Standard Deviation

= Actual Mean

âˆ‘fx2 = Sum total of the deviations of every mid-value of the class intervals multiplied by frequency

N = Number of pair of observations

Ïƒ =

Where,

Ïƒ = Standard Deviation

âˆ‘fd2 = Sum total of the squared deviations multiplied by frequency

âˆ‘fd = Sum total of deviations multiplied by frequency

N = Number of pair of observations

Where,

Ïƒ = Standard Deviation

= Sum total of the squared step deviations multiplied by frequency

= Sum total of step deviations multiplied by frequency

C = Common Factor

N = Number of pair of observations

24. Coefficient of Standard Deviation

Where,

Ïƒ = Standard Deviation

= Arithmetic Mean

25. Combined Standard Deviation

Where,

= Combined Standard Deviation of two groups

=Standard Deviation of first group

= Standard Deviation of second group

= Combined Arithmetic Mean of two groups

= Arithmetic Mean of first group

= Arithmetic Mean of second group

= Number of Observations in the first group

= Number of Observations in the second group

26. Variance

Variance = Ïƒ2

Where,

Ïƒ = Standard Deviation

27. Coefficient of Variation

Where,

C.V. = Coefficient of Variation

Ïƒ = Standard Deviation

= Arithmetic Mean

29. 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

= Standard Deviation of X series

= Standard Deviation of Y series

r = Coefficient of Correlation

Where,

âˆ‘xy = Sum of Product of Deviation of X series and Y series from their respective Means

âˆ‘x = Sum of squares of Deviation of X Series

âˆ‘y = Sum of squares of Deviation of Y Series

r = Coefficient of Correlation

N = Number of Pair of Observations

Where,

âˆ‘XY = Sum of Product of X Series and Y Series

âˆ‘X = Sum of Series X

âˆ‘Y = Sum of Series Y

âˆ‘X = Sum of squares of Series X

âˆ‘Y = Sum of squares of Series Y

r = Coefficient of Correlation

N = Number of Pair of Observations

Where,

N = Number of pair of observations

âˆ‘dx = Sum of deviations of X values from assumed mean

âˆ‘dy = Sum of deviations of Y values from assumed mean

âˆ‘dx2 = Sum of squared deviations of X values from assumed mean

âˆ‘dy2 = Sum of squared deviations of Y values from assumed mean

âˆ‘dxdy = Sum of the products of deviations dx and dy

Where,

N = Number of pair of observations

= Sum of deviations of X values from assumed mean

= Sum of deviations of Y values from assumed mean

= Sum of squared deviations of X values from assumed mean

= Sum of squared deviations of Y values from assumed mean

= Sum of the products of deviations  and

30. Karl Pearson’s Coefficient of Correlation and Covariance

Where,

COV(X,Y) = Covariance of X and Y

âˆ‘xy = Sum of Product of Deviation of X series and Y series from their respective Means

N = Number of pair of observations

= Arithmetic Mean of Series X

= Arithmetic Mean of Series Y

31. Spearman’s Rank Correlation Coefficient

• When ranks are not equal

Where,

rk = Coefficient of rank correlation

D = Rank differences

N = Number of variables

• When ranks are equal

Here,

m1, m2, ……. are the number of times a value has repeated in the given X, Y, …….. series respectively.

32. Unweighted or Simple Index Numbers

Where,

P01 = Index Number of the Current Year

âˆ‘p1 = Total of the current year’s price of all commodities

âˆ‘p0 = Total of the base year’s price of all commodities

33. 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

Where,

âˆ‘RW = Sum of product of Price Relatives (R) and Value Weights (W)

âˆ‘W = Sum of Value Weights

34. Methods of Constructing Consumer Price Index

• Aggregate Expenditure Method

Where,

p1 = Price of goods in the current year

p0 = Price of goods in the base year

q0 = Quantity of goods at base year

• Family Budget Method

Where,

âˆ‘RW = Sum of product of Price Relatives (R) and Value Weights (W)

âˆ‘W = Sum of Value Weights