Important Formulas in Statistics for Economics | Class 11

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Chapter: Organisation of Data

1. Width of Class Interval

$Widht~of~Class~Interval=\frac{Largest~Observation-Smallest~Observation}{Number~of~Classes~Desired}$

2. Mid-Point or Mid-Value

$Mid-Point/Mid-Value=\frac{Lower~Class~Limit+Upper~Class~Limit}{2}$

Chapter: Diagrammatic Presentation of Data

1. Conversion of percentage into degrees in Pie Diagram

$1\%=\frac{360\degree}{100}=3.6\degree$

2. Adjustment Factor for any Class (Histogram)

$Adjustment~Factor~for~any~Class=\frac{Class~Interval~of~the~Concerned~Class}{Lowest~Class~Interval}$

$Adjustment~Factor~for~any~Class=\frac{Width~of~the~Class}{Width~of~the~Lowest~Class}$

Chapter: Measures of Central Tendency: Arithmetic Mean

1. Arithmetic Mean

i) Individual Series:

Direct Method

$\bar{X}=\frac{\sum{X}}{N}$

Where,

$\bar{X}=Arithmetic~Mean$

$\sum{X}=Sum~of~all~the~values~of~items$

N = Total Number of Items

Short-cut Method

$\bar{X}=A+\frac{\sum{d}}{N}$

Where,

$\bar{X}=Arithmetic~Mean$

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

Step Deviation Method

$\bar{X}=A+\frac{\sum{d^\prime}}{N}\times{C}$

Where,

$\bar{X}=Arithmetic~Mean$

A = Assumed Mean

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

$d^\prime=\frac{X - A}{C}$ (Step Deviations; i.e., deviations divided by common factor)

$\sum{d^\prime}=Sum~of~Step~Deviations$

C = Common Factor

N = Total Number of Items

ii) Discrete Series:

Direct Method

$\bar{X}=\frac{\sum{fX}}{\sum{f}}$

Where,

$\bar{X}=Arithmetic~Mean$

∑fX = Sum of the product of variables with the respective frequencies

∑f = Total Number of Items

Short-cut Method

$\bar{X}=A+\frac{\sum{fd}}{\sum{f}}$

Where,

$\bar{X}=Arithmetic~Mean$

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

Step Deviation Method

$\bar{X}=A+\frac{\sum{fd^\prime}}{\sum{f}}\times{C}$

Where,

$\bar{X}=Arithmetic~Mean$

A = Assumed Mean

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

$d^\prime=\frac{X - A}{C}$ (Step Deviations; i.e., deviations divided by common factor)

$\sum{fd^\prime}=Sum~of~product~of~Step~Deviations~with~the~respective~frequencies$

C = Common Factor

∑f = Total Number of Items

iii) Continuous Series:

Direct Method

$\bar{X}=\frac{\sum{fm}}{\sum{f}}$

Where,

$\bar{X}=Arithmetic~Mean$

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

∑f = Total Number of Items

Short-cut Method

$\bar{X}=A+\frac{\sum{fd}}{\sum{f}}$

Where,

$\bar{X}=Arithmetic~Mean$

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

Step Deviation Method

$\bar{X}=A+\frac{\sum{fd^\prime}}{\sum{f}}\times{C}$

Where,

$\bar{X}=Arithmetic~Mean$

A = Assumed Mean

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

$d^\prime=\frac{m - A}{C}$ (Step Deviations; i.e., deviations divided by common factor)

$\sum{fd^\prime}=Sum~of~product~of~Step~Deviations~with~the~respective~frequencies$

C = Common Factor

∑f = Total Number of Items

2. Charlier’s Accuracy Check

For Short-cut Method

∑f(d + 1) = ∑fd + ∑f

For Step Deviation Method

$\sum{f(d^\prime+1)}=\sum{fd^\prime}+\sum{f}$

3. Missing Value

i) Individual Series:

$\bar{X}=\frac{X_1+X_2+.....................+X_{n-1}+X_n}{N}$

Where,

X₁, X₂, ………………… X_n-1 = Given Values

X_n = Missing Value

ii) Discrete Series:

$\bar{X}=\frac{\sum{fX}}{\sum{f}}$

iii) Continuous Series:

$\bar{X}=\frac{\sum{fm}}{\sum{f}}$

4. Combined Mean

$\bar{X}_{1,2}=\frac{N_1\bar{X}_1+N_2\bar{X}_2}{N_1+N_2}$

Where,

$\bar{X}_{1,2}=Combined~Mean$

$\bar{X}_1=Arithmetic~mean~of~first~distribution$

$\bar{X}_2=Arithmetic~mean~of~second~distribution$

N₁ = Number of Items of first distribution

N₂ = Number of Items of second distribution

5. Corrected Mean

$Correct~\bar{X}=\frac{\sum{X}(Wrong)+Correct~Value-Incorrect~Value}{N}$

6. Weighted Arithmetic Mean

$\bar{X_w}=\frac{\sum{XW}}{\sum{W}}$

Where,

$\bar{X_w} =Weighted~Mean$

∑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:

$Median(M)=Size~of~[\frac{N+1}{2}]^{th}~item$

Where,

N = Number of Items

If the Number of Items is Even

$Median(M)=\frac{Size~of~[\frac{N}{2}]^{th}~item+Size~of~[\frac{N}{2}+1]^{th}~item}{2}$

ii) Discrete Series:

$Median(M)=Size~of~[\frac{N+1}{2}]^{th}~item$

Where,

N = Total of Frequency

Find out the value of $[\frac{N+1}{2}]^{th}~item$ Locate the cumulative frequency which is equal to higher than $[\frac{N+1}{2}]^{th}~item$ and then find the value corresponding to this cf. This value will be the Median value of the series.

iii) Continuous Series:

$Median=l_1+\frac{\frac{N}{2}-c.f.}{f}\times{i}$

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:

$Lower~Quartile(Q_1)=Size~of~[\frac{N+1}{4}]^{th}~item$

$Upper~Quartile(Q_3)=Size~of~3[\frac{N+1}{4}]^{th}~item$

Where,

N = Number of Items

ii) Discrete Series:

$Lower~Quartile(Q_1)=Size~of~[\frac{N+1}{4}]^{th}~item$

$Upper~Quartile(Q_3)=Size~of~3[\frac{N+1}{4}]^{th}~item$

Where,

N = Cumulative Frequency

iii) Continuous Series:

$Lower~Quartile(Q_1)=l_1+\frac{\frac{N}{4}-c.f.}{f}\times{i}$

$Upper~Quartile(Q_3)=l_1+\frac{\frac{3N}{4}-c.f.}{f}\times{i}$

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:

$Z=l_1+\frac{f_1-f_0}{2f_1-f_0-f_2}\times{i}$

Where,

Z = Value of Mode

$l_1$ = lower limit of the modal class

$f_1$ = frequency of modal class

f_0 = frequency of pre-modal class

$f_2$ = frequency of the next higher class or post-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

$Coefficient~of~Range=\frac{Largest~Item(L)-Smallest~Item(S)}{Largest~Item(L)+Smallest~Item(S)}$

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

$Quartile~Deviation=\frac{Q_3-Q_1}{2}$

Where,

Q₃ = Upper Quartile (Size of $3[\frac{N+1}{4}]^{th}$ item)

Q₁ = Lower Quartile (Size of $[\frac{N+1}{4}]^{th}$ item)

4. Coefficient of Quartile Deviation

$Coefficient~of~Quartile~Deviation=\frac{Q_3-Q_1}{Q_3+Q_1}$

Where,

Q₃ = Upper Quartile (Size of $3[\frac{N+1}{4}]^{th}$ item)

Q₁ = Lower Quartile (Size of $[\frac{N+1}{4}]^{th}$ item)

5. Mean Deviation

i) Individual Series:

Mean Deviation from Mean

$Mean~Deviation~from~Mean(MD_{\bar{X}})=\frac{\sum{|X-\bar{X}|}}{N}=\frac{\sum{|D|}}{N}$

Mean Deviation from Median

$Mean~Deviation~from~Median(MD_M)=\frac{\sum{|X-M|}}{N}=\frac{\sum{|D|}}{N}$

Alternate Method

$Mean~Deviation~from~Assumed~Mean=\frac{\sum|D|+(\bar{X}-A)(\sum{f_B}-\sum{f_A})}{N}$

Where,

∑|D| = Sum of absolute deviations from Assumed Mean

$\bar{X}=Actual~Mean$

A = Assumed Mean

∑f_B = Number of Values from actual mean

∑f_A = Number of values below actual mean including actual mean

N = Number of Observations

ii) Discrete Series:

Mean Deviation from Mean

$Mean~Deviation~from~Mean(MD_{\bar{X}})=\frac{\sum{f|X-\bar{X}|}}{N}=\frac{\sum{f|D|}}{N}$

Mean Deviation from Median

$Mean~Deviation~from~Median(MD_M)=\frac{\sum{f|X-M|}}{N}=\frac{\sum{f|D|}}{N}$

Where,

M = Median

iii) Continuous Series:

Mean Deviation from Mean

$Mean~Deviation~from~Mean(MD_{\bar{X}})=\frac{\sum{f|m-\bar{X}|}}{N}=\frac{\sum{f|D|}}{N}$

Mean Deviation from Median

$Mean~Deviation~from~Median(MD_M)=\frac{\sum{f|m-M|}}{N}=\frac{\sum{f|D|}}{N}$

Where,

M = Median

6. Coefficient of Mean Deviation

Coefficient of Mean Deviation from Mean

$Coefficient~of~Mean~Deviation~from~Mean(\bar{X})=\frac{MD_{\bar{X}}}{\bar{X}}$

Coefficient of Mean Deviation from Median

$Coefficient~of~Mean~Deviation~from~Median(M)=\frac{MD_{M}}{M}$

7. Standard Deviation

i) Individual Series:

Actual Mean Method

$\sigma=\sqrt{\frac{\sum{x^2}}{N}}$

Where,

σ = Standard Deviation

∑x² = Sum total of the squares of deviations from the actual mean

N = Number of pairs of observations

Direct Method

$\sigma=\sqrt{\frac{\sum{X^2}}{N}-(\bar{X})^2}$

$=\sqrt{\frac{\sum{X^2}}{N}-(\frac{\sum{X}}{N})^2}$

Where,

σ = Standard Deviation

∑X² = Sum total of the squares of observations

$\bar{X}$ = Actual Mean

N = Number of Observations

Short-cut or Assumed Mean Method

$\sigma=\sqrt{\frac{\sum{d^2}}{N}-(\frac{\sum{d}}{N})^2}$

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:

Actual Mean Method

$\sigma=\sqrt{\frac{\sum{fx^2}}{N}}$

Where,

σ = Standard Deviation

∑fx² = Sum total of the squared deviations multiplied by frequency

N = Number of pairs of observations

Direct Method

$\sigma=\sqrt{\frac{\sum{fX^2}}{N}-(\bar{X})^2}$

$=\sqrt{\frac{\sum{fX^2}}{N}-(\frac{\sum{fX}}{N})^2}$

Where,

σ = Standard Deviation

∑fx² = Sum total of the squared deviations multiplied by frequency

$\bar{X}$ = Actual Mean

N = Number of Observations

Short-cut Method or Assumed Mean Method

$\sigma=\sqrt{\frac{\sum{fd^2}}{N}-(\frac{\sum{fd}}{N})^2}$

$=\sqrt{\frac{\sum{fd^2}}{N}-(\frac{\sum{fd}}{N})^2}$

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

Step Deviation Method

$\sigma=\sqrt{\frac{\sum{fd^\prime{^2}}}{N}-(\frac{\sum{fd^\prime}}{N})^2}\times{C}$

Where,

σ = Standard Deviation

$\sum{fd^\prime{^2}}$ = Sum total of the squared step deviations multiplied by frequencies

$\sum{fd^\prime}$ = Sum total of step deviations multiplied by frequencies

N = Number of pairs of observations

iii) Continuous Series:

Direct Method

σ = $\sqrt{\frac{\sum{\\fx^{2}}}{N}}$

$\sqrt{\frac{\sum f(X - \bar{X})^{2}}{N}}$

Where,

σ = Standard Deviation

$\bar{X}$ = Actual Mean

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

N = Number of pair of observations

Short-cut Method

σ = $\sqrt{{\frac{\sum{\\fd^{2}}}{N}}-({\frac{\sum{fd}}{N}})^2}$

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

Step Deviation Method

$σ=\sqrt{{\frac{\sum{\\fd'^{2}}}{N}}-({\frac{\sum{fd'}}{N}})^2}\times{C}$

σ = Standard Deviation

$\sum{fd^\prime}^2$ = Sum total of the squared step deviations multiplied by frequency

$\sum{fd^\prime}$ = Sum total of step deviations multiplied by frequency

C = Common Factor

N = Number of pair of observations

8. Coefficient of Standard Deviation

$Coefficient~of~Standard~Deviation(\sigma)=\frac{\sigma}{\bar{X}}$

9. Combined Standard Deviation

$\sigma_{1,2}=\sqrt{\frac{N_1\sigma_1^2+N_2\sigma_2^2+N_1d_1^2+N_2d_2^2}{N_1+N_2}}$

Where,

$\sigma_{1,2}$ = Combined Standard Deviation of two groups

$\sigma_{1}$ =Standard Deviation of first group

$\sigma_{2}$ = Standard Deviation of second group

$d_1^2=(\bar{X_1}-\bar{X}_{1,2})^2$

$d_2^2=(\bar{X_2}-\bar{X}_{1,2})^2$

$\bar{X}_{1,2}$ = Combined Arithmetic Mean of two groups

$\bar{X}_1$ = Arithmetic Mean of first group

$\bar{X}_2$ = Arithmetic Mean of second group

$N_1$ = Number of Observations in the first group

$N_2$ = Number of Observations in the second group

10. Variance

Variance = σ²

11. Coefficient of Variation

$Coefficient~of~Variation(C.V.)=\frac{\sigma}{\bar{X}}\times{100}$

Where,

C.V. = Coefficient of Variation

σ = Standard Deviation

$\bar{X}$ = Arithmetic Mean

Chapter: Correlation

1. Degree of Correlation

2. Karl Pearson’s Coefficient of Correlation

$Karl~Pearson's~Coefficient~of~Correlation(r)=\frac{Sum~of~Products~of~Deviations~from~their~respective~means}{Number~of~Pairs\times{Standard~Deviations~of~both~Series}}$

Or, $r=\frac{\sum{xy}}{N\times{\sigma_x}\times{\sigma_y}}$

Or, $r=\frac{\sum{xy}}{N}\times{\frac{1}{\sigma_x}}\times{\frac{1}{\sigma_y}}$

Or, $r=\frac{\sum{xy}}{N\times{\sqrt{\frac{\sum{x^2}}{N}}}\times{\sqrt{\frac{\sum{y^2}}{N}}}}$

Or, $r=\frac{\sum{xy}}{\sqrt{\sum{x^2}\times{\sum{y^2}}}}$

Where,

N = Number of Pair of Observations

x = Deviation of X series from Mean $(X-\bar{X})$

y = Deviation of Y series from Mean $(Y-\bar{Y})$

$\sigma_x$ = Standard Deviation of X series $(\sqrt{\frac{\sum{x^2}}{N}})$

$\sigma_y$ = Standard Deviation of Y series $(\sqrt{\frac{\sum{y^2}}{N}})$

r = Coefficient of Correlation

Actual Mean Method

$r=\frac{\sum{xy}}{\sqrt{\sum{x^2}\times{\sum{y^2}}}}$

Direct Method

$r=\frac{N\sum{XY}-\sum{X}.\sum{Y}}{\sqrt{N\sum{X^2}-(\sum{X})^2}{\sqrt{N\sum{Y^2}-(\sum{Y})^2}}}$

Short-Cut Method/Assumed Mean Method

$r=\frac{N\sum{dxdy}-\sum{dx}.\sum{dy}}{\sqrt{N\sum{dx^2}-(\sum{dx})^2}{\sqrt{N\sum{dy^2}-(\sum{dy})^2}}}$

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

Step Deviation Method

$r=\frac{N\sum{dx^\prime{dy^\prime}}-\sum{dx^\prime}.\sum{dy^\prime}}{\sqrt{N\sum{dx^\prime{^2}}-(\sum{dx^\prime})^2}{\sqrt{N\sum{dy^\prime{^2}}-(\sum{dy^\prime})^2}}}$

Where,

N = Number of pair of observations

$\sum{dx^\prime}$ = Sum of deviations of X values from assumed mean

$\sum{dy^\prime}$ = Sum of deviations of Y values from assumed mean

$\sum{dx^\prime{^2}}$ = Sum of squared deviations of X values from assumed mean

$\sum{dy^\prime{^2}}$ = Sum of squared deviations of Y values from assumed mean

$\sum{dx^\prime{dy^\prime}}$ = Sum of the products of deviations $(dx^\prime)$ and $(dy^\prime)$

3. Karl Pearson’s Coefficient of Correlation and Covariance

$COV(X,~Y)=\frac{\sum{(X-\bar{X})(Y-\bar{Y})}}{N}=\frac{\sum{xy}}{N}$

With Covariance formula, the formula for r (coefficient of correlation) can be written as:

$r=\frac{\sum{xy}}{N\times{\sigma_x}\times{\sigma_y}}$

Or, $r=\frac{\sum{xy}}{N}\times{\frac{1}{\sigma_x}}\times{\frac{1}{\sigma_y}}$

Or, $r=\frac{\sum{xy}}{N\times{\sqrt{\frac{\sum{x^2}}{N}}}\times{\sqrt{\frac{\sum{y^2}}{N}}}}$

Or, $r=\frac{\sum{xy}}{\sqrt{\sum{x^2}\times{\sum{y^2}}}}$

4. Spearman’s Rank Correlation Coefficient

When ranks are not equal

$r_k=1-\frac{6\sum{D^2}}{N^3-N}$

Where,

r_k= Coefficient of rank correlation

D = Rank differences

N = Number of variables

When ranks are equal

$r_k=1-\frac{6[\sum D^2+\frac{1}{12}(m_1^3-m_1)+\frac{1}{12}(m_2^3-m_2)+...]}{N^3-N}$

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

Simple Aggregative Method

$P_{01}=\frac{\sum{p_1}}{\sum{p_0}}\times100$

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

Simple Average of Price Relatives Method

$P_{01}=\frac{\sum{(\frac{p_1}{p_0}\times100)}}{N}$

2. Weighted Index Numbers

i) Weighted Aggregative Method

Laspeyre’s Method

$Laspeyre's~Price~Index~(P_{01})=\frac{\sum{p_1q_0}}{\sum{p_0q_0}}\times{100}$

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

Pasche’s Method

$Pasche's~Index~Number~(P_{01})=\frac{\sum{p_1q_1}}{\sum{p_0q_1}}\times{100}$

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

$Fisher's~Price~Index~(P_{01})=\sqrt{\frac{\sum{p_1q_0}}{\sum{p_0q_0}}\times{\frac{\sum{p_1q_1}}{\sum{p_0q_1}}}}\times{100}$

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

$P_{01}=\frac{\sum{RW}}{\sum{W}}$

3. Methods of Constructing Consumer Price Index

Aggregate Expenditure Method

$Consumer~Price~Index=\frac{\sum{p_1q_0}}{\sum{p_0q_0}}\times100$

Family Budget Method

$Consumer~Price~Index=\frac{\sum{RW}}{\sum{W}}$

4. Purchasing Power

$Purchasing~Power=\frac{1}{Consumer~Price~Index}\times100$

5. Real Wages

$Real~Wages=\frac{Money~Wages}{Consumer~Price~Index}\times100$

My Personal Notes

Last Updated : 06 Apr, 2023

Related Articles

Important Formulas in Statistics for Economics | Class 11

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)

Chapter: Measures of Central Tendency: Arithmetic Mean

1. Arithmetic Mean

2. Charlier’s Accuracy Check

3. Missing Value

4. Combined Mean

5. Corrected Mean

6. Weighted Arithmetic Mean

Chapter: Measures of Central Tendency: Median and Mode

1. Median

2. Quartiles

3. Mode

4. Relationship between Mean, Median, and Mode

Chapter: Measures of Dispersion

1. Range

2. Coefficient of Range

3. Quartile Deviation

4. Coefficient of Quartile Deviation

5. Mean Deviation

6. Coefficient of Mean Deviation

7. Standard Deviation

8. Coefficient of Standard Deviation

9. Combined Standard Deviation

10. Variance

11. Coefficient of Variation

Chapter: Correlation

1. Degree of Correlation

2. Karl Pearson’s Coefficient of Correlation

3. Karl Pearson’s Coefficient of Correlation and Covariance

4. Spearman’s Rank Correlation Coefficient

Chapter: Index Number

1. Unweighted or Simple Index Numbers

2. Weighted Index Numbers

3. Methods of Constructing Consumer Price Index

4. Purchasing Power

5. Real Wages

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