How To Find The Sample Correlation Coefficient

Correlation Coefficient Formula

In statistics, correlation is a way of establishing the relationship/association between two variables. In other words, the correlation coefficient formula helps in calculating the correlation coefficient which measures the dependency of one variable on the other variable. Correlation is measured numerically using the correlation coefficient. The correlation coefficient lies between -1 and 1. A negative correlation coefficient indicates that the human relationship between two variables is inverse. A positive correlation coefficient indicates that the value of ane variable depends on the other variable directly. A cypher-correlation coefficient indicates that in that location is no correlation between both variables. There are many types of correlation coefficients, among them, the Pearson Correlation Coefficient (PCC) is the almost common ane. Permit us explore how to calculate the correlation coefficient formula for a given population or sample beneath.

What Is the Correlation Coefficient Formula?

The correlation coefficient is a statistical concept. Information technology establishes a relation between predicted and actual values obtained at the end of a statistical experiment. The correlation coefficient formula helps to calculate the relationship betwixt two variables and thus the upshot so obtained explains the carefulness between the predicted and actual values.

Correlation Coefficient Formula

Pearson Correlation Coefficient Formula:

1. Sample Correlation Coefficient

The formula for pearson correlation coefficient for population of sizeN(written equallyρ_{X, Y} ) is given as:

\(\rho_{10, Y}=\frac{\operatorname{cov}(X, Y)}{\sigma_{10} \sigma_{Y}}=\dfrac{\sum_{i=1}^{north}\left(X_{i}-\bar{X}\correct)\left(Y_{i}-\bar{Y}\correct)}{\sqrt{{\Sigma}_{\Sigma_{i}=i}^{north}\left(X_{i}-\bar{Ten}\right)^{two}} \sqrt{{\sum}_{\Sigma_{i}=1}^{n}\left(Y_{i}-\bar{Y}\right)^{two}}}\)

where covis the covariance and (cov(Ten,Y)= \(\frac{\sum_{i=ane}^{Northward}\left(X_{i}-\bar{Ten}\right)\left(Y_{i}-\bar{Y}\right)}{Northward}\), σ_X is standard divergence of Xandσ_Y is standard divergence of Y.

Given X and Y are two random variables.

2. Population Correlation Coefficient

The formula for pearson correlation coefficient for sample of sizedue north (written asr_xy ) is given as:

\(r_{x, y}=\frac{\sum_{i=1}^{north}\left(x_{i}-\bar{x}\correct)\left(y_{i}-\bar{y}\correct)}{\sqrt{\Sigma}_{\Sigma_{i}=1}^{northward}\left(x_{i}-\bar{x}\correct)^{2} \sqrt{\sum}_{\Sigma_{i}=1}^{n}\left(y_{i}-\bar{y}\right)^{2}}\)

wherenis the sample size,x_i&y_iare the i^th sample points andx̄&ȳare the sample ways for the random variables X and Y respectively.

Given X and Y are ii random variables.

3. Linear Correlation Coefficient

It uses pearson's correlation coefficient to determine the linear relationship between two variables. Its value lies between -1 and 1. It is given every bit:

\(r=\frac{n(\Sigma x y)-(\Sigma ten)(\Sigma y)}{\sqrt{\left[n \Sigma ten^{2}-(\Sigma x)^{2}\right]\left[n \Sigma y^{2}-(\Sigma y)^{two}\correct]}}\)

wherenis the sample size,x_i&y_iare the i^thursday sample points andx̄&ȳare the sample means for the random variables x and y respectively.

The sign of r indicates the strength of the linear relationship between the variables.

If r is almost 1, and then the ii variables have a potent linear relationship.
If r is near 0, and then the two variables have no linear relation.
If r is near -1, then the two variables have a weak (negative) linear human relationship.

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Let us see the applications of the correlation coefficient formula in the post-obit section.

Examples using Correlation Coefficient Formula

Case 1.Given the following population data. Find the Pearson correlation coefficient between x and y for this data. (Accept 1√7 equally 0.378)

x	600	800	thou
y	1200	thou	2000

Solution:

To simplify the calculation, we divide both x and y past 100.

x/100	y/100	\(x_{i}-\bar{x}\)	\(y_{i}-\bar{y}\)	\(\left(\mathrm{ten}_{\mathrm{i}}-\overline{\mathrm{x}}\right)^{ii}\)	\(\left(\mathrm{y}_{\mathrm{i}}-\overline{\mathrm{y}}\right)^{2}\)	\((x_i - \bar{x})(y_i - \bar{y})\)
6	12	-two	-ii	four	4	4
viii	10	0	-4	0	16	0
10	20	2	vi	4	36	12
\(\bar{x}=8\)	\(\bar{y}=14\)			\(\Sigma\left(x_{i}-\bar{ten}\correct)^{2}=8\)	\(\Sigma\left(y_{i}-\bar{y}\correct)^{2}=56\)	\(\Sigma(x_i-\bar{x})(y_i - \bar{y}) = 16\)

Using the correlation coefficient formula,

Pearson correlation coefficient for population = \(\frac{\Sigma\left(x_{i}-\bar{x}\right)\left(y_{i}-\bar{y}\right)}{\sqrt{\Sigma\left(x_{i}-\bar{10}\right)^{2} \sum\left(y_{i}-\bar{y}\right)^{2}}}\) = \(\frac{sixteen}{\sqrt{8} \sqrt{56}}\) = \(\frac{ii}{\sqrt{vii}}\) = 0.756

Respond: Pearson correlation coefficient = 0.756

Example 2. A survey was conducted in your city. Given is the following sample data containing a person'southward historic period and their corresponding income. Detect out whether the increment in age has an effect on income using the correlation coefficient formula. (Use i√181 as 0.074 and 1√2091 as 0.07)

Age	25	30	36	43
Income	30000	44000	52000	70000

Solution:

To simplify the calculation, we divide y by 1000.

Age (x_i)	Income/one thousand (y_i/yard)	\(x_{i}-\bar{10}\)	\(y_{i}-\bar{y}\)	\(\left(\mathrm{10}_{\mathrm{i}}-\overline{\mathrm{x}}\right)^{2}\)	\(\left(\mathrm{y}_{\mathrm{i}}-\overline{\mathrm{y}}\right)^{two}\)	\((x_i - \bar{10})(y_i - \bar{y})\)
25	30	-8.5	-19	72.25	361	161.5
30	44	-3.5	-5	12.25	25	17.5
36	52	2.5	3	half dozen.25	9	7.5
43	70	nine.five	21	90.25	441	199.5
\(\bar{x}=33.5\)	\(\bar{y}=49\)			\(\Sigma\left(x_{i}-\bar{ten}\right)^{2}=181\)	\(\Sigma\left(y_{i}-\bar{y}\right)^{2}=836\)	\(\Sigma(x_i-\bar{x})(y_i - \bar{y}) = 386\)

Pearson correlation coefficient for sample = \(\frac{\Sigma\left(x_{i}-\bar{10}\right)\left(y_{i}-\bar{y}\right)}{\sqrt{\Sigma\left(x_{i}-\bar{x}\correct)^{two} \sum\left(y_{i}-\bar{y}\right)^{2}}}\) = \(\frac{386}{\sqrt{181} \sqrt{836}}\) = \(\frac{193}{\sqrt{181} \sqrt{209}}\) = 0.99

Answer: Yes, with the increase in age a person's income increases as well, since the Pearson correlation coefficient between age and income is very close to one.

Example 3: Calculate the Correlation coefficient of given data.

ten	41	42	43	44	45
y	3.2	3.3	3.4	3.5	three.six

Solution:

Hither northward = 5

Permit us find ∑x , ∑y, ∑xy, ∑10 ⁱⁱ, ∑yⁱⁱ

10	y	xy	x²	y^two
41	3.2	131.2	1681	10.24
42	3.3	138.half-dozen	1764	10.89
43	iii.iv	146.2	1849	xi.56
44	3.5	154	1936	12.25
45	3.6	162	2025	12.96
∑x = 215	∑y = 17	∑xy = 732	∑x² = 9255	∑yⁱⁱ = 57.9

values:

∑x = 215

∑x² = 9255

x̄ = 43

∑(x - x̄)² = σσ₁₀=10

Y values:

∑y = 17

∑yⁱⁱ = 57.ix

∑(y - ȳ)² =σσ_y= 0.1

X and Y combined

N = 5

∑((x - x̄)(y - ȳ)) = one

∑xy = 732

R calculation:

r = ∑((x - x̄)(y - ȳ))/√((σσ_x)(σσ_y))

r = one/√((10)(0.ane)) = one

Since r = 1, this indicates pregnant relation between x and y.

FAQs on Correlation Coefficient Formula

What Is Correlation Coefficient Formula in Statistics?

The correlation coefficient formula determines the relationship between two variables in a dataset and thus checks for the exactness between the predicted and actual values.

How To Utilize Correlation Coefficient Formula?

Nosotros can use the coefficient correlation formula to summate the Pearson product-moment correlation,

Stride i: Determine the covariance of the ii given variables.
Stride ii: Summate the standard difference of each variable.
Step 3: Divide the covariance by the product of the standard deviations of ii variables.

What Is due north in the Correlation Coefficient Formula?

In the coefficient correlation formula, n refers to the sample size.

Sample Correlation Coefficient: \(\rho_{X, Y}=\frac{\operatorname{cov}(10, Y)}{\sigma_{X} \sigma_{Y}}=\frac{\Sigma_{i=1}^{Due north}\left(X_{i}-\bar{10}\correct)\left(Y_{i}-\bar{Y}\right)}{\sqrt{\Sigma_{i=1}^{N}\left(X_{i}-\bar{X}\right)^{2}}^{Northward} \sum_{\Sigma_{i}=1}^{N}\left(Y_{i}-\bar{Y}\right)^{2}}\)
Population Correlation Coefficient: \(r_{x, y}=\frac{\sum_{i=i}^{n}\left(x_{i}-\bar{x}\right)\left(y_{i}-\bar{y}\correct)}{\sqrt{\Sigma}_{\Sigma_{i}=1}^{n}\left(x_{i}-\bar{x}\correct)^{2} \sqrt{\sum}_{\Sigma_{i}=1}^{northward}\left(y_{i}-\bar{y}\right)^{2}}\)
Linear Correlation Coefficient: \(r=\frac{northward(\Sigma x y)-(\Sigma x)(\Sigma y)}{\sqrt{\left[north \Sigma x^{2}-(\Sigma x)^{2}\right]\left[n \Sigma y^{ii}-(\Sigma y)^{2}\right]}}\)

What Are the Applications of Correlation Coefficient Formula?

Given below are the most important applications of the coefficient correlation formula:

The coefficient correlation formula helps in the analysis of the given data past quantifying the caste to which two variables are related which further depicts a linear human relationship between two variables.
It is used for financial analysis as it determines the human relationship between data sets in business and thus, in a way support determination making.
It helps a lot in decision-making in various fields every bit information technology helps to understand the strength of the relationship between two different variables.