Question

The value of covariance of X and Y is 1, standard deviation of X is 5, standard deviation of Y is 2, then what is the value of Karl
Pearson's coefficient of correlation?​

Answer 1

Answer:

Karl Pearson Correlation Coefficient Formula

The coefficient of correlation rxy between two variables x and y, for the bivariate dataset (xi,yi) where i = 1,2,3…..N; is given by –

r(x,y)=cov(x,y)σxσy

where,

⇒ cov(x,y): the covariance between x and y

– ΣNi=1(xi–x¯)(yi–y¯)N=ΣxiyiN–x¯y¯

Here, barx and y¯ are simply the respective means of the distributions of x and y.

⇒ σx and σy are the standard deviations of the distributions x and y.

– σx=Σ(xi–x¯)2N−−−−−−√=Σx2iN–x¯2−−−−−−√

– σy=Σ(yi–y¯)2N−−−−−−√=Σy2iN–y¯2−−−−−−√

Alternate Formula

If some data is given in the form of a class-distributed frequency distribution, you may use the following formulae –

⇒ cov(x,y): the covariance between x and y

– Σi,jxiyifijN–x¯y¯

Here, x¯ and y¯ are simply the respective means of the distributions of x and y.

⇒ σx and σy are the standard deviations of the distributions x and y.

– σx=Σifiox2iN–x¯2−−−−−−−−√

– σy=Σjfioy2iN–y¯2−−−−−−−−√

where,

xi: The central value of the i’th class of x

yj: The central value of the j’th class of y

fio,fij: Marginal Frequencies of x and y

fij: Frequency of the (i,j)th cell

In any case, the following equality must always hold:

Total frequency = N = Σi,jfij = Σifio = Σjfjo

A Single Formula for Discrete Datasets –

rxy=NΣxiyi–ΣxiΣyiNΣx2i–(Σxi)2−−−−−−−−−−−√NΣy2i–(Σyi)2−−−−−−−−−−−√

Let us understand more about Scatter Diagram here

Properties of the Pearson’s Correlation Coefficient

⇒ r is unit-less. Thus, we may use it to compare association between totally different bivariate distributions as well. For eg – you may compare how much of you not going for a movie is related to your friends not joining you, and to you not being much interested for the movie yourself, both at the same time, with the Pearson’s correlation coefficients obtained from both the cases. In economics therefore, where the cost price or the market shares depend on lots of different factors, this parameter is of utmost importance in ascertaining the connection between various quantities.

⇒ The value of r always lies between +1 and -1. Depending on its exact value, we see the following degrees of association between the variables

Step-by-step explanation:

I HOPE IT WILL BE HELP YOU MARK AS BRAINEST