The following information is obtained to study the relationship between the advertisement cost and the sales o

Question 34 of 42

The following information is obtained to study the relationship between the advertisement cost and the sales of electric fans of the companies manufacturing electric fans. Find the correlation coefficient between advertisement cost and sales by Karl-Pearson's method :

Company	A	B	C	D	E	F
Advertisement cost (lakh ₹)	140	120	80	100	80	180
Sales of electric fans (crore ₹)	35	45	15	40	20	50

Gujarat GsebGujarat Board (GSEB) HSC Commerce Board 2025Subjective· 5mImportance★★★★★

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Using the raw-sum formula, $r = \dfrac{n\Sigma xy - \Sigma x \Sigma y}{\sqrt{[n\Sigma x^2 - (\Sigma x)^2][n\Sigma y^2 - (\Sigma y)^2]}} = \dfrac{13100}{16226.21} \approx 0.81$ .

GSEB Class-12 Statistics, Linear Correlation (Karl Pearson):

Let $x$ = advertisement cost, $y$ = sales, $n = 6$ .

Company	$x$	$y$	$x^2$	$y^2$	$xy$
A	140	35	19600	1225	4900
B	120	45	14400	2025	5400
C	80	15	6400	225	1200
D	100	40	10000	1600	4000
E	80	20	6400	400	1600
F	180	50	32400	2500	9000
Total	700	205	89200	7975	26100

Karl Pearson's coefficient:

$r = \frac{n\Sigma xy - \Sigma x \, \Sigma y}{\sqrt{[\,n\Sigma x^2 - (\Sigma x)^2\,]\,[\,n\Sigma y^2 - (\Sigma y)^2\,]}}$

Numerator:

$n\Sigma xy - \Sigma x \Sigma y = 6(26100) - (700)(205) = 156600 - 143500 = 13100$

Denominator terms: …

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Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

CBSE 2026Set MARCH1 mark
$Q. Define correlation coefficient.$
›Reveal solutionSolution

$r=\dfrac{\text{Cov}(x,y)}{\sigma_x\sigma_y}$ measures the degree and direction of linear relationship, $-1\le r\le 1$ .

Karl Pearson's coefficient of correlation between two variables $X$ and $Y$ is defined as the ratio of their covariance to the product of their standard deviations:

$r=\frac{\text{Cov}(x,y)}{\sigma_x\,\sigma_y}=\frac{\Sigma(x-\bar x)(y-\bar y)}{\sqrt{\Sigma(x-\bar x)^2\,\Sigma(y-\bar y)^2}}.$ …

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CBSE 2024Set MARCH1 markMCQ
$Q. Correlation co-efficient lies between :$
(a) $-1$ to $0$
(b) $0$ to $\infty$
(c) $-1$ to $\infty$
(d) $-1$ to $+1$
›Reveal solutionSolution

The correlation coefficient always lies between $-1$ and $+1$ .

Karl Pearson's coefficient of correlation $r$ measures the strength and direction of the linear relationship between two variables and is a pure number with no unit. It is bounded:

$-1\le r\le +1.$ …

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CBSE 2023Set MARCH1 markMCQ
$Q. What does the numerator indicate in the formula for calculating correlation coefficient by Karl Pearson's method?$
(a) (A) Product of variance of $X$ and $Y$
(b) (B) Covariance of $X$ and $Y$
(c) (C) Variance of $X$
(d) (D) Variance of $Y$
›Reveal solutionSolution

Karl Pearson's formula is $r=\dfrac{\text{cov}(x,y)}{S_x\,S_y}$ , so the numerator is the covariance of $X$ and $Y$ . Option (B).

The Karl Pearson coefficient of correlation is defined as

$r=\frac{\text{cov}(x,y)}{S_x\cdot S_y}=\frac{\tfrac{1}{n}\sum (x-\bar x)(y-\bar y)}{S_x\,S_y}.$ …

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CBSE 2022Set MARCH1 markMCQ
$Q. If the values of two variables move in opposite direction then the correlation is said to be :$
(a) Perfect positive
(b) Negative
(c) No correlation
(d) Positive
›Reveal solutionSolution
Opposite-direction movement of two variables indicates negative correlation.

The direction of movement decides the sign of correlation:

Same direction (both increase or both decrease together): positive correlation. …
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CBSE 2022Set MARCH1 markMCQ
$Q. Example for positive correlation is :$
(a) Repayment period and EMI
(b) Income and expenditure
(c) Weight and Income
(d) Price and demand
›Reveal solutionSolution
Income and expenditure move in the same direction, an example of positive correlation.

A positive correlation exists when both variables increase (or decrease) together. Examine the options:

(a) Repayment period and EMI: a longer period lowers the EMI — opposite direction (negative).

(b) Income and expenditure: as income rises, expenditure usually rises too — same direction (positive).

(c) Weight and Income: no logical relationship — essentially no correlation. …
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CBSE 2020Set MARCH1 markMCQ
$Q. If two variables move in decreasing direction then the correlation is :$
(a) negative
(b) positive
(c) perfect negative
(d) no correlation
›Reveal solutionSolution
If two variables move together in the same direction — both increasing or, as here, both decreasing — the correlation between them is positive.

The sign of correlation is decided by the direction in which the variables move together:

Positive correlation: both variables change in the same direction (both increase together, or both decrease together). …
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