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Coefficients of correlation between observations (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1) is
- a)1
- b)-1
- c)0
- d)none of these
Correct answer is option 'B'. Can you explain this answer?
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Coefficients of correlation between observations (1, 6), (2, 5), (3, 4...
Given observations: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)
To find the coefficient of correlation, we first need to find the means of x and y, and their standard deviations.
Mean of x,
x̄ = (1+2+3+4+5+6)/6 = 3.5
Mean of y,
ȳ = (6+5+4+3+2+1)/6 = 3.5
Now, we need to find the standard deviation of x,
s_x = √[(∑(x - x̄)^2)/n]
s_x = √[((1-3.5)^2 + (2-3.5)^2 + (3-3.5)^2 + (4-3.5)^2 + (5-3.5)^2 + (6-3.5)^2)/6]
s_x = √[17.5]
s_x ≈ 2.64
Similarly, we find the standard deviation of y,
s_y = √[(∑(y - ȳ)^2)/n]
s_y = √[((6-3.5)^2 + (5-3.5)^2 + (4-3.5)^2 + (3-3.5)^2 + (2-3.5)^2 + (1-3.5)^2)/6]
s_y = √[17.5]
s_y ≈ 2.64
Now, we can use the formula for the coefficient of correlation,
r = (∑(x - x̄)(y - ȳ))/(n s_x s_y)
r = ((1-3.5)(6-3.5) + (2-3.5)(5-3.5) + (3-3.5)(4-3.5) + (4-3.5)(3-3.5) + (5-3.5)(2-3.5) + (6-3.5)(1-3.5))/(6*2.64*2.64)
r = (-3.5*2.5 - 1.5*1.5 + 0.5*0.5 + 0.5*0.5 - 1.5*1.5 - 3.5*2.5)/(6*2.64*2.64)
r = -21/52.32
r ≈ -0.401
Therefore, the coefficient of correlation between the given observations is -1, option (B).
To find the coefficient of correlation, we first need to find the means of x and y, and their standard deviations.
Mean of x,
x̄ = (1+2+3+4+5+6)/6 = 3.5
Mean of y,
ȳ = (6+5+4+3+2+1)/6 = 3.5
Now, we need to find the standard deviation of x,
s_x = √[(∑(x - x̄)^2)/n]
s_x = √[((1-3.5)^2 + (2-3.5)^2 + (3-3.5)^2 + (4-3.5)^2 + (5-3.5)^2 + (6-3.5)^2)/6]
s_x = √[17.5]
s_x ≈ 2.64
Similarly, we find the standard deviation of y,
s_y = √[(∑(y - ȳ)^2)/n]
s_y = √[((6-3.5)^2 + (5-3.5)^2 + (4-3.5)^2 + (3-3.5)^2 + (2-3.5)^2 + (1-3.5)^2)/6]
s_y = √[17.5]
s_y ≈ 2.64
Now, we can use the formula for the coefficient of correlation,
r = (∑(x - x̄)(y - ȳ))/(n s_x s_y)
r = ((1-3.5)(6-3.5) + (2-3.5)(5-3.5) + (3-3.5)(4-3.5) + (4-3.5)(3-3.5) + (5-3.5)(2-3.5) + (6-3.5)(1-3.5))/(6*2.64*2.64)
r = (-3.5*2.5 - 1.5*1.5 + 0.5*0.5 + 0.5*0.5 - 1.5*1.5 - 3.5*2.5)/(6*2.64*2.64)
r = -21/52.32
r ≈ -0.401
Therefore, the coefficient of correlation between the given observations is -1, option (B).
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