How is Karl Pearson's correlation coefficient calculated?

Multiply the deviations dx = X − X̄ and dy = Y − Ȳ for each pair, sum these products, and divide by the square root of the product of the sums of dx² and dy². The document applies this to a seven-pair data set to get r ≈ 0.95.

What is the range of the regression coefficient k in this problem?

Because the product of the regression coefficients equals r² and r² lies between 0 and 1, k must satisfy 0 ≤ k ≤ 0.25.

What does Chebyshev's Theorem guarantee?

For any random variable with mean μ and variance σ², the probability that the value lies within k standard deviations of the mean is at least 1 − 1/k².

Karl Pearson Correlation Coefficient, Regression Bounds and Chebyshev's Theorem

Summary

This study document works through three classic statistics problems: calculating Karl Pearson's correlation coefficient from paired data, showing that the regression coefficient k lies between 0 and 0.25, and stating and proving Chebyshev's Theorem. A student will see how the correlation formula is applied step by step with a calculation table, how the product of regression coefficients relates to r squared, and how Chebyshev's inequality bounds the probability of values falling within k standard deviations of the mean.

Key points

Karl Pearson's coefficient is r = Σdx·dy / √(Σdx² × Σdy²), where dx = X − X̄ and dy = Y − Ȳ.
For the given data, the means are X̄ = 10 and Ȳ = 9, and the resulting correlation is r ≈ 0.95.
The product of the two regression coefficients equals r², so with bxy = 4 and byx = k, we get 4k = r².
Since 0 ≤ r² ≤ 1, it follows that 0 ≤ k ≤ 0.25.
Chebyshev's Theorem states that P(|X − μ| < kσ) ≥ 1 − 1/k², proved here using Markov's inequality.
Within 2 standard deviations at least 75% of values lie, and within 3 standard deviations at least 88.89%.

Test yourself

Calculating Karl Pearson Correlation Coefficient

Question 11: Calculate the coefficient of correlation from the following data:

X	12	9	8	10	11	13	7
Y	14	8	6	9	11	12	3

To calculate Karl Pearson's coefficient of correlation (r), we use the formula:

r = ∑dx·dy / √(∑dx² × ∑dy²)

Where:

dx = X - X̄
dy = Y - Ȳ

Step 1: Find the Means

First, calculate the mean of X (X̄) and Y (Ȳ):

X̄ = (12 + 9 + 8 + 10 + 11 + 13 + 7) / 7 = 70 / 7 = 10

Ȳ = (14 + 8 + 6 + 9 + 11 + 12 + 3) / 7 = 63 / 7 = 9

Step 2: Prepare the Calculation Table

X	Y	dx = X - 10	dy = Y - 9	dx²	dy²	dx·dy
12	14	2	5	4	25	10
9	8	-1	-1	1	1	1
8	6	-2	-3	4	9	6
10	9	0	0	0	0	0
11	11	1	2	1	4	2
13	12	3	3	9	9	9
7	3	-3	-6	9	36	18
Total				28	84	46

Step 3: Substitute into the Formula

Substitute the sums from the table into the correlation formula:

r = 46 / √(28 × 84)
r = 46 / √(2352)
r = 46 / 48.50 ≈ 0.949

Answer

The coefficient of correlation is r ≈ 0.95.

Proving Regression Line Coefficients Range

Question 12: Given that x = 4y + 5 and y = kx + 4 are the regression lines of x on y and y on x respectively, show that 0 ≤ k ≤ 0.25.

Solution

The regression coefficients are:

Regression coefficient of x on y: b_xy = 4
Regression coefficient of y on x: b_yx = k

The product of the two regression coefficients is equal to the square of the correlation coefficient:

b_xy × b_yx = r²

Hence,

4k = r²

Since the correlation coefficient satisfies -1 ≤ r ≤ 1, we have:

0 ≤ r² ≤ 1

Therefore,

0 ≤ 4k ≤ 1

Dividing throughout by 4, we get:

0 ≤ k ≤ 1/4 or 0 ≤ k ≤ 0.25.

State and Prove Chebyshev's Theorem

Question 6: State and prove Chebyshev's Theorem.

Proof

Let X be a random variable with mean μ and variance σ².

By Markov's Inequality, for any non-negative random variable:

P((X - μ)² ≥ k²σ²) ≤ E[(X - μ)²] / (k²σ²)

Since E[(X - μ)²] = σ², we get:

P((X - μ)² ≥ k²σ²) ≤ σ² / (k²σ²) = 1 / k²

But we know that:

(X - μ)² ≥ k²σ² ⇔ |X - μ| ≥ kσ

Hence,

P(|X - μ| ≥ kσ) ≤ 1 / k²

Taking the complement of the probability:

P(|X - μ| < kσ) = 1 - P(|X - μ| ≥ kσ) ≥ 1 - 1 / k²

Thus,

P(|X - μ| < kσ) ≥ 1 - 1 / k²

Hence, Chebyshev's Theorem is proved.

Important Results

Within 2 standard deviations (k = 2): 1 - 1/2² = 3/4 = 75%
Within 3 standard deviations (k = 3): 1 - 1/3² = 8/9 ≈ 88.89%