Math Problem Statement

Calculation the co efficient of the correlation of the following data X- 2,3,4,5,6,7,8 Y-4,5,6,8,9,7,1

Solution

To calculate the coefficient of correlation (denoted as rr) for the given data, we can use the formula for the Pearson correlation coefficient:

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

Where:

  • xx and yy are the values of the variables.
  • nn is the number of data points (in this case, 7).

Step 1: Calculate the necessary sums.

Given data:

  • X=(2,3,4,5,6,7,8)X = (2, 3, 4, 5, 6, 7, 8)
  • Y=(4,5,6,8,9,7,1)Y = (4, 5, 6, 8, 9, 7, 1)

Let's compute the following:

  • x\sum{x}
  • y\sum{y}
  • x2\sum{x^2}
  • y2\sum{y^2}
  • xy\sum{xy}

Step 2: Perform the calculations.

We first calculate these step by step:

  1. Sum of X (x\sum{x}): 2+3+4+5+6+7+8=352 + 3 + 4 + 5 + 6 + 7 + 8 = 35

  2. Sum of Y (y\sum{y}): 4+5+6+8+9+7+1=404 + 5 + 6 + 8 + 9 + 7 + 1 = 40

  3. Sum of x2x^2 (x2\sum{x^2}): 22+32+42+52+62+72+82=4+9+16+25+36+49+64=2032^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 + 8^2 = 4 + 9 + 16 + 25 + 36 + 49 + 64 = 203

  4. Sum of y2y^2 (y2\sum{y^2}): 42+52+62+82+92+72+12=16+25+36+64+81+49+1=2724^2 + 5^2 + 6^2 + 8^2 + 9^2 + 7^2 + 1^2 = 16 + 25 + 36 + 64 + 81 + 49 + 1 = 272

  5. Sum of xyxy (xy\sum{xy}): 2×4+3×5+4×6+5×8+6×9+7×7+8×1=8+15+24+40+54+49+8=1982 \times 4 + 3 \times 5 + 4 \times 6 + 5 \times 8 + 6 \times 9 + 7 \times 7 + 8 \times 1 = 8 + 15 + 24 + 40 + 54 + 49 + 8 = 198

Step 3: Substitute into the formula.

Now, using the values in the correlation formula:

[ r = \frac{7 \times 198 - 35 \times 40}{\sqrt{[7 \times 203 - (35)^2][7 \times 272 - (40)^2]}} ]

Step 4: Simplify the numerator and denominator.

  • Numerator: 7×198=1386,35×40=14007 \times 198 = 1386, \quad 35 \times 40 = 1400 Numerator=13861400=14\text{Numerator} = 1386 - 1400 = -14

  • Denominator: 7×203=1421,352=122514211225=1967 \times 203 = 1421, \quad 35^2 = 1225 \quad \Rightarrow \quad 1421 - 1225 = 196 7×272=1904,402=160019041600=3047 \times 272 = 1904, \quad 40^2 = 1600 \quad \Rightarrow \quad 1904 - 1600 = 304 Denominator=196×304=59664244.17\text{Denominator} = \sqrt{196 \times 304} = \sqrt{59664} \approx 244.17

Step 5: Calculate the final value.

r=14244.170.0573r = \frac{-14}{244.17} \approx -0.0573

Thus, the correlation coefficient is approximately -0.057.


Would you like a detailed breakdown of any specific part of the process, or is there another concept you'd like to explore?


Here are 5 related questions:

  1. What does a negative correlation coefficient tell us about the relationship between X and Y?
  2. How do you interpret a correlation coefficient close to 0?
  3. What is the significance of a Pearson correlation coefficient in statistical analysis?
  4. Can correlation be used to prove causation? Why or why not?
  5. How do you calculate the slope of a regression line from a correlation coefficient?

Tip: Always double-check the units of your data. A correlation tells you about the relationship between two variables, but not their absolute values.

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Math Problem Analysis

Mathematical Concepts

Statistics
Correlation Coefficient
Pearson Correlation

Formulas

Pearson correlation coefficient formula: r = (nΣxy - ΣxΣy) / √[(nΣx² - (Σx)²)(nΣy² - (Σy)²)]

Theorems

Pearson's correlation theorem

Suitable Grade Level

Grades 10-12