Calculating r for a Small Dataset

Find the correlation coefficient for the data: $(1, 2), (2, 4), (3, 5), (4, 4), (5, 5)$

Find the means $\bar{x}$ and $\bar{y}$: $\bar{x} = \frac{1+2+3+4+5}{5} = 3$, $\bar{y} = \frac{2+4+5+4+5}{5} = 4$ = $\bar{x} = 3$, $\bar{y} = 4$

Calculate deviations $(x_i - \bar{x})$ and $(y_i - \bar{y})$: Point $(1,2)$: $(-2, -2)$ Point $(2,4)$: $(-1, 0)$ Point $(3,5)$: $(0, 1)$ Point $(4,4)$: $(1, 0)$ Point $(5,5)$: $(2, 1)$ = Deviations calculated

Calculate products $(x_i - \bar{x})(y_i - \bar{y})$: $(-2)(-2) = 4$ $(-1)(0) = 0$ $(0)(1) = 0$ $(1)(0) = 0$ $(2)(1) = 2$ Sum $= 4 + 0 + 0 + 0 + 2 = 6$ = $\sum(x_i - \bar{x})(y_i - \bar{y}) = 6$

Calculate $\sum(x_i - \bar{x})^2$: $(-2)^2 + (-1)^2 + 0^2 + 1^2 + 2^2 = 4 + 1 + 0 + 1 + 4 = 10$ = $\sum(x_i - \bar{x})^2 = 10$

Calculate $\sum(y_i - \bar{y})^2$: $(-2)^2 + 0^2 + 1^2 + 0^2 + 1^2 = 4 + 0 + 1 + 0 + 1 = 6$ = $\sum(y_i - \bar{y})^2 = 6$

Apply the formula: $r = \frac{6}{\sqrt{10 \cdot 6}} = \frac{6}{\sqrt{60}} = \frac{6}{7.746} \approx 0.775$ = $r \approx 0.77$

Answer: $r \approx 0.77$ (strong positive correlation)

Interpreting Correlation Values

A study finds $r = -0.82$ between hours of TV watched per day and exam scores. Interpret this result.

Identify the sign: $r = -0.82$ is negative = Negative correlation (as one increases, the other decreases)

Assess the strength using $|r|$: $|{-0.82}| = 0.82$, which is in the range $0.80 - 1.00$ = Very strong correlation

Interpret in context: Students who watch more TV tend to score lower on exams = Strong inverse relationship

Add appropriate caveats: Correlation does not imply causation. Other factors may explain both variables. = Association, not causation

Answer: There is a very strong negative correlation ($r = -0.82$). Students who watch more TV tend to have lower exam scores, but this association does not prove that TV causes lower scores.

Finding Coefficient of Determination

If $r = 0.80$ between study hours and test scores, what percentage of variance in test scores is explained by study hours?

Square the correlation coefficient: $r^2 = (0.80)^2 = 0.64$ = $r^2 = 0.64$

Convert to percentage: $0.64 \times 100 = 64\%$ = 64% of variance explained

Interpret the result: 64% of variation in test scores can be attributed to study hours = Strong explanatory power

Answer: $r^2 = 0.64 = 64\%$. This means 64% of the variance in test scores is explained by the linear relationship with study hours. The remaining 36% is due to other factors.

Mistake: Assuming correlation implies causation

Why: A strong correlation between $A$ and $B$ doesn't mean $A$ causes $B$. There could be a third variable affecting both, or the relationship could be coincidental.

Correct: Always say "associated with" or "correlated with" rather than "causes." Look for experimental evidence to establish causation.

Mistake: Ignoring outliers when interpreting $r$

Why: A single outlier can dramatically change the correlation coefficient. $r$ is sensitive to extreme values.

Correct: Always plot the data first. Look for outliers and consider calculating $r$ with and without them.

Mistake: Using $r$ for non-linear relationships

Why: $r$ only measures linear relationships. A perfect curved pattern could have $r = 0$.

Correct: Examine the scatter plot. If the pattern is curved, $r$ is not the right measure. Consider transformations or non-linear models.

Stock Market Diversification

Investment analysts use correlation to build diversified portfolios. Stocks with low or negative correlation balance risk.

If Stock A and Stock B have $r = -0.3$, when A drops, B tends to rise slightly, reducing portfolio volatility.

Medical Research

Researchers use correlation to identify potential risk factors for diseases.

A study finds $r = 0.68$ between blood pressure and risk of heart disease, indicating a strong positive relationship.

Q: What is the difference between correlation and causation?

A: Correlation shows that two variables move together, but causation means one variable directly causes changes in another. Ice cream sales and drowning rates are correlated (both increase in summer), but ice cream doesn't cause drowning - hot weather affects both.

Q: Can $r$ be greater than 1 or less than -1?

A: No. By mathematical definition, $r$ is always between $-1$ and $1$. If you calculate a value outside this range, there's an error in your calculations.

Q: What does $r = 0$ mean?

A: $r = 0$ means there is no linear relationship between the variables. However, there could still be a non-linear relationship (like a parabola or sine curve).