Correlation

1. Bounded Range: The numerical value of $r$ lies strictly between $-1$ and $+1$ ($-1 \le r \le +1$). A value of $+1$ indicates perfect positive linear correlation; $-1$ indicates perfect negative linear correlation; $0$ indicates no linear correlation.
   2. Pure Unit-Free Number: $r$ is a dimensionless ratio completely independent of the physical units of measurement used for $X$ and $Y$.
   3. Independence of Origin and Scale: Changing the origin (adding or subtracting a constant from $X$ or $Y$) or changing the scale (multiplying or dividing $X$ or $Y$ by a positive constant) leaves the correlation coefficient $r$ completely unchanged.

Using Karl Pearson's actual mean deviation formula:

$$r = \frac{\Sigma x y}{\sqrt{\Sigma x^2 \cdot \Sigma y^2}}$$

Substitute the given values:

$$r = \frac{120}{\sqrt{144 \times 225}} = \frac{120}{12 \times 15} = \frac{120}{180} = \frac{2}{3} \approx \mathbf{+0.67}$$

Comment: There is a moderate-to-high degree of positive linear correlation between the two variables.

• (a) Untied Ranks:

$$r_s = 1 - \frac{6 \Sigma D^2}{N(N^2 - 1)}$$

Where $D = R_1 - R_2$ (difference between ranks), and $N$ is the number of pairs.
• (b) Tied / Repeated Ranks:

$$r_s = 1 - \frac{6 \left[ \Sigma D^2 + \Sigma \frac{m(m^2 - 1)}{12} \right]}{N(N^2 - 1)}$$

Where $m$ is the number of times an individual rank value is repeated.

• Correlation: Measures the mathematical degree and direction of simultaneous linear co-variation between two variables. It simply observes that when $X$ changes, $Y$ also tends to change.
• Causation: Proves that change in variable $X$ directly causes the change in variable $Y$ through an established causal mechanism.
• Spurious Correlation Example: High positive correlation between ice cream sales and drowning deaths. Neither causes the other; both variables increase independently due to a common third factor—hot summer weather.

A Scatter Diagram is a graphical representation where paired observations $(X_i, Y_i)$ are plotted as individual dots on a Cartesian coordinate plane.
• (a) Perfect Positive ($r = +1$): All dots lie exactly on a single straight line sloping upward from lower-left to upper-right.
• (b) Perfect Negative ($r = -1$): All dots lie exactly on a single straight line sloping downward from upper-left to lower-right.
• (c) Zero Correlation ($r = 0$): Dots are scattered chaotically across the entire graph in a circular or diffuse cloud with zero linear slope.

The new correlation coefficient will be exactly $r = 0.85$.
Reason: Karl Pearson's coefficient of correlation is mathematically independent of change of origin (adding 10 to $X$) and independent of change of scale (multiplying $Y$ by 5). Since 5 is a positive constant, neither the magnitude nor the sign of $r$ is altered.

Spearman's Rank Correlation is preferred when:
1. Data consists of qualitative attributes (e.g., honesty, beauty, leadership, intelligence) that cannot be directly measured on a physical numerical scale but can easily be ranked.
2. The dataset contains extreme outliers that would severely distort Karl Pearson's parametric calculations.
3. The relationship between variables is non-linear but monotonic.

Differences in ranks ($D = R_1 - R_2$):
• Candidate 1: $1 - 5 = -4 \implies D^2 = 16$
• Candidate 2: $2 - 4 = -2 \implies D^2 = 4$
• Candidate 3: $3 - 3 = 0 \implies D^2 = 0$
• Candidate 4: $4 - 2 = +2 \implies D^2 = 4$
• Candidate 5: $5 - 1 = +4 \implies D^2 = 16$

$$\Sigma D^2 = 16 + 4 + 0 + 4 + 16 = 40, \quad N = 5$$

Formula: $r_s = 1 - \frac{6 \Sigma D^2}{N(N^2 - 1)} = 1 - \frac{6 \times 40}{5(25 - 1)} = 1 - \frac{240}{5 \times 24} = 1 - \frac{240}{120} = 1 - 2 = \mathbf{-1}$
Conclusion: There is perfect disagreement (perfect negative rank correlation) between the two judges!