(iii) non-existent
Reason: Correlation coefficient ($r$) is a pure number and is independent of the units of measurement.

(ii) minus one to plus one
Reason: The value of $r$ always lies between -1 and +1, i.e., $-1 \leq r \leq +1$.

(i) When Y increases X increases
Reason: Positive correlation implies that both variables move in the same direction.

(ii) not linearly related
Reason: $r=0$ strictly indicates the absence of a linear relationship. They might still be related in a non-linear (curvilinear) way.

(iii) Scatter diagram
Reason: A scatter diagram plots all points visually, revealing linear, curvilinear, or irregular patterns. Pearson and Spearman only measure linear/monotonic relationships. [Image of scatter diagram showing non-linear relationship]

(i) more accurate than rank correlation coefficient
Reason: Simple (Pearson’s) correlation uses the actual magnitude of the data, whereas Rank correlation uses only the order (ranks), resulting in some loss of information.

Covariance depends on the units of the variables (e.g., kg-meters), making it difficult to compare relationships between different datasets. Correlation coefficient ($r$) is preferred because it is unit-free (a pure number), allowing for easy comparison of the strength of relationships across different contexts.

No. The mathematical derivation of the correlation coefficient ensures that its value always lies within the range $[-1, 1]$, regardless of the type of data.

No. Correlation only indicates that two variables move together; it does not prove that one causes the other. They could both be influenced by a third variable (spurious correlation). For example, ice cream sales and shark attacks are correlated (both rise in summer), but one does not cause the other.

Rank correlation is more appropriate and precise when the data is qualitative (attributes like beauty, honesty, intelligence) which cannot be measured quantitatively but can be ranked in order of merit.

Not necessarily. Zero correlation ($r=0$) specifically means there is no linear relationship. The variables could still be dependent in a non-linear way (e.g., a circular relationship $X^2 + Y^2 = a^2$). However, if variables are independent, correlation is definitely zero.

No. The simple (Pearson’s) correlation coefficient measures only the strength of the linear relationship. It cannot accurately measure curvilinear or other complex non-linear relationships.

(Student Activity)
Guidance:

• Select two vegetables (e.g., Potato and Onion).
• Record their prices for 7 days.
• Use Pearson’s formula to find $r$.
• If $r > 0$, prices move together (e.g., general inflation). If $r < 0$, one becomes cheaper as the other gets expensive.

(Student Activity)
Guidance:

• Variable X: Height of student.
• Variable Y: Height of benchmate.
• Usually, students of similar heights sit together or become friends, so you might find a positive correlation.

(Student Activity)
Guidance:

• Collect data for National Income (X) and Exports (Y) for the last 10 years from the Economic Survey of India.
• Calculate $r$.
• You will likely find a high positive correlation (near +1) as exports generally grow with national income.

Accurate quantitative measurement is difficult for qualitative variables (attributes) that involve subjective human behavior or characteristics.
Examples:

• Beauty or Attractiveness
• Intelligence (IQ is an approximation, not a perfect measure)
• Honesty and Integrity
• Efficiency
• Social Welfare

• $r = +1$: Perfect Positive Correlation. The variables move in the same direction in a constant proportion. All points on the scatter diagram lie on a straight line rising from left to right.
• $r = -1$: Perfect Negative Correlation. The variables move in opposite directions in a constant proportion. All points lie on a straight line falling from left to right.
• $r = 0$: No Linear Correlation. There is no linear relationship between the variables. They are linearly independent.

They differ because of the nature of data they utilize:

• Pearson’s ($r$): Considers the actual magnitude (numerical value) of the observations. It is sensitive to the gap between values.
• Rank Correlation ($r_k$): Considers only the order (rank) of observations. It ignores the magnitude of differences between values.

Example: If student A scores 90 and B scores 89, their difference is 1. If A scores 90 and B scores 40, the difference is 50. Pearson captures this huge gap; Rank correlation simply treats A as Rank 1 and B as Rank 2 in both cases.

Data Summary: $\sum X = 534$, $\sum Y = 540$, $N=8$.
$\bar{X} = 66.75$, $\bar{Y} = 67.5$.
Using Karl Pearson’s Formula: $r = \frac{\sum xy}{\sqrt{\sum x^2 \times \sum y^2}}$
(where $x = X – \bar{X}$, $y = Y – \bar{Y}$)

Calculation Values:
$\sum x^2 = 155.5$
$\sum y^2 = 174$
$\sum xy = 99$

Result:
$r = \frac{99}{\sqrt{155.5 \times 174}} = \frac{99}{\sqrt{27057}} = \frac{99}{164.49} = \mathbf{0.603}$
Interpretation: There is a moderate positive correlation.

Data:
X: -3, -2, -1, 1, 2, 3
Y: 9, 4, 1, 1, 4, 9

Calculation:
$\sum X = 0$, $\sum Y = 28$
$\sum XY = (-27) + (-8) + (-1) + 1 + 8 + 27 = 0$
Since the sum of the product of deviations (numerator) is zero, the coefficient is zero.

Result: $\mathbf{r = 0}$
Comment: There is no linear correlation. However, there is a perfect non-linear relationship ($Y = X^2$), which Pearson’s coefficient does not capture.

Data:
X: 1, 3, 4, 5, 7, 8
Y: 2, 6, 8, 10, 14, 16

Calculation:
Notice that for every pair, $Y = 2X$.
This indicates a perfect linear relationship where points lie exactly on a straight line.
If we calculate, all variations in Y are fully explained by X.

Result: $\mathbf{r = +1}$
Comment: There is a perfect positive correlation between X and Y.