Understanding Correlation Coefficient with Age

The question asks about the coefficient of correlation between the ages of husbands and wives at two different points in time: at the time of marriage and 25 years later. We are given that the coefficient of correlation at the time of marriage is 0.7 for a set of 100 couples.

What is the Coefficient of Correlation?

The coefficient of correlation, often denoted by \(r\), is a statistical measure that describes the strength and direction of a linear relationship between two variables. Its value ranges from -1 to +1.

• A value close to +1 indicates a strong positive linear relationship (as one variable increases, the other tends to increase).
• A value close to -1 indicates a strong negative linear relationship (as one variable increases, the other tends to decrease).
• A value close to 0 indicates a weak or no linear relationship.

In this case, a correlation of 0.7 between the ages of husband and wife at marriage suggests a moderately strong positive linear relationship, meaning older husbands tend to marry older wives, and younger husbands tend to marry younger wives.

Analyzing the Change After 25 Years

Let's denote the age of a husband at marriage as \(X\) and the age of his wife at marriage as \(Y\). The coefficient of correlation between \(X\) and \(Y\) is given as \(r_{XY} = 0.7\).

After 25 years, the husband's age will be \(X' = X + 25\) and the wife's age will be \(Y' = Y + 25\). We need to find the coefficient of correlation between these new ages, \(r_{X'Y'}\).

Effect of Adding a Constant on Correlation

A key property of the coefficient of correlation is that it is invariant under changes of origin. This means that if you add or subtract a constant value from all observations of a variable, the correlation coefficient between the variables remains unchanged. Mathematically, if \(X' = aX + b\) and \(Y' = cY + d\), where \(a, b, c, d\) are constants and \(a\) and \(c\) have the same sign, then \(r_{X'Y'} = r_{XY}\).

In this scenario, the new ages are \(X' = X + 25\) and \(Y' = Y + 25\). Here, the transformation is of the form \(X' = 1 \cdot X + 25\) and \(Y' = 1 \cdot Y + 25\). We are adding a constant (25) to both variables. According to the property, adding a constant does not change the correlation coefficient.

Consider the formula for the correlation coefficient:

\(r_{XY} = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{\sqrt{\sum (X_i - \bar{X})^2 \sum (Y_i - \bar{Y})^2}}\)

If we consider the new variables \(X'_i = X_i + c\) and \(Y'_i = Y_i + d\), the new means will be \(\bar{X}' = \bar{X} + c\) and \(\bar{Y}' = \bar{Y} + d\).

The deviations from the mean for the new variables are:

• \(X'_i - \bar{X}' = (X_i + c) - (\bar{X} + c) = X_i - \bar{X}\)
• \(Y'_i - \bar{Y}' = (Y_i + d) - (\bar{Y} + d) = Y_i - \bar{Y}\)

Notice that the deviations remain the same. Therefore, the numerator \(\sum (X'_i - \bar{X}')(Y'_i - \bar{Y}') = \sum (X_i - \bar{X})(Y_i - \bar{Y})\) and the denominator terms \(\sum (X'_i - \bar{X}')^2 = \sum (X_i - \bar{X})^2\) and \(\sum (Y'_i - \bar{Y}')^2 = \sum (Y_i - \bar{Y})^2\) also remain the same.

Thus, the correlation coefficient between \(X'\) and \(Y'\) is:

\(r_{X'Y'} = \frac{\sum (X'_i - \bar{X}')(Y'_i - \bar{Y}')}{\sqrt{\sum (X'_i - \bar{X}')^2 \sum (Y'_i - \bar{Y}')^2}} = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{\sqrt{\sum (X_i - \bar{X})^2 \sum (Y_i - \bar{Y})^2}} = r_{XY}\)

In this specific problem, \(c = 25\) and \(d = 25\). The ages of all husbands increased by 25, and the ages of all wives increased by 25. This is simply a shift in the origin for both sets of data. The relationship between their ages relative to each other remains exactly the same.

Conclusion on Correlation and Age

Since the original coefficient of correlation between the ages of husband and wife at marriage was 0.7, adding 25 years to both their ages will not change this linear relationship measure. The coefficient of correlation at the time of their silver jubilee (25 years later) will still be 0.7.

Therefore, the coefficient of correlation at that point of time will be 0.7.

Revision Table: Key Concepts for Correlation and Age

Additional Information: Properties of Correlation Coefficient

Besides being invariant under change of origin, the correlation coefficient also has other important properties:

• It is invariant under change of scale (multiplying/dividing by a positive constant). If \(X' = aX\) and \(Y' = cY\) where \(a, c > 0\), then \(r_{X'Y'} = r_{XY}\). If \(a\) and \(c\) have opposite signs, \(r_{X'Y'} = -r_{XY}\).
• It is a dimensionless quantity; it does not have any units.
• It is symmetric: the correlation between X and Y is the same as the correlation between Y and X (\(r_{XY} = r_{YX}\)).
• It only measures linear relationships. Non-linear relationships, even strong ones, might have a correlation close to zero.
• Correlation does not imply causation. A high correlation between two variables does not necessarily mean that one causes the other.

Understanding these properties helps in interpreting correlation coefficients correctly and applying them in various scenarios, such as studying the relationship between ages over time.