What went wrong when the king rode under the arch?

It was too low and knocked off his crown

How does the chain of blame unfold in the poem?

Each accused person blames another, ending with the king's own changes

What does correlation measure?

Correlation measures the direction and the intensity of the relationship between two variables, that is, their covariation, but it does not by itself establish causation.

Distinguish between positive and negative correlation.

In positive correlation both variables move in the same direction, like income and consumption; in negative correlation they move in opposite directions, like the price of a good and its demand.

Why does correlation not imply causation?

Two variables may move together by mere coincidence (spurious correlation) or because both are influenced by a hidden third variable, so a high correlation alone cannot prove that one causes the other.

What is a scatter diagram and what does it show?

A scatter diagram plots the paired values of two variables as points and reveals the direction and closeness of their relationship; if all points lie on a straight line the correlation is perfect.

Write the formula for Karl Pearson's coefficient of correlation.

\(r=\dfrac{\sum xy}{N\,\sigma_x \sigma_y}\), where \(x=X-\bar{X}\), \(y=Y-\bar{Y}\), and \(\sigma_x\), \(\sigma_y\) are the standard deviations of the two variables.

State the properties of Karl Pearson's coefficient of correlation.

It has no unit, lies between \(-1\) and \(+1\), is unaffected by a change of origin and scale, and equals zero when there is no linear relationship and \(\pm1\) for a perfect linear one.

What does the sign and magnitude of r indicate?

A positive r shows the variables move together and a negative r shows they move oppositely, while the closer the magnitude is to one the stronger the linear relationship and the closer to zero the weaker.

When is Spearman's rank correlation used?

When variables can be ranked but not measured, when the characteristics are qualitative like beauty or honesty, when the relationship is clearly non-linear, and when the data contain extreme values.

Write the formula for Spearman's rank correlation.

\(r_s=1-\dfrac{6\sum D^2}{n^3-n}\), where \(D\) is the difference between the ranks of paired items and \(n\) is the number of observations.

How are repeated ranks handled in Spearman's method?

Tied items are given the mean of the ranks they would otherwise occupy, and a correction factor \(\dfrac{m^3-m}{12}\) is added for each group of \(m\) repeated ranks.

Correlation

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Summary

Correlation analysis studies the relationship between two variables, measuring both the direction and the intensity of their co-movement. It measures covariation, not causation: a correlation may reflect a cause-and-effect link, a coincidence (spurious correlation, like shoe size and pocket money), or the influence of a hidden third factor (temperature behind both ice-cream sales and drowning deaths). Correlation is positive when the variables move in the same direction (income and consumption) and negative when they move in opposite directions (price of apples and their demand). Three techniques measure correlation. A scatter diagram plots the paired values and shows the direction and closeness of the relationship visually but gives no number; if all points lie on a line, correlation is perfect. Karl Pearson's coefficient of correlation gives a precise numerical value of the degree of a linear relationship: \(r=\dfrac{\sum xy}{N\,\sigma_x \sigma_y}\), where \(x=X-\bar{X}\) and \(y=Y-\bar{Y}\). It has no unit, lies between \(-1\) and \(+1\), and is unaffected by change of origin and scale; \(r=0\) means no linear relation and \(r=\pm1\) means a perfect linear one. Spearman's rank correlation, \(r_s=1-\dfrac{6\sum D^2}{n^3-n}\), is used when items can only be ranked (qualities like beauty or honesty), when the relationship is clearly non-linear, or when data have extreme values, since it is not distorted by them; a correction factor is added for repeated ranks. Both coefficients lie between \(-1\) and \(+1\).

Correlation

Key terms

Correlation

The statistical relationship between two variables, measuring the direction and intensity of their co-movement (covariation, not causation).

Positive correlation

A relationship in which two variables move in the same direction, such as income and consumption.

Negative correlation

A relationship in which two variables move in opposite directions, such as price and demand.

Scatter diagram

A graph of paired values of two variables that shows the direction and closeness of their relationship without a numerical value.

Karl Pearson's coefficient

A numerical measure of the degree of linear correlation, lying between \(-1\) and \(+1\) and free of units.

Spearman's rank correlation

A measure of correlation based on the ranks of items, useful for qualitative data and data with extreme values.

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Correlation measures the direction and intensity of:

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Correlation

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