Question

Correlation conclusions I. The correlation between Age and Income as measured on 100 people is \(r=0.75 .\) Explain whether or not each of these possible conclusions is justified: a) When Age increases, Income increases as well. b) The form of the relationship between Age and Income is straight. c) There are no outliers in the scatterplot of Income vs. Age. d) Whether we measure Age in years or months, the correlation will still be \(0.75\).

Short answer

a) Justified, b) Justified, c) Not justified, d) Justified.

Step-by-step solution

Analyze correlation meaning for increase

The correlation coefficient, denoted as \( r \), indicates the strength and direction of a linear relationship between two variables. A positive \( r = 0.75 \) suggests a strong positive linear relationship between Age and Income. Therefore, it is justified to conclude that as Age increases, Income tends to increase as well.

Determine the form of the relationship

The value of \( r = 0.75 \) indicates a linear relationship between Age and Income, but the correlation coefficient does not itself depict the exact form of the relationship beyond its linear nature. However, it's reasonable to interpret \( r = 0.75 \) as suggesting a strong linear (straight) pattern in the context of scatterplots.

Assess presence of outliers

The presence of outliers in a dataset can affect the value of \( r \), but the correlation coefficient alone cannot confirm the absence of outliers. To identify outliers, a scatterplot of Age vs. Income would need to be visually examined. Therefore, a conclusion about no outliers is not justified using only the correlation value.

Evaluate effect of measurement scale

Correlation is a dimensionless measure, meaning it does not change with the units of measurement. Therefore, whether Age is measured in years or months, the correlation coefficient will still be \( r = 0.75 \), making this conclusion justified.

Key concepts

Linear Relationship

The concept of a linear relationship in statistics refers to a straight-line association between two variables. In this context, the linear relationship between Age and Income, as indicated by a correlation coefficient of \( r = 0.75 \), suggests that as Age increases, Income likely increases too. This value reflects a strong and positive linear relationship. A strong positive correlation near 1 means that when one variable goes up, the other typically does as well. Conversely, if the value was negative, it would imply that as one variable goes up, the other goes down. A correlation around zero suggests no linear relationship between the variables. To fully understand the linearity, it is essential to interpret the data in a scatterplot, as others factors might influence this relationship.

Scatterplot

A scatterplot is a valuable visual tool in statistics that shows the relationship between two variables. For Age and Income, each point on the scatterplot represents an individual's age and corresponding income. This visualization helps identify patterns, trends, and potential relationships between the variables. In a scatterplot, if data points cluster around a line, it hints at a linear relationship. The correlation coefficient \( r = 0.75 \) suggests there is such a pattern between Age and Income; it's a strong indication of a linear trend. However, the scatterplot is necessary to confirm the linear nature beyond the mathematical statement of correlation. Viewing the plotted data offers direct insight into whether unusual patterns or deviations, such as outliers, exist.

Outliers

Outliers are data points that significantly differ from other observations in a dataset. In the context of Age and Income, outliers might be individuals whose income doesn’t follow the general trend related to their age. These outliers can skew the data. They might increase or decrease the correlation coefficient if present. This means that while \( r = 0.75 \) is our calculated strength of relationship, it could be influenced by such outliers. A scatterplot is essential to visually identify these anomalies. It shows points that lie far from the general cluster or trend-line. Determining outliers solely based on the correlation coefficient is impossible, as the number itself doesn't provide specific information on individual data points.

Dimensionless Measure

The term "dimensionless measure" in correlation means that the correlation coefficient \( r \) does not depend on the units of the variables it describes. Whether Age is recorded in years, months, or even minutes, the value of \( r = 0.75 \) remains the same. This is because correlation measures the ratio of covariance between the two variables to the product of their standard deviations, which does not change with the units of measurement. This property is highly beneficial because it allows for comparison of relationships between datasets with different units. Understanding that correlation is dimensionless helps in maintaining the consistency of data interpretation, regardless of the units used or the scale of measurement.