Bivariate Statistics

In univariate statistics, we analyze a single variable at a time. Bivariate statistics extends this analysis to explore the relationship between two variables. By examining pairs of data, we can investigate patterns, determine the nature and strength of their relationship, and use this relationship to make pictions. This chapter focuses on the relationship between two quantitative variables.

Bivariate data consists of pairs of values for two quantitative variables, recorded for each individual in a dataset. We typically denote these variables as \((x, y)\), where:

• \(x\) is the independent (or explanatory) variable.
• \(y\) is the dependent (or response) variable.

A teacher records the hours each student studied (\(x\)) and their final exam score (\(y\)).

Hours Studied (\(x\))	5	10	8	15
Exam Score (\(y\))	50	85	75	95

Each pair of values, such as \((5, 50)\), is a single bivariate data point.

A scatter plot is a graph that displays bivariate data as a collection of points in the Cartesian plane. The independent (explanatory) variable is plotted on the horizontal axis (\(x\)-axis), and the dependent (response) variable is plotted on the vertical axis (\(y\)-axis).

A scatter plot is the primary tool for visually identifying a potential relationship, or correlation, between two quantitative variables.

1. Identify Variables: Determine which variable is independent (\(x\)) and which is dependent (\(y\)).
2. Set Up the Axes: Draw and label the horizontal axis for the \(x\)-variable and the vertical axis for the \(y\)-variable. Choose appropriate scales for both axes that cover the range of the data.
3. Plot the Points: For each pair of (\(x, y\)) values in your dataset, plot a single point on the graph at the corresponding coordinates.

A teacher recorded the number of hours students studied and their corresponding exam scores. The data is shown below:

Hours Studied (\(x\))	5	10	8	15
Exam Score (\(y\))	50	85	75	95

Construct a scatter plot to visualize this data.

Correlation describes the nature of the relationship between two quantitative variables.

The direction describes the overall trend of the data.

• Positive: As the independent variable (\(x\)) increases, the dependent variable (\(y\)) tends to increase. The points trend upward.
• Negative: As the independent variable (\(x\)) increases, the dependent variable (\(y\)) tends to decrease. The points trend downward.

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The form of the relationship is linear if the data points appear to follow a straight-line pattern. If they follow a curve other than a straight line, the form is non-linear.

The strength of a correlation describes how closely the data points adhere to the identified form.

An outlier is a data point that deviates significantly from the main pattern of the data.

When asked to describe the relationship shown in a scatter plot, you should always comment on all four features in a concise statement.

1. Direction: Is it positive or negative?
2. Form: Is it linear or non-linear?
3. Strength: Is it strong, moderate, or weak?
4. Outliers: Are there any notable outliers?

Describe the correlation between hours studied and exam scores shown in this scatter plot.

Observing a statistical relationship (correlation) between two variables, \(x\) and \(y\), is not sufficient evidence to conclude that a change in \(x\) causes a change in \(y\).

Causation exists only if a change in the independent variable is shown to directly cause a change in the dependent variable. Proving causation requires a carefully designed controlled experiment, not just observational data.

Often, a correlation between two variables (\(x\) and \(y\)) is actually caused by a third, unobserved factor known as a confounding variable (\(z\)). This variable influences both \(x\) and \(y\), creating an apparent but misleading relationship between them.

Data shows a strong positive correlation between ice cream sales and the number of people who get sunburned.
Does this mean eating ice cream causes sunburn? If not, identify the relationships and the likely confounding variable.

While scatter plots allow us to visually describe a correlation, this assessment is subjective. To provide a precise and objective measure of the strength and direction of a linear relationship, we use numerical coefficients.

The Pearson correlation coefficient, denoted by \(r\), is a measure of the linear correlation between two sets of data. It is defined as the ratio of the covariance of the two variables to the product of their standard deviations:$$ r = \frac{\text{cov}(x,y)}{\sigma_x \sigma_y} $$where \(\text{cov}(x,y)\) is the covariance and \(\sigma_x\), \(\sigma_y\) are the standard deviations of \(x\) and \(y\).

In practice, calculating \(r\) by hand is tedious for large datasets. We use a Graphic Display Calculator (GDC) or statistical software.

1. Enter the bivariate data into two lists (e.g., List 1 for \(x\) and List 2 for \(y\)).
2. Select the linear regression calculation (often labeled \texttt{LinReg(ax+b)} or similar).
3. Read the value of \(r\) displayed on the screen. (Note: On some calculators, you may need to turn "Diagnostics On" to see \(r\)).

The Pearson's correlation coefficient (\(r\)) is a value in the range \([-1, 1]\) that quantifies the direction and strength of a linear relationship between two quantitative variables.

• The sign of \(r\) indicates the direction (positive or negative).
• The magnitude (absolute value) of \(r\) indicates the strength. An \(|r|\) value close to 1 implies a strong linear correlation, while a value close to 0 implies a weak or no linear correlation.

Value of \(|r|\)	Strength of Correlation
\(|r| = 1\)	Perfect
\(0.9 \le |r| < 1\)	Very Strong
\(0.7 \le |r| < 0.9\)	Strong
\(0.5 \le |r| < 0.7\)	Moderate
\(0.3 \le |r| < 0.5\)	Weak
\(0 \le |r| < 0.3\)	Very Weak or None

When a scatter plot indicates a linear correlation between two variables, we can model this relationship using a straight line. This line, known as the regression line, can be used to make predictions. The reliability of this model is often assessed using the coefficient of determination (\(r^2\)). A high \(r^2\) value indicates that a large proportion of the variance in the dependent variable is explained by the independent variable, suggesting the linear model is a good fit for the data.

The least squares regression line, written as \(y = ax + b\), is the unique line of best fit that models the linear relationship between \(x\) and \(y\). It is calculated by minimizing the sum of the squares of the residuals.
A residual is the vertical distance between an observed data point \((x_i, y_i)\) and the predicted point on the regression line \((x_i, \hat{y}_i)\).$$ \text{Residual} = \text{observed } y - \text{predicted } y = y_i - \hat{y}_i $$

In practice, specifically for large datasets, the equation of the regression line is determined using the statistical functions of a calculator (GDC) or software. However, it is possible to calculate the coefficients manually using summary statistics.

A fundamental property of the least squares regression line is its relationship with the arithmetic means of the data sets. This property is frequently used to find missing variables without needing the full dataset.

The least squares regression line (\(y = ax + b\)) passes through the point defined by the mean of the \(x\)-values and the mean of the \(y\)-values: \((\bar{x}, \bar{y})\).$$ \bar{y} = a\bar{x} + b $$

The average weekly study time and the final mathematics test score for a selection of students are shown below:

Study Time (\(x\) hours)	2	5	1	7	4	9	6	3
Test Score (\(y\) marks)	45	65	30	85	60	95	70	55

1. Construct a scatter diagram to illustrate the data.
2. Find the equation of the regression line \(y\) on \(x\). State and interpret its gradient in the context of the problem.
3. Estimate the test score for a student who studies for 8 hours per week.
4. Estimate the weekly study time for a student who scores 40 marks.