Scatterplots
A scatterplot is a type of graph used to display the relationship between two quantitative variables measured on the same individual or object. Each point on the graph represents a single observation, with one variable plotted on the horizontal axis (x-axis) and the other on the vertical axis (y-axis). Because we are dealing with two variables simultaneously, this type of data is called bivariate quantitative data. Scatterplots are one of the most powerful tools in exploratory data analysis. By examining the pattern of points, statisticians can quickly identify whether a relationship exists between variables, describe the direction and strength of that relationship, and spot unusual observations called outliers. For example, a scatterplot of study hours versus exam scores would likely reveal a positive trend: students who study more tend to score higher. Understanding scatterplots is foundational to regression analysis, correlation, and prediction. Before computing any numerical summaries, creating a scatterplot allows you to verify whether a linear model is appropriate, identify influential points, and gain an intuitive sense of the data. In real-world applications, scatterplots appear everywhere — from economics and medicine to environmental science and sports analytics.
- Identify Explanatory and Response Variables
- Plot the Ordered Pairs
- Describe the Direction
- Describe the Form
- Describe the Strength
- Identify Outliers
Identify Explanatory and Response Variables
The explanatory variable (also called the independent variable) is plotted on the x-axis. It is the variable you think might explain or predict changes in the other variable. The response variable (dependent variable) is plotted on the y-axis. For example, if studying the effect of fertilizer amount on plant height, fertilizer amount is explanatory (x) and plant height is response (y). Identifying these roles before graphing keeps your analysis logically consistent.
Plot the Ordered Pairs
Each observation in your dataset becomes an ordered pair (x, y). Locate the x-value on the horizontal axis and the y-value on the vertical axis, then place a point at their intersection. For instance, if a student studied for 3 hours and scored 78, you plot the point (3, 78). Repeat this process for every data point in your dataset.
Describe the Direction
Once all points are plotted, examine the overall pattern. A positive association means that as x increases, y tends to increase — the cloud of points slopes upward from left to right. A negative association means that as x increases, y tends to decrease — the cloud slopes downward. No association means there is no discernible trend; the points appear scattered randomly with no slope.
Describe the Form
The form describes the shape of the pattern. A linear form means the points roughly follow a straight line. A curved or nonlinear form means the relationship follows a curve, such as a parabola or exponential curve. Always check the form before applying a linear model — using a straight line to model curved data leads to poor predictions.
Describe the Strength
Strength refers to how tightly the points cluster around the underlying pattern. In a strong association, points cluster closely together with little scatter, making the trend easy to see. In a weak association, points are widely spread and the trend is harder to identify. Strength is quantified numerically by the correlation coefficient r, which ranges from -1 to +1.
Identify Outliers
An outlier in a scatterplot is a point that falls far outside the overall pattern. Some outliers have extreme x or y values; others fall far from the linear trend even though their individual x and y values are not extreme — these are called influential points. Always note outliers and consider whether they result from data entry errors, unusual circumstances, or genuine variation.
The Key Equation
- r — correlation coefficient (unitless): Measures the strength and direction of the linear association; ranges from -1 to +1.
- n — sample size (count): The number of data points (ordered pairs) in the dataset.
- x_i, y_i — individual data values (varies): The x and y values for the i-th observation.
- \bar{x}, \bar{y} — sample means (varies): The arithmetic mean of all x-values and all y-values respectively.
- s_x, s_y — sample standard deviations (varies): The standard deviation of the x-values and y-values respectively, measuring their spread.
Worked Example
Problem. A student records the number of hours studied (x) and the exam score (y) for 5 classmates: (1, 55), (2, 60), (3, 72), (4, 78), (5, 90). Describe the association shown in the scatterplot and calculate the correlation coefficient r.
Answer. r ≈ 0.991; strong, positive, linear association between hours studied and exam score.
Origins · 1 minFrancis Galton and how this was discovered
Francis Galton was a Victorian-era polymath who pioneered the use of scatterplots and the concept of statistical correlation. While studying the relationship between parents' heights and their children's heights, Galton created some of the earliest scatter diagrams and developed the idea of regression toward the mean. His work laid the groundwork for the formal correlation coefficient later perfected by Karl Pearson. Galton was so obsessed with measuring things that he once attempted to mathematically determine the most beautiful face and created a 'beauty map' of Britain by secretly rating the attractiveness of women he passed on the street during his travels.
Related Concepts
- Correlation Coefficient — The correlation coefficient r quantifies the strength and direction of the linear relationship visible in a scatterplot, making it the natural numerical companion to this graphical tool.
- Least-Squares Regression Line — After identifying a linear pattern in a scatterplot, the least-squares regression line models that pattern mathematically, enabling predictions of y from x.
- Univariate Data Distributions — Understanding single-variable distributions (histograms, boxplots) provides the foundation for moving to two-variable analysis with scatterplots.
- Residuals and Residual Plots — Residual plots extend scatterplot analysis by displaying the leftover variation after fitting a model, helping verify whether a linear fit is appropriate.