Scatterplot

Scatterplot

A graph of paired data in which the data values are plotted as (x, y) points.

Scatterplot with x and y axes showing approximately 12 data points scattered in a loosely upward-trending pattern.

Key Formula

r = \frac{\displaystyle\sum_{i=1}^{n}(x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\displaystyle\sum_{i=1}^{n}(x_i - \bar{x})^2 \;\cdot\; \displaystyle\sum_{i=1}^{n}(y_i - \bar{y})^2}}

Where:

$r$ = Correlation coefficient, measuring the strength and direction of the linear relationship (ranges from −1 to 1)
$n$ = Number of data pairs
$x_i$ = The x-value of the i-th data point
$y_i$ = The y-value of the i-th data point
$\bar{x}$ = Mean of all x-values
$\bar{y}$ = Mean of all y-values

Worked Example

Problem: A teacher records the number of hours five students studied and their test scores. Plot the data as a scatterplot and describe the association. Student A: (1, 50), Student B: (2, 60), Student C: (3, 65), Student D: (4, 80), Student E: (5, 90)

Step 1: Identify the variables. Hours studied is the independent variable (x-axis) and test score is the dependent variable (y-axis).

x = \text{hours studied}, \quad y = \text{test score}

Step 2: Set up the axes. The x-axis runs from 0 to 6. The y-axis runs from 40 to 100.

Step 3: Plot each ordered pair. Place a dot at (1, 50), (2, 60), (3, 65), (4, 80), and (5, 90) on the coordinate plane.

Step 4: Describe the association. As hours increase, scores also increase. The points roughly follow an upward line, indicating a strong positive linear association.

Step 5: Compute the correlation coefficient to quantify the relationship. The means are:

\bar{x} = \frac{1+2+3+4+5}{5} = 3, \quad \bar{y} = \frac{50+60+65+80+90}{5} = 69

Answer: The scatterplot shows a strong positive linear association between hours studied and test score. Calculating further yields r ≈ 0.99, confirming a nearly perfect positive linear relationship.

Another Example

This example differs from the first by showing data with no apparent association, illustrating that not every scatterplot reveals a trend. It reinforces the idea that correlation near zero means the variables lack a linear relationship.

Problem: A researcher records the age (in years) and daily screen time (in hours) for six people: (10, 4), (20, 5), (30, 3), (40, 2), (50, 6), (60, 3). Plot a scatterplot and describe what you see.

Step 1: Assign age to the x-axis and screen time to the y-axis. The x-axis runs from 0 to 70, and the y-axis from 0 to 7.

Step 2: Plot the six points: (10, 4), (20, 5), (30, 3), (40, 2), (50, 6), (60, 3).

Step 3: Examine the pattern. The points do not follow any clear upward or downward trend. They appear scattered without a consistent direction.

Step 4: Compute the means and correlation coefficient. The means are:

\bar{x} = \frac{10+20+30+40+50+60}{6} = 35, \quad \bar{y} = \frac{4+5+3+2+6+3}{6} \approx 3.83

Step 5: After computing the full formula, the correlation coefficient is approximately r ≈ 0.07, which is very close to zero.

r \approx 0.07

Answer: The scatterplot shows no clear linear association between age and daily screen time. The correlation coefficient r ≈ 0.07 confirms virtually no linear relationship.

Frequently Asked Questions

What is the difference between a scatterplot and a line graph?

A scatterplot displays individual data points without connecting them, showing the general relationship between two variables. A line graph connects consecutive data points with straight segments, typically used when the x-variable represents a sequence like time. Scatterplots are better for exploring whether a relationship exists, while line graphs emphasize change over an ordered variable.

How do you tell if a scatterplot shows a positive, negative, or no correlation?

If the points trend upward from left to right, the scatterplot shows a positive correlation—as x increases, y tends to increase. If the points trend downward, there is a negative correlation. If the points show no discernible upward or downward pattern, there is little to no correlation. The correlation coefficient r quantifies this: values near +1 indicate strong positive, near −1 indicate strong negative, and near 0 indicate no linear relationship.

When do you use a scatterplot instead of a bar chart?

Use a scatterplot when both variables are quantitative (numerical) and you want to explore the relationship between them. Use a bar chart when one variable is categorical (like names of countries or types of fruit). Scatterplots are designed for continuous numerical data, while bar charts compare counts or values across distinct categories.

Scatterplot vs. Line Graph

	Scatterplot	Line Graph
Definition	Plots individual (x, y) data points without connecting them	Plots data points and connects them with line segments
Data type	Two quantitative variables, not necessarily ordered	Typically one ordered variable (like time) on the x-axis
Purpose	Explore relationships, identify patterns, detect outliers	Show trends and changes over a continuous or sequential variable
Connecting points	Points are left unconnected (or a trend line is added separately)	Points are connected in order from left to right
Best for	Determining if two variables are correlated	Tracking how a single measurement changes over time

Why It Matters

Scatterplots are one of the first tools you encounter in statistics courses and standardized tests (SAT, ACT, AP Statistics) for analyzing bivariate data. In science classes, you use scatterplots to identify experimental relationships—like whether increasing temperature affects reaction rate. Beyond school, scatterplots are fundamental in data science and research for detecting trends, spotting outliers, and deciding whether to fit a regression model.

Common Mistakes

Mistake: Connecting the dots in a scatterplot as if it were a line graph.

Correction: Leave the points unconnected. A scatterplot shows the overall pattern of the data, not a point-to-point path. If you want to show a trend, add a separate best-fit (regression) line instead.

Mistake: Placing the independent variable on the y-axis and the dependent variable on the x-axis.

Correction: The independent variable (the one you control or that serves as the predictor) belongs on the x-axis, and the dependent variable (the outcome you measure) belongs on the y-axis. Reversing them makes the graph misleading and any regression analysis incorrect.

Related Terms

Paired Data — The type of data plotted in a scatterplot
Least-Squares Regression Line — Best-fit line drawn through scatterplot points
Least-Squares Regression Equation — Equation of the line fitted to scatterplot data
Regression — Method for modeling relationships shown in scatterplots
Linear Fit — Describes how well a line matches scatterplot data
Graph of an Equation or Inequality — Broader concept of graphing on a coordinate plane
Point — Each data pair is represented as a point