Read scatterplots, describe the form, direction, and strength of an association, identify clusters and outliers, and interpret two-way frequency tables.

What this topic is asking

The Statistics and Probability category requires reading two-variable data (the S-ID standards). On the Grade 10 MCAS you interpret scatterplots, describing the form, direction, and strength of an association, identify clusters and outliers, and read two-way frequency tables. These are interpretation tasks more than calculation, so the vocabulary (positive, negative, strong, weak, linear) must be precise.

Reading a scatterplot

A scatterplot shows paired $(x, y)$ data as points, revealing whether the two variables move together. Three features describe the pattern:

• Form: is the pattern roughly linear (a straight-line trend), curved, or formless?
• Direction: is it positive (as $x$ increases, $y$ tends to increase, points rising left to right) or negative (as $x$ increases, $y$ tends to decrease, points falling)?
• Strength: are the points tightly clustered around the trend (strong) or widely scattered (weak)?

So a scatterplot of study hours versus score that rises in a narrow band is a strong positive linear association.

Clusters and outliers

Beyond the overall trend, look for:

• Clusters: groups of points bunched in one region, which can signal subgroups in the data.
• Outliers: individual points far from the main pattern. In two-variable data, an outlier may not be extreme in either variable alone but lies off the trend (for example, a point that is high in $x$ but low in $y$ when the rest rise together).

The MCAS may ask you to identify an outlier on a scatterplot or to consider how removing it would affect the trend line.

Two-way frequency tables

A two-way frequency table cross-tabulates two categorical variables, with counts in each cell. For "plays a sport" by "owns a pet", the four cells are the combinations, and the margins give the totals for each category. To answer "how many own a pet", add every cell where pet ownership is yes, across both sport categories.

Two-way tables also support relative frequencies: dividing a cell by a row total, a column total, or the grand total gives a proportion, which the MCAS may ask you to compare (for example, the fraction of sport-players who own a pet versus the fraction of non-players who do).

Linear versus nonlinear patterns

Not every association is linear. A scatterplot may curve, rising then leveling off, or rising at an increasing rate. The MCAS asks you to judge whether a straight line is a reasonable model or whether the pattern is nonlinear. If the points bend consistently, a line will systematically miss them, and a curved model fits better. Recognizing that a relationship is positive but curved (not linear) is a valid and tested description, and it warns against forcing a line of best fit onto data that does not support one.

From a scatterplot to a prediction

When a scatterplot shows a strong linear pattern, drawing a trend line through the middle of the points lets you estimate a $y$-value for a given $x$, reading it off the line. This is the visual version of using a line of best fit. The estimate is only as trustworthy as the pattern is tight, and only within the range of the data, which connects this topic directly to linear regression. A loose, scattered cloud of points does not support a confident prediction.

Try this

Q1. A scatterplot shows no pattern, points scattered everywhere. What is the association?

• Cue. No association (or very weak).

Q2. In a two-way table, 12 students like math and science, 8 like math only. How many like math?

• Cue. $12 + 8 = 20$.