If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

## AP®︎/College Statistics

### Course: AP®︎/College Statistics>Unit 5

Lesson 1: Representing the relationship between two quantitative variables

# Clusters in scatter plots

AP.STATS:
DAT‑1 (EU)
,
DAT‑1.A (LO)
,
DAT‑1.A.6 (EK)
CCSS.Math:
Learn what a cluster in a scatter plot is!

## What are clusters in scatter plots?

Sometimes the data points in a scatter plot form distinct groups. These groups are called clusters.
A scatterplot plots Sodium per serving in milligrams on the y-axis, versus Calories per serving on the x-axis. 16 points rise diagonally in a relatively narrow pattern with a cluster of 8 points between (135, 350) and (155, 360) and another cluster of 8 points between (170, 450) and (195, 500). Both clusters are labeled a different color.
Data source: Consumer Reports, June 1986, pp. 366-367
Consider the scatter plot above, which shows nutritional information for 16 brands of hot dogs in 1986. (Each point represents a brand.) The points form two clusters, one on the left and another on the right.
The left cluster is of brands that tend to be start color #1fab54, start text, l, o, w, space, i, n, space, c, a, l, o, r, i, e, s, space, a, n, d, space, l, o, w, space, i, n, space, s, o, d, i, u, m, end text, end color #1fab54.
The right cluster is of brands that tend to be start color #11accd, start text, h, i, g, h, space, i, n, space, c, a, l, o, r, i, e, s, space, a, n, d, space, h, i, g, h, space, i, n, space, s, o, d, i, u, m, end text, end color #11accd.

## Practice problems

To better wrap our minds around the idea of clusters, let's try a couple of practice problems.

### Problem 1: Male and female fish

Adult male Lamprologus callipterus (a type of fish) are much bigger than their female counterparts. They weigh about 13 times as much. Also, while females reach a length of 6 centimeters, males reach a length of 15 centimeters.
Which of the plots shown below might describe measurements of a group of adult Lamprologus callipterus?

### Problem 2: SAT test scores

Some high school students in the U.S. take a test called the SAT before applying to colleges. The scatter plot below shows what percent of each state's college-bound graduates participated in the SAT in 2009, start text, negative, end text, 2010, along with that state's average score on the math section.
Data from National Center for Education Statistics
There is a cluster of states with start color #1fab54, start text, l, o, w, e, r, space, p, a, r, t, i, c, i, p, a, t, i, o, n, end text, end color #1fab54, and a cluster of states with start color #11accd, start text, h, i, g, h, e, r, space, p, a, r, t, i, c, i, p, a, t, i, o, n, end text, end color #11accd.
What is the best interpretation of these clusters?

## Why do clusters exist in data?

Explaining why clusters exist in a particular data set can be difficult. This article presented three data sets, each using data from the real world. Only in the fish data set was there a clear explanation behind the clusters.
If you have a theory that explains the clusters in either of the other data sets, please share your thoughts in the comments below.

## Want to join the conversation?

• up vote for a cookie
• The ingredients in the hot dogs can effect their ratings
• Perhaps states with lower participation but higher math shoes on SAT is correlated to ambition. For example, in states with lower educational progress, the majority of those students might not have the ambition of higher education as a priority. For that reason, the best performing of the state's students (the minority) would be the more likely to want to go to college, therefore participate in SAT testing. Or perhaps, certain states place more value on teaching every student SAT skills, whereas other states focus on individual students who already perform well and encourage those students to sit for the test. Or yet another scenario, perhaps the state's with the best education and smartest students have less participation due to some other factor (i.e. derision for the educational system as a whole, lack of ambition due to laziness from being accustomed to learning being so easy, boredom, etc).
• Bro it was a math question 💀
(1 vote)
• This was very helpful for me, but I am a little confused on #2, could someone please explain to me?
• The problem is categorizing the dots into either lower or higher participation. It is asking you to find the best fitting answer, but in this case, it is basically just true or false.

The first answer choice says, "The states with lower participation typically had lower math scores." If you look at the graph, all the green dots, which represent lower participation, are higher on the y-axis. The y-axis' variable is test scores. This means this is untrue.

The second choice says, "The states with lower participation typically had higher math scores." Check the graph again. The green dots are high on the y-axis, representing higher test scores. This statement is true.
• Im not ready for college LET ALONE THIS
• I think that for the SAT problem, clusters might be present because in the states with lower participation because only the students that feel like taking the SAT is "worth it" or have confidence in their abilities take the test. This theory makes sense because of the math scores being higher for the students in states with lower participation. I'm not sure if this is really the reason, but I gave this a shot!
• also some dont make a lot sense like the college SAT's
• There is college sat's ∞
(1 vote)
• The hotdog brand clusters seem to be an example of competitive positioning in marketing. Hotdog brands need to be able to compete with other brands either by being healthier or by being tastier. Any brands with mid-range levels of sodium and calories will be cornered out of the market on all sides.