SAT & ACT

ACT Math: Guide to Data Analysis & Representation

tl;dr: In the ACT Math section, you'll learn about measures of center, measures of spread, normal distributions, linear and quadratic regressions, 2-way tables, and scatter plots.

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In this ACT Math section, we take a break from standard algebra and geometry and head into the wondrous world of data! By the end of this study guide, youโ€™ll be able to create and interpret different types of models ๐Ÿ“Š that will help you predict how different pieces of information relate to each other.

These types of problems are found throughout the ACT Math section. The majority of these topics are a part of statistics, but some may also show up in the algebra section! They are part of Integrating Essential Skills, which make up about 40-43% of the exam. For a detailed overview of the ACT as a whole, check out this article.

Distribution Measures ๐Ÿ“ค

Measures of Center ๐ŸŽฏ

There are many ways to describe a data set. Measures of center help describe individual values within the data set, such as mean and median.

Need a refresher on how to find these values? Take a look at our Stats & Probability study guide!

Example

The average of five different test scores has the same value as the median of the five scores. The sum of the five test scores is 465. What is the sum of the six scores that are NOT the median?

In this problem, we will first find the average of the test scores: 465/5 = 93. The mean and median are the same, so the median is also 93. Since the scores are all distinct, the median value does not repeat anywhere in the data set. Thus, the sum of the 6 other scores is 530 - 93 = 437.

Measures of Spread ๐Ÿž

Measures of spread describe the data's variation and/or distribution within the set, such as range and standard deviation.

Standard Deviation

The standard deviation is the mean distance between the mean of the data set and a single data point. The greater the standard deviation, the larger the spread of data will be since the datapoint will be further from the mean.

You will likely be asked to find these values (except standard deviation) from a single data set or compare them between multiple data sets.

Normal distribution ๐Ÿ›Ž๏ธ

The Normal Distribution is a special kind of distribution. The data is centered with no left or right skew and shaped like a โ€œbell curveโ€ ๐Ÿ””

The mean, median, and mode are all the same in a normal distribution, making the distribution symmetrical.

Standard Deviation within the Normal Distribution

In a normal distribution, the standard deviations follow a pattern known as 68-95-99:

  • 68% of values in the data set are 1๏ธโƒฃ standard deviation from the mean.
  • 95% of the values in the data set are 2๏ธโƒฃ standard deviations from the mean.
  • 99.7% of the values in the data set are 3๏ธโƒฃ standard deviations from the mean.

This information is helpful to help understand where a data value is located within the distribution!

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Image from The Science of Machine Learning

The simplest and most common form is simple linear regression, in which a linear equation ๐Ÿ“ˆ can be created from a data set. On the ACT, this will most likely be used to predict future data values.

Associations between two variables ๐Ÿ‘‹

Correlation ๐Ÿค

Association, most commonly indicated by the word correlation, can help us find how two variables relate to each other. Although finding the correlation values may be a bit advanced for the ACT, you might be asked to look at a graph or a dataset and determine if there is a strong or weak correlation between the two variables.

Regression ๐ŸŒ 

Regression is a method to determine the equation that โ€œbest fitsโ€ data from 2 variables.

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Image from Towards Data Science

If the data doesnโ€™t seem linearly related, another form is quadratic regression, which produces a quadratic equation that โ€œbest fitsโ€ the presented data.

The fastest and easiest way to perform any regression is on a calculator๐Ÿ–ฉ If you are uncomfortable with using your calculator, check out these resources!

Linear Regression

Non-linear Regression

2-way tables ๐Ÿ“‹

A 2-way table tells you how many data points fit both categories of variables. Some common examples are gender vs. political affiliation, age, ice-cream flavor, grade level, and mode of transportation to school.

2-way table problems are like fun logic puzzles! Generally, youโ€™re given certain data points and the total number of data points and will have to use arithmetic to find a missing value.

Example

High school upperclassmen are surveyed to determine their favorite subjects: math, history, English, and science. Some results are shown in the two-way frequency table below.

๐Ÿงช What percentage of seniors chose science?

ย  Math English Science History Total
Juniors ย  12 55 107 214
Seniors 22 16 ย  92 ย 
Total 62 28 ย  199 398

At first glance, you may want to fill out all of the missing information completely. However, the ACT math section is timed! Only fill out the information that you need to get to the answer. To figure out how many seniors chose science, we need to figure out 1 of 2 things: how many people took science in total, or how many seniors there are in total.

Letโ€™s try to solve it with the first method. To find out how many total science-lovers there are, we must add up the other subjects' totals and subtract that from the total students. Then, weโ€™ll subtract the 55 juniors from that number to get the seniors: 398 - (62 + 28+ 199) = 109, 109 - 55 = 54 seniors.

In the second method, weโ€™ll find the total number of seniors by subtracting the total number of juniors from all students. Then, weโ€™ll add up the other subjects for seniors and subtract that from all seniors to get just the ones that like science: 398 - 214 = 184, 184 - (22 + 16 + 92) = 54 seniors!

Scatter plots โš‡

Scatter plots use dots to represent each data point in the set. They will help you determine correlations between two variables depending on their shape and direction and will help you determine if linear regression needs to be done to find a line of best fit!

Interpreting models ๐Ÿค”

You can create linear models using both linear regression and algebraic manipulation. Graphing linear models can help us determine how useful or representative the linear model actually is! As mentioned previously, accurate models can help us predict values beyond the dataset.

Cross-referencing scatter plot data with the line of best fit will help you determine if a linear model is appropriate โœ”๏ธ If the scatter plot seems to have no shape or is trending in a different direction than the linear model, it may not โŒ be the model to use!

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Image from Chartio

Conclusion ๐Ÿฅณ

Congrats, you made it to the end of this guide! Use the tips and tricks you learned here and you'll be able to ace the ACT data problems.

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Need more ACT practice?

Fiveable has you covered! Check out these articles that tell you all you need to know about each ACT Subject including the ACT essay. With consistent practice, you'll be on your way toward a 36 on your next ACT exam!

For more ACT prep guides, visit the ACT subject page. We at Fiveable are rooting for your ACT success on your upcoming exam!

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