ACT Math: Essential Skills: Data Analysis & Representations

Introduction to Data Analysis on the ACT Math Section

In the ACT Math section, you will encounter a variety of problems that test your ability to analyze and interpret data. While many students focus heavily on algebra and geometry, mastering data analysis is equally important. By the end of this guide, you’ll be equipped with the skills necessary to create and interpret different types of data models, helping you predict how various pieces of information relate to one another.

Data analysis questions are scattered throughout the ACT Math section and primarily draw from topics in statistics, though you may also see them in algebra-related questions. These questions fall under the category of “Integrating Essential Skills,” which make up about 40-43% of the exam. Understanding how to analyze and represent data is crucial for success on the ACT, and this guide will walk you through the essential concepts and techniques you need to know.

Key Concepts in Data Analysis

1. Distribution Measures

Distribution measures are a fundamental aspect of data analysis. They help you understand the characteristics of a data set, such as its central tendency and variability.

Measures of Central Tendency

Measures of central tendency (also referred to as measures of center or central location) describe the center of a data set, offering insights into typical values within that set. The three primary measures are:

• Mean: The average of a data set, calculated by adding all the values together and dividing by the number of values.
• Median: The middle value of a data set when the values are arranged in order from least to greatest. If the number of data items is odd, the median is the middle value when the data values are listed in order.  If the number of data items is even, then the median is the average of the two middle values. 
• Mode: The value that occurs most frequently in a data set. Data sets can have two modes (bimodal) or more. Some data sets have no mode. 

Example:

Suppose you have five test scores: 85, 90, 93, 88, and 94. To find the mean, you would add all the scores together (85 + 90 + 93 + 88 + 94 = 450) and then divide by the number of scores (450 ÷ 5 = 90). The median would be 90, as it is the middle value when the scores are rewritten in order. The mode would not exist in this set, as no number repeats.

Measures of Spread

Measures of spread describe how much the data values vary and how they are distributed across the range. The two main measures of spread are:

• Range: The difference between the highest and lowest values in a data set.
• Standard Deviation: The average distance of each data point from the mean, which indicates how spread out the data points are.

Example:

Consider the prices of three different items in a store: $2, $5, and $8. The range is calculated as the difference between the highest and lowest values, so the range here is $8 – $2 = $6. Standard deviation would require a more detailed calculation, but it helps to understand that a higher standard deviation indicates more variability in the data.

2. Normal Distribution

The normal distribution is a specific type of data distribution that is symmetrical, with no skew to the left or right. It is often referred to as a “bell curve” because of its shape.

In a normal distribution:

• The mean, median, and mode are all the same.
• Data is distributed evenly around the mean.

Standard Deviation in Normal Distribution

In a normal distribution, standard deviations follow the 68-95-99.7 rule:

• 68.26% of data values fall within one standard deviation of the mean.
• 95.44% of data values fall within two standard deviations of the mean.
• 99.72% of data values fall within three standard deviations of the mean.

Understanding this distribution is essential for interpreting where specific data points fall in relation to the overall data set.

Example:

If the mean score on a test is 75 with a standard deviation of 5, then about 68.26% of students scored between 70 and 80 (within one standard deviation of the mean). Similarly, 95.44% of students scored between 65 and 85 (within two standard deviations).

3. Associations Between Two Variables

When analyzing data, it’s often important to understand the relationship between two variables. This relationship can be explored through correlation and regression.

Correlation

Correlation measures the strength and direction of a relationship between two variables. A strong correlation means that as one variable increases or decreases, the other does so as well. Correlation can be positive, negative, or zero.

• Positive correlation: Both variables increase or decrease together.
• Negative correlation: As one variable increases, the other decreases.
• Zero correlation: No apparent relationship between the variables.

Example:

If you find that as study time increases, test scores also increase, this indicates a positive correlation between study time and test scores.

Regression

Regression is a statistical method used to determine the equation that best fits the data. The most common form on the ACT is linear regression, which models the relationship between two variables with a straight line.

• Linear Regression: Models a linear relationship between two variables.
• Quadratic Regression: Used when the relationship between variables follows a parabolic shape.

Example:

Given a data set showing the number of hours studied and the corresponding test scores, you could use linear regression to create an equation that predicts test scores based on study hours. If the equation is

where

is the number of study hours,

then studying for 6 hours would predict a score of

4. Two-Way Tables

Two-way tables are used to organize data that fall into two different categories. They help you analyze relationships between the categories and answer questions about the data.

Example:

Suppose a survey asks students about their favorite subjects, and the results are organized into a two-way table showing the number of students preferring math, science, and English, split by grade level. You might be asked to calculate the percentage of seniors who prefer science, which would involve analyzing the data presented in the table.

Example:

A two-way table shows the preferences of students for different types of sports across different age groups. If the table indicates that 60 out of 200 students prefer basketball and are aged 15-18, you can use this information to determine the percentage of students in that age group who prefer basketball.

5. Scatter Plots

Scatter plots are graphical representations of data points, where each point represents the values of two variables. Scatter plots are useful for identifying correlations and patterns within the data.

Example:

Consider a scatter plot that shows the relationship between hours of exercise per week and weight loss. If the points on the scatter plot form an upward trend, it suggests that more exercise is associated with greater weight loss. You might also be asked to determine if a linear model would be appropriate for the data.

6. Interpreting Models

Creating linear models using regression or algebraic manipulation is only part of the process; interpreting these models is crucial to understanding how well they represent the data.

Cross-referencing scatter plot data with the line of best fit can help you determine whether a linear model is appropriate. If the scatter plot shows a clear trend that aligns with the line of best fit, then the linear model is likely a good representation of the data. However, if the data points are scattered with no clear pattern, a different model might be more appropriate.

Example:

If a scatter plot shows a curved pattern, a linear regression model might not be the best fit. Instead, you might consider a quadratic model to better represent the relationship between the variables.

Conclusion

Data analysis and representation are critical skills for the ACT Math section. Mastering these concepts will not only help you answer questions more accurately but will also enhance your overall understanding of how data is used in mathematical contexts.

By practicing with various types of data representations—whether they involve measures of central tendency, understanding distributions, analyzing correlations, or interpreting regression models—you’ll be well-equipped to tackle the data analysis questions on the ACT.

Remember, the key to success is practice. Familiarize yourself with these concepts, work through practice problems, and review any areas where you feel less confident. With consistent effort, you’ll be ready to excel in the data analysis portion of the ACT Math section.