Mathematics Study Guide for the ACT

Statistics

The study of statistics deals with the collection, analysis, interpretation, and organization of data. We use statistics to make sense of large amounts of data and draw conclusions from them. Two of the main branches of statistics are:

descriptive statistics—This involves the collection, organization, and presentation of data. Measures such as mean, median, mode, range, and standard deviation are part of descriptive statistics.
inferential statistics—This uses data from a sample to make inferences about a population. It involves techniques such as hypothesis testing and regression analysis.

Finding the Average

The average of a data set is a commonly used statistical measure that represents the data set’s central tendency (a value that attempts to describe a data set by finding its middle). Average is also known as the mean. To find the average of a set of numbers, you add all the given numbers and then divide this sum by the total number of values in the data set.

Mean, Median, and Mode

The mean, median, and mode are three measures of central tendency of a data set that provide insights into the data distribution.

mean—This is the average of a set of numbers. You can calculate the mean by adding all the numbers in a data set and then dividing the total by the number of numbers.

median—This is the middle value in a set of data when the values are arranged in ascending or descending order. If there is an odd number of values, the median is the middle value. If there is an even number of values, the median is the average of the two middle values.

mode—This is the most frequently occurring value in a data set. A data set can have one mode (unimodal), two modes (bimodal), or more than two modes (multimodal).

We can choose the most appropriate measure of central tendency based on the data distribution, presence of outliers, and data scale.

Mean is best used for symmetrical data sets without outliers (observations that deviate significantly from the overall pattern or trend of the data, exhibiting an exceptionally high or low value).
Median is ideal for skewed data (a data distribution that is not symmetrical, with an imbalance toward one tail of the data) or data with outliers.
Mode is suitable for identifying the most frequently occurring category in categorical data.

Using Chart Data

Charts and graphs are visual representations of data that provide valuable insights for data analysis. When using chart data to perform computations, it’s important to consider factors such as rounding and precision, as well as to translate between different types of graphs representing the same data, utilize Venn diagrams for counting, and draw meaningful conclusions from charts and tables. These skills are essential in accurately interpreting and analyzing chart data.

Rounding Data

It’s crucial to carefully consider rounding and the level of precision required when working with chart data. Rounded data in charts may not always yield the most accurate results, as rounding errors can accumulate and affect the final calculations. Being mindful of rounding and precision is essential to ensure accurate computation and interpretation of chart data, especially when calculating percentages or performing other computations that require high precision.

Translate between Types of Graphs of the Same Data

Charts come in different forms, including bar charts, line charts, and pie charts. Each type of chart and graph has its own unique characteristics. Understanding how to translate between different types of graphs that represent the same data is essential. For example, knowing how to convert data from a bar chart to a line chart or vice versa can help you accurately compare and analyze data. Familiarity with different types of graphs and their interpretation is crucial in effectively utilizing chart data for computation and analysis.

Venn Diagrams

Venn diagrams are graphical illustrations that depict the connections and intersections between different sets or groups of data. They consist of circles that are used to represent each set, with overlapping areas indicating common elements or data points shared by multiple sets.

41 Venn Diagram.png

To use Venn diagrams for counting, one can follow these steps:

Identify the sets. Determine the different sets that you want to compare, usually two or three.
Draw the circles.
Label the circles with names or letters.
Mark the overlaps.
Count the number of elements.

Drawing Conclusions from Graphs and Tables

When interpreting graphs, such as bar, line, or pie graphs, it is essential to analyze trends and patterns. For example, comparing bar heights in bar graphs, identifying upward or downward trends in line graphs, and analyzing the proportion of different categories in pie charts can provide meaningful insights into the data set.

Tables also require careful reading and analysis to extract meaningful conclusions. You need to understand the labels, headings, and units of measurement used in the table. You should be able to interpret the data presented in rows and columns.

Probability

Probability deals with the likelihood of events happening. It is used to predict outcomes, make informed decisions, and analyze data. The probability of something is a numerical measure that represents the chances of an event. It ranges from \(0\) to \(1\), where \(0\) means an impossible event and \(1\) represents a certain event. Probability is calculated as the ratio of the number of favorable outcomes to the total number of possible outcomes.

Events

Events are the results of a particular situation or experiment. Events can be described using the words “and”, “or”, and “not” to denote the relationship between multiple events.

and—When two events occur together, it is denoted as the intersection of the events. For example, if event \(A\) and event \(B\) both occur, it is represented as \(A \cap B\). This symbol indicates the outcomes where both events \(A\) and \(B\) happen simultaneously.

or—When at least one of the two events occur, it is denoted as the union of the events. For example, if event \(A\) or event \(B\) occurs, it is represented as \(A \cup B\). This symbol indicates the outcomes where either event \(A\) or event \(B\) happens.

not—The complement of an event means the outcomes that are not part of the event. It is denoted as \(A’\), where \(A\) represents the event itself.

Computing Probability

When computing probability, we determine the chances of an event occurring based on given information. There are two methods for calculating probability, depending on the nature of the event and amount of data available.

theoretical probability—This method involves using mathematical calculations to find the probability. It is based on the assumption of equally likely outcomes and is often used in simple situations where all possible outcomes are known and have the same chance of occurring. The formula is shown below:

\[P(E) = \frac{\text{number of favorable outcomes}}{\text{number of total possible outcomes}}\]

empirical probability—This method involves finding probability based on observed data or past occurrences. It is often used when actual data is available and can be used to estimate the probability of an event occurring in the future. The formula is shown below:

\[P(E) = \frac{\text{number of times event E occurs}}{\text{total number of observations}}\]

The Complement of a Probability

The complement of a probability refers to the chances of an event not happening. It is basically the opposite of the event being considered. It is denoted by the symbol \(P(E’)\), where \(E\) represents the event and the apostrophe (\(’\)) denotes the complement. The formula for this is:

\[P(E’) = 1 - P(E)\]

So, the probability of an event not occurring is equal to \(1\) minus the probability of the event happening. The concept of the complement of a probability is useful in solving problems where it is easier to find the probability of an event not occurring rather than directly finding the probability of the event itself. It is also commonly used in situations where there are only two possible outcomes and finding the probability of one outcome automatically gives the probability of the other outcome.

Types of Probability

There are different types of probability but we are going to focus on three important ones:

conditional probability—This refers to the probability of an event occurring given that another event has already occurred. It is denoted as:

\[P(A \vert B)\]

where \(A\) represents the event of interest and \(B\) represents the condition.

Conditional probability is used when the probability of an event is influenced by the occurrence of another event.

independent probability—This refers to the probability of two or more events occurring without an influence on each other. The occurrence of one event does not affect the occurrence of the other event. It is calculated using the multiplication rule of probability:

\[P(A \, \text{and} \, B) = P(A) \times P(B)\]

where \(P(A \, \text{and} \, B)\) represents the probability of both events \(A\) and \(B\) happening, and \(P(A)\) and \(P(B)\) represent the probabilities of events \(A\) and \(B\) occurring individually.

joint probability—This refers to the probability of two or more events occurring together. It is denoted as:

\[P(A \, \text{and} \, B)\]

where \(A\) and \(B\) represent the events.

Joint probability is used when considering the likelihood of two or more events occurring simultaneously. It is commonly used in situations where events are interrelated or have dependencies.