Statistics & Probability

Master data analysis and chance to boost your ACT Math score!

0%

Statistics & Probability Fundamentals

Descriptive Statistics

Descriptive statistics summarize and organize data. Understanding these concepts is essential for the ACT Math section.

Key ACT Concept: Measures of Central Tendency

  • Mean (Average): The sum of all values divided by the number of values
  • Median: The middle value when data is arranged in order (or average of two middle values if there's an even number of data points)
  • Mode: The value that appears most frequently in the data set

Example:

Find the mean, median, and mode of the following data set: 4, 7, 10, 8, 4, 9, 7

Step 1: Calculate the mean.

\[ \text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}} = \frac{4 + 7 + 10 + 8 + 4 + 9 + 7}{7} = \frac{49}{7} = 7 \]

Step 2: Find the median by arranging the data in order.

Ordered data: 4, 4, 7, 7, 8, 9, 10

The middle value (4th out of 7 values) is 7.

Step 3: Identify the mode.

The values 4 and 7 both appear twice, while all other values appear once. So the modes are 4 and 7.

Key ACT Concept: Measures of Dispersion

  • Range: The difference between the maximum and minimum values in a data set
  • Variance: The average of the squared differences from the mean
  • Standard Deviation: The square root of the variance, indicating how spread out the data is

Example:

Find the range of the data set: 12, 15, 9, 18, 11

Step 1: Identify the maximum and minimum values.

Maximum value = 18

Minimum value = 9

Step 2: Calculate the range.

\[ \text{Range} = \text{Maximum} - \text{Minimum} = 18 - 9 = 9 \]

Data Representation

The ACT often includes questions about interpreting data from various types of graphs and charts.

Key ACT Concept: Types of Data Displays

  • Bar Graphs: Used to compare categories
  • Line Graphs: Show trends over time or continuous data
  • Pie Charts: Display parts of a whole as percentages
  • Scatter Plots: Show relationships between two variables
  • Box Plots (Box-and-Whisker Plots): Display the distribution of data based on quartiles
  • Histograms: Show frequency distributions of continuous data

Example:

The box plot below shows the distribution of test scores for a class. What is the median score?

Step 1: Identify the median from the box plot.

In a box plot, the vertical line inside the box represents the median.

From the box plot, we can see that the median is 78.

Probability

Probability measures the likelihood of an event occurring. It's a common topic on the ACT Math section.

Key ACT Concept: Basic Probability

  • Probability of an Event: P(Event) = Number of favorable outcomes / Total number of possible outcomes
  • Probability Range: 0 ≤ P(Event) ≤ 1
    • P(Event) = 0: The event is impossible
    • P(Event) = 1: The event is certain
  • Complementary Events: P(not Event) = 1 - P(Event)

Example:

A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. If a marble is drawn at random, what is the probability of drawing a red marble?

Step 1: Count the total number of marbles.

Total number of marbles = 5 + 3 + 2 = 10

Step 2: Calculate the probability.

\[ P(\text{red}) = \frac{\text{Number of red marbles}}{\text{Total number of marbles}} = \frac{5}{10} = \frac{1}{2} = 0.5 \]

The probability of drawing a red marble is 0.5 or 50%.

Key ACT Concept: Compound Probability

  • Independent Events: The outcome of one event does not affect the outcome of the other
    • P(A and B) = P(A) × P(B)
  • Mutually Exclusive Events: Events that cannot occur simultaneously
    • P(A or B) = P(A) + P(B)
  • Non-Mutually Exclusive Events: Events that can occur simultaneously
    • P(A or B) = P(A) + P(B) - P(A and B)

Example:

A die is rolled twice. What is the probability of rolling a 6 on the first roll and an even number on the second roll?

Step 1: Identify the individual probabilities.

P(rolling a 6 on first roll) = 1/6

P(rolling an even number on second roll) = 3/6 = 1/2 (even numbers are 2, 4, 6)

Step 2: Since these are independent events, multiply the probabilities.

\[ P(\text{6 on first AND even on second}) = P(\text{6 on first}) \times P(\text{even on second}) = \frac{1}{6} \times \frac{1}{2} = \frac{1}{12} \]

The probability is 1/12 or approximately 0.083 or 8.3%.

Counting Principles

Counting principles help determine the number of possible outcomes in various scenarios.

Key ACT Concept: Fundamental Counting Principle

If there are m ways to do one task and n ways to do another task, then there are m × n ways to do both tasks.

Example:

A restaurant offers 4 appetizers, 6 main courses, and 3 desserts. How many different three-course meals can be created?

Step 1: Apply the fundamental counting principle.

\[ \text{Number of possible meals} = \text{Number of appetizers} \times \text{Number of main courses} \times \text{Number of desserts} = 4 \times 6 \times 3 = 72 \]

There are 72 different possible three-course meals.

Key ACT Concept: Permutations and Combinations

  • Permutations: Arrangements where order matters
    • P(n,r) = n! / (n-r)! = Number of ways to arrange r items from n distinct items
  • Combinations: Selections where order doesn't matter
    • C(n,r) = n! / [r!(n-r)!] = Number of ways to select r items from n distinct items

Example:

In how many ways can a committee of 3 people be formed from a group of 8 people?

Step 1: Since the order doesn't matter (a committee is a selection, not an arrangement), use the combination formula.

\[ C(8,3) = \frac{8!}{3!(8-3)!} = \frac{8!}{3!5!} = \frac{8 \times 7 \times 6 \times 5!}{3 \times 2 \times 1 \times 5!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = \frac{336}{6} = 56 \]

There are 56 different possible committees of 3 people from a group of 8 people.

Experimental Probability and Expected Value

The ACT may include questions about experimental probability and expected value.

Key ACT Concept: Experimental vs. Theoretical Probability

  • Theoretical Probability: Based on the possible outcomes
  • Experimental Probability: Based on observed results from experiments or trials
    • P(Event) = Number of times event occurs / Total number of trials

Example:

A coin is flipped 200 times, and heads appears 112 times. What is the experimental probability of getting heads?

Step 1: Calculate the experimental probability.

\[ P(\text{heads}) = \frac{\text{Number of times heads appears}}{\text{Total number of flips}} = \frac{112}{200} = 0.56 \]

The experimental probability of getting heads is 0.56 or 56%.

Key ACT Concept: Expected Value

Expected value is the average outcome of a random variable over many trials.

Expected Value = Sum of (Value × Probability)

Example:

In a game, you roll a die. If you roll a 1 or 2, you win $5. If you roll a 3 or 4, you win $10. If you roll a 5 or 6, you lose $15. What is the expected value of this game?

Step 1: Identify the values and their probabilities.

Value of rolling 1 or 2: $5 with probability 2/6 = 1/3

Value of rolling 3 or 4: $10 with probability 2/6 = 1/3

Value of rolling 5 or 6: -$15 with probability 2/6 = 1/3

Step 2: Calculate the expected value.

\[ \text{Expected Value} = (5 \times \frac{1}{3}) + (10 \times \frac{1}{3}) + (-15 \times \frac{1}{3}) = \frac{5}{3} + \frac{10}{3} - \frac{15}{3} = \frac{15 - 15}{3} = 0 \]

The expected value is $0, which means that, on average, you neither win nor lose money in this game over many plays.

Ready to Practice?

Now that you've learned the fundamentals of statistics and probability, test your knowledge with practice problems!

Practice Problems

Test your understanding with these practice problems. Select a difficulty level to begin.

Select a difficulty level to load practice problems.

Statistics & Probability Challenges

Put your statistics and probability skills to the test with these timed challenges. Complete them to earn badges and level up your character!

Data Analysis Challenge

100 XP

Analyze 5 data sets in 5 minutes.

5 minutes
5 problems

Probability Challenge

150 XP

Solve 5 probability problems in 7 minutes.

7 minutes
5 problems

Counting & Combinations Challenge

200 XP

Solve 5 counting problems in 10 minutes.

10 minutes
5 problems

Statistics & Probability ACT Test Simulation

Take a timed test that simulates the statistics and probability questions you'll see on the ACT. This will help you identify your strengths and areas for improvement.

Test Details

  • Time Limit: 15 minutes
  • Questions: 10 statistics and probability questions in ACT format
  • Topics Covered: Descriptive statistics, data representation, probability, counting principles, and expected value
  • Scoring: Instant results with detailed explanations

Ready to Test Your Skills?

This simulation will help you prepare for the actual ACT math section.