How to Solve: ACT Math – Probability and Statistics (Mean, Median, Mode, Expected Value)

How to Solve: ACT Math – Probability and Statistics (Mean, Median, Mode, Expected Value)

Introduction

"Mastering mean, median, mode, and expected value doesn’t just boost your ACT Math score—it unlocks 5-7 easy points on test day, often in questions most students skip. These concepts appear in 10-15% of ACT Math problems, and if you follow this exact method, you’ll solve them in under 60 seconds—guaranteed."

What You Need To Know First

1. Basic arithmetic (addition, division, multiplication).
2. Ordering numbers (smallest to largest).
3. Simple probability (fraction = favorable outcomes / total outcomes).

Key Vocabulary

Formulas To Know

1. Mean (Average)
   [
   \text{Mean} = \frac{\text{Sum of all numbers}}{\text{Number of items}}
   ]

2. Memorise This. (not given on ACT formula sheet).

3. Median

4. Odd number of items: Middle number.
5. Even number of items: Average of two middle numbers.

6. Memorise This. (not given on ACT formula sheet).

7. Mode

8. The number that appears most frequently.

9. Memorise This. (not given on ACT formula sheet).

10. Expected Value
    [
    \text{Expected Value} = \sum (x \times P(x))
    ]

11. (x) = outcome value
12. (P(x)) = probability of outcome

13. Memorise This. (not given on ACT formula sheet).

14. Probability of an Event
    [
    P(\text{Event}) = \frac{\text{Favorable outcomes}}{\text{Total possible outcomes}}
    ]

15. Memorise This. (not given on ACT formula sheet).

Step-by-Step Method

How to Find the Mean

1. Add all numbers together.
2. Count how many numbers there are.
3. Divide the sum by the count.

How to Find the Median

1. Order the numbers from smallest to largest.
2. If odd count: Pick the middle number.
3. If even count: Average the two middle numbers.

How to Find the Mode

1. Count how many times each number appears.
2. Pick the number with the highest count.
3. If tied, list all modes (can have multiple).

How to Find Expected Value

1. List all possible outcomes and their probabilities.
2. Multiply each outcome by its probability.
3. Add all these products together.

Worked Examples

Example 1 – Basic (Mean, Median, Mode)

Problem: Find the mean, median, and mode of: 3, 7, 2, 7, 6.

Step-by-Step Solution: 1. Mean:
- Sum = 3 + 7 + 2 + 7 + 6 = 25
- Count = 5
- Mean = 25 / 5 = 5

1. Median:
2. Order: 2, 3, 6, 7, 7

3. Middle number = 6

4. Mode:

5. Counts: 2 (1), 3 (1), 6 (1), 7 (2)
6. Mode = 7 (appears most)

What we did and why: - Mean = average, so we summed and divided. - Median = middle number, so we ordered first. - Mode = most frequent, so we counted repeats.

Example 2 – Medium (Even Count Median, Outlier)

Problem: Find the median of: 12, 5, 20, 8, 15, 3.

Step-by-Step Solution: 1. Order: 3, 5, 8, 12, 15, 20 2. Even count (6 numbers), so average the 3rd and 4th:
- 3rd = 8, 4th = 12
- Median = (8 + 12) / 2 = 10

What we did and why: - Ordered first because median requires sorted data. - Even count = average two middle numbers.

Example 3 – Exam-Style (Expected Value)

Problem: A game costs $5 to play. You roll a die: - Roll 1 or 2 → Win $10 - Roll 3, 4, or 5 → Win $3 - Roll 6 → Win $0 What is the expected value of playing?

Step-by-Step Solution: 1. List outcomes and probabilities:
- $10 win: P = 2/6 = 1/3
- $3 win: P = 3/6 = 1/2
- $0 win: P = 1/6

1. Calculate expected winnings:
2. $10 × (1/3) = $3.33
3. $3 × (1/2) = $1.50
4. $0 × (1/6) = $0

5. Total expected winnings = $3.33 + $1.50 + $0 = $4.83

6. Subtract cost to play:

7. Expected value = $4.83 - $5 = -$0.17

What we did and why: - Expected value = average outcome over time. - Subtracted cost because the problem asked for net value.

Common Mistakes

Exam Traps

1-Minute Recap

"Night before the ACT? Here’s the crash course: 1. Mean = sum ÷ count. Add ‘em up, divide. 2. Median = middle number. Order first, then pick. 3. Mode = most frequent. Count repeats. 4. Expected value = (outcome × probability) for each, then add. 5. Watch for outliers—mean changes, median stays strong. 6. If a problem gives the mean and asks for a missing number, plug into the formula and solve for x. 7. Always read: ‘with replacement’ or ‘without’? That changes the total!

You’ve got this. Now go crush those 5-7 points!