Statistics and Probability (ACT Math)

Statistics on the ACT

Measures of Central Tendency

Mean (Average):

$\text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}}$

Example: Find the mean of 3, 7, 9, 12, 14

$\text{Mean} = \frac{3 + 7 + 9 + 12 + 14}{5} = \frac{45}{5} = 9$

Median (Middle Value):

Steps:

1. Order the data from least to greatest
2. If odd number of values: middle value
3. If even number of values: average of two middle values

Example 1 (odd): 3, 5, 7, 9, 11 → Median = 7

Example 2 (even): 2, 5, 8, 10 → Median = $\frac{5 + 8}{2} = 6.5$

Mode:

The value that appears most frequently

Example: 2, 3, 3, 5, 7, 7, 7, 9 → Mode = 7

Note: Can have multiple modes or no mode

Measures of Spread

Range:

$\text{Range} = \text{Maximum} - \text{Minimum}$

Example: For data 3, 7, 12, 15, 20

$\text{Range} = 20 - 3 = 17$

Standard Deviation:

Measures how spread out the data is from the mean

• Small standard deviation: Data clustered near mean
• Large standard deviation: Data spread out

ACT Tip: You won't calculate standard deviation by hand — just understand what it means!

Box Plots (Box-and-Whisker Plots)

Five-number summary:

1. Minimum: Smallest value
2. Q1 (First Quartile): Median of lower half
3. Q2 (Median): Middle value
4. Q3 (Third Quartile): Median of upper half
5. Maximum: Largest value

Interquartile Range (IQR): $\text{IQR} = Q3 - Q1$

Example: Data: 2, 4, 6, 8, 10, 12, 14, 16, 18

• Minimum: 2
• Q1: 5 (median of 2, 4, 6, 8)
• Median: 10
• Q3: 15 (median of 12, 14, 16, 18)
• Maximum: 18
• IQR: $15 - 5 = 10$

Outliers

Definition: Data points significantly different from others

Rule: A value is an outlier if:

• Less than $Q1 - 1.5 \times \text{IQR}$, OR
• Greater than $Q3 + 1.5 \times \text{IQR}$

Example: With $Q1 = 5$, $Q3 = 15$, $\text{IQR} = 10$:

• Lower fence: $5 - 1.5(10) = -10$
• Upper fence: $15 + 1.5(10) = 30$
• Any value < -10 or > 30 is an outlier

Probability on the ACT

Basic Probability

$P(\text{event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$

Requirements:

• $0 \leq P(\text{event}) \leq 1$
• Probability of 0 = impossible
• Probability of 1 = certain
• Often expressed as fraction, decimal, or percent

Example: What's the probability of rolling a 4 on a standard die?

$P(4) = \frac{1}{6}$

Complementary Events

Complement rule: $P(\text{not } A) = 1 - P(A)$

Example: If probability of rain is 0.3, probability of no rain is: $P(\text{no rain}) = 1 - 0.3 = 0.7$

Multiple Events

Independent Events: One event doesn't affect the other

Multiplication rule for independent events: $P(A \text{ and } B) = P(A) \times P(B)$

Example: Flip a coin and roll a die. What's the probability of heads AND rolling a 5?

$P(\text{heads and 5}) = \frac{1}{2} \times \frac{1}{6} = \frac{1}{12}$

Dependent Events: First event affects the second

Example: Draw 2 cards from a deck without replacement. What's the probability both are aces?

First card: $P(\text{ace}) = \frac{4}{52}$

Second card: $P(\text{ace}|\text{first was ace}) = \frac{3}{51}$ (only 3 aces left in 51 cards)

Both aces: $\frac{4}{52} \times \frac{3}{51} = \frac{12}{2652} = \frac{1}{221}$

"OR" Probabilities

Addition rule for mutually exclusive events (can't happen together): $P(A \text{ or } B) = P(A) + P(B)$

Example: Drawing a 5 OR a 6 from a standard deck: $P(5 \text{ or } 6) = \frac{4}{52} + \frac{4}{52} = \frac{8}{52} = \frac{2}{13}$

If NOT mutually exclusive: $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$

Example: Drawing a heart OR a king: $P(\text{heart or king}) = \frac{13}{52} + \frac{4}{52} - \frac{1}{52} = \frac{16}{52} = \frac{4}{13}$

(Subtract $\frac{1}{52}$ because king of hearts is counted twice)

Data Interpretation

Tables and Charts

ACT will give you data in tables — read carefully!

Example: Survey of 100 students

| | Freshman | Sophomore | Total | |-----------|----------|-----------|-------| | Own car | 5 | 20 | 25 | | No car | 45 | 30 | 75 | | Total | 50 | 50 | 100 |

Questions:

• Probability a randomly selected student is a freshman who owns a car:
  $P = \frac{5}{100} = 0.05$

• Probability a student owns a car, given they're a sophomore:
  $P = \frac{20}{50} = 0.4$

Scatterplots

Correlation types:

Positive correlation: As $x$ increases, $y$ increases
Negative correlation: As $x$ increases, $y$ decreases
No correlation: No clear relationship

Strong vs weak:

• Strong: Points close to a line
• Weak: Points scattered

ACT Question Types

Type 1: Calculate Mean, Median, Mode

Strategy:

• Mean: Add all, divide by count
• Median: Order data, find middle
• Mode: Find most frequent

If they add a value and ask new mean: $\text{New mean} = \frac{\text{Old sum} + \text{New value}}{\text{New count}}$

Type 2: Box Plot Interpretation

Strategy:

• Know what each part represents
• Q1, Q2 (median), Q3 are marked
• IQR = $Q3 - Q1$

Type 3: Basic Probability

Strategy:

• Count favorable outcomes (numerator)
• Count total possible outcomes (denominator)
• Simplify fraction

Type 4: Complementary Probability

Strategy:

• If asked "at least one," use complement
• $P(\text{at least one}) = 1 - P(\text{none})$

Example: Probability at least one head in 3 coin flips?

$P(\text{at least one H}) = 1 - P(\text{all T}) = 1 - \left(\frac{1}{2}\right)^3 = 1 - \frac{1}{8} = \frac{7}{8}$

Type 5: Multiple Events

Strategy:

• Identify if independent or dependent
• If independent: multiply probabilities
• If dependent: adjust second probability

Type 6: Conditional Probability

"Given that" or "if" signals conditional probability

Strategy: Use only the subset that meets the condition

Example: Given a student is a sophomore (from table above), probability they own a car:

$P(\text{car}|\text{sophomore}) = \frac{20}{50} = 0.4$

Use sophomore column as your total (50), not whole table (100)

Common ACT Mistakes

❌ Forgetting to order data before finding median
❌ Dividing by wrong number for mean (count ALL values)
❌ Adding probabilities for "AND" (should multiply for independent)
❌ Not adjusting for dependent events (deck gets smaller after first card)
❌ Using whole population instead of subset for conditional probability
❌ Confusing range with IQR (range = max - min; IQR = Q3 - Q1)

Quick Tips for ACT

✓ Mean is affected by outliers — median is more resistant
✓ Complement rule saves time for "at least one" problems
✓ AND = multiply, OR = add (for mutually exclusive)
✓ Dependent events: Adjust denominator and numerator
✓ Conditional probability: Focus only on the given condition
✓ Box plots: Middle line is MEDIAN, not mean
✓ Probability is never > 1 — if you get > 1, you made an error

Formula Quick Reference

| Concept | Formula | |---------|---------| | Mean | $\frac{\sum x}{n}$ | | Range | Max - Min | | IQR | $Q3 - Q1$ | | Basic Probability | $\frac{\text{favorable}}{\text{total}}$ | | Complement | $P(\text{not } A) = 1 - P(A)$ | | Independent AND | $P(A \cap B) = P(A) \times P(B)$ | | Mutually Exclusive OR | $P(A \cup B) = P(A) + P(B)$ |

Practice Approach

1. Read carefully — is it asking for mean, median, or mode?
2. Organize data if needed (order for median)
3. Identify probability type — basic, complement, AND, OR?
4. Set up fraction with favorable/total
5. Simplify — ACT usually wants simplified fractions or decimals
6. Check reasonableness — probability should be between 0 and 1

Remember: Statistics and probability on the ACT test core concepts. Know your formulas, understand the difference between mean/median, and practice probability rules!