📊 Data and Statistics Formulas

🎯 Measures of Center

Mean (Average)

$$\text{Mean} = \frac{\sum x_i}{n} = \frac{x_1 + x_2 + \cdots + x_n}{n}$$

Usage: Find the average of a set of numbers Micro-example: Mean of 2, 4, 6 is $\frac{2+4+6}{3} = 4$

Median

Odd number of values: Middle value when ordered Even number of values: Average of two middle values

Usage: Find the middle value of ordered data Micro-example: Median of 1, 3, 7, 9, 12 is 7; Median of 2, 4, 6, 8 is $\frac{4+6}{2} = 5$

Mode

The value that appears most frequently in the data set.

Usage: Find the most common value Micro-example: Mode of 2, 3, 3, 4, 5 is 3

📏 Measures of Spread

Range

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

Usage: Find how spread out the data is Micro-example: Range of 5, 8, 12, 15 is $15 - 5 = 10$

Standard Deviation

$$\sigma = \sqrt{\frac{\sum(x_i - \mu)^2}{n}}$$

Usage: Measure how spread out data is from the mean Note: Usually calculated with calculator for MATHCOUNTS

🎲 Probability Formulas

Basic Probability

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

Usage: Find probability of a simple event Micro-example: Probability of rolling a 3 on a die is $\frac{1}{6}$

Complementary Events

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

Usage: Find probability of the opposite event Micro-example: If $P(\text{rain}) = 0.3$, then $P(\text{no rain}) = 0.7$

Independent Events

$$P(\text{A and B}) = P(\text{A}) \times P(\text{B})$$

Usage: Find probability of both events occurring Micro-example: Probability of rolling two 3’s is $\frac{1}{6} \times \frac{1}{6} = \frac{1}{36}$

📊 Weighted Averages

Weighted Mean

$$\text{Weighted Mean} = \frac{\sum(w_i \cdot x_i)}{\sum w_i}$$

Usage: Find average when different values have different weights Micro-example: Grades 80, 90 with weights 2, 3: $\frac{2 \cdot 80 + 3 \cdot 90}{2 + 3} = \frac{430}{5} = 86$

🔢 Counting Formulas

Factorial

$$n! = n \times (n-1) \times (n-2) \times \cdots \times 1$$

Usage: Count arrangements of distinct objects Micro-example: $4! = 4 \times 3 \times 2 \times 1 = 24$

Permutations

$$P(n,r) = \frac{n!}{(n-r)!}$$

Usage: Count arrangements of r objects from n objects Micro-example: $P(5,3) = \frac{5!}{2!} = \frac{120}{2} = 60$

Combinations

$$C(n,r) = \frac{n!}{r!(n-r)!}$$

Usage: Count ways to choose r objects from n objects Micro-example: $C(5,3) = \frac{5!}{3!2!} = \frac{120}{6 \cdot 2} = 10$

💡 Quick Reference

⚠️ Common Pitfalls

• Median: Always order data first
• Weighted average: Don’t forget to divide by total weight
• Probability: Count carefully, check your work
• Factorial: Remember $0! = 1$
• Permutations vs Combinations: Order matters in permutations