The average (arithmetic mean) of a set of numbers is the sum of the numbers divided by how many numbers there are.
Explanation — Arithmetic mean (most common)
- Formula: $$\bar{x}=\frac{1}{n}\sum_{i=1}^n x_i$$
- Steps:
- Add all the values: $S=\sum_{i=1}^n x_i$.
- Count how many values there are: $n$.
- Divide the sum by the count: $\bar{x}=S/n$.
Example:
- Numbers: 4, 7, 9
- Sum: $S=4+7+9=20$
- Count: $n=3$
- Mean: $$\bar{x}=\frac{20}{3}\approx6.67$$
Other common “averages”
- Median — the middle value when data are sorted. If there are two middle values, the median is their mean. Use median when data are skewed or have outliers.
- Example: For 2, 3, 9, 50 → median is $(3+9)/2=6$.
- Mode — the most frequently occurring value(s). Useful for categorical data.
- Example: In 1, 2, 2, 3 → mode is 2.
- Weighted average — use when values have different importance/weights.
- Formula: $$\bar{x}=\frac{\sum_{i} w_i x_i}{\sum_{i} w_i}$$
- Example: Grades: 80 (homework, weight 30%), 90 (exam, weight 70%): $$\bar{x}=\frac{0.3\cdot80+0.7\cdot90}{0.3+0.7}=87$$
- Geometric mean — for growth rates or multiplicative processes:
- $$\text{GM}=\left(\prod_{i=1}^n x_i\right)^{1/n}$$
- Harmonic mean — for rates (e.g., average speed when time per segment is equal):
- $$\text{HM}=\frac{n}{\sum_{i=1}^n \frac{1}{x_i}}$$
Tips
- Choose the measure that fits the situation: median for skewed data, mean for symmetric numeric data, mode for categorical.
- Watch units and rounding: keep appropriate precision based on context.
If you want, tell me a specific dataset and I’ll compute its mean/median/mode and explain which average suits it best.