Mean, Median, Mode, and Range

Mean (Average)

Add all values and divide by the number of values:

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

Example: Data: $4, 7, 9, 3, 12$ $\text{Mean} = \frac{4 + 7 + 9 + 3 + 12}{5} = \frac{35}{5} = 7$

Median (Middle Value)

1. Put values in order from least to greatest
2. Find the middle value

Odd number of values: $3, 4, 7, 9, 12$ → Median = $7$

Even number of values: $3, 4, 7, 9$ → Median = $\frac{4 + 7}{2} = 5.5$

Mode (Most Common)

The value that appears most often.

Example: $2, 3, 3, 5, 7, 3, 8$ → Mode = $3$

A data set can have:

• One mode
• More than one mode (bimodal, multimodal)
• No mode (all values appear once)

Range (Spread)

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

Example: $3, 4, 7, 9, 12$ → Range = $12 - 3 = 9$

Which Measure to Use?

| Situation | Best Measure | |-----------|-------------| | Symmetric data, no outliers | Mean | | Skewed data or outliers | Median | | Categorical data | Mode | | How spread out | Range |

Effect of Outliers

An outlier is a value much larger or smaller than the rest.

Data: $5, 6, 7, 8, 100$

• Mean = $\frac{126}{5} = 25.2$ (pulled up by outlier)
• Median = $7$ (not affected)

Key idea: The mean is sensitive to outliers; the median is resistant to outliers.