Averages and Measures of Spread

1. Types of Averages (Central Tendency)

Averages provide a single value that represents a data set.

Mean

The sum of all values divided by the total number of values.

  • Non-grouped data: $\text{Mean} = \frac{\sum x}{n}$.
  • Frequency data: $\text{Mean} = \frac{\sum (f \cdot x)}{\sum f}$ (where $f$ is frequency and $x$ is the value/midpoint).

Median

The middle value when data is ordered from least to greatest.

  • Odd number of values: The exact middle term.
  • Even number of values: The average of the two middle terms.

Mode

The value that appears most frequently.

  • A data set can be unimodal (one mode), bimodal (two modes), or have no mode.

2. Measures of Spread

Describes how dispersed the data is around the average.

Range

The difference between the maximum and minimum values. $$\text{Range} = \text{Maximum} - \text{Minimum}$$

Interquartile Range (IQR)

The range of the middle 50% of the data. $$\text{IQR} = Q_3 - Q_1$$

3. Quartiles

Values that divide the ordered data into four equal parts.

  • Lower Quartile ($Q_1$): The median of the lower half of the data (25th percentile).
  • Upper Quartile ($Q_3$): The median of the upper half of the data (75th percentile).

Box plot showing Q1, Median, Q3, and IQR

4. Grouped Data (Extended)

When data is given in classes (e.g., $0 < x \le 10$), we use estimates.

Estimated Mean

Since exact values are unknown, use the midpoint of each class.

  1. Find midpoint $m$ for each class.
  2. Calculate $\sum (f \cdot m)$.
  3. Divide by total frequency $\sum f$.

The class interval with the highest frequency.