12.1 Measures of Central Tendency
A measure of central tendency is a single value that represents a whole set of data. The three main measures of central tendency are Mode, Mean, and Median.
1. Mode
The mode is the value or category with the highest frequency in a set of data.
- A data set can have no mode, one mode (unimodal), or more than one mode (bimodal/multimodal).
- Modal Class: In grouped data or frequency tables, the class with the highest frequency is called the modal class.
2. Mean
The mean is the sum of all data values divided by the total number of data values.
- For Ungrouped Data:
$$\text{Mean } (\bar{x}) = \frac{\sum x}{N}$$
where $\sum x$ is the sum of all data values and $N$ is the total number of data values.
- For Data in a Frequency Table:
$$\text{Mean } (\bar{x}) = \frac{\sum (f \cdot x)}{\sum f}$$
where $f$ is the frequency and $x$ is the data value or midpoint.
3. Median
The median is the middle value when a set of data is arranged in ascending or descending order.
- If total data points ($N$) is odd: The median is the middle item at position $\frac{N + 1}{2}$.
- If total data points ($N$) is even: The median is the average of the two middle items at positions $\frac{N}{2}$ and $\left(\frac{N}{2} + 1\right)$.
4. Effects of Extreme Values (Outliers)
- Mean: Heavily affected by extreme values/outliers.
- Median: Not affected by extreme values; best measure when extreme values exist.
- Mode: Useful for categorical data (e.g., shoe sizes, favorite colors).