Basic Statistical Measures
Understanding how to summarize, describe, and interpret data using measures of central tendency and dispersion.
1. Measures of Central Tendency
Measures of central tendency describe the center or typical value of a dataset. The three primary measures are the mean, median, and mode.
Mean (Arithmetic Average)
Definition: The sum of all data values divided by the number of values.
$$ \bar{x} = \frac{\sum_{i=1}^{n} x_i}{n} = \frac{x_1 + x_2 + \cdots + x_n}{n} $$
Properties:
- Uses every data point
- Sensitive to outliers
- The "balance point" of the data
Median
Definition: The middle value when data is arranged in ascending order.
If $n$ is odd: $$ \text{Median} = x_{\frac{n+1}{2}} $$
If $n$ is even: $$ \text{Median} = \frac{x_{\frac{n}{2}} + x_{\frac{n}{2}+1}}{2} $$
Properties:
- Resistant to outliers
- Better for skewed data
- Divides data into two equal halves
Mode
Definition: The value(s) that appear most frequently in the dataset.
Types:
- Unimodal: One mode
- Bimodal: Two modes
- Multimodal: More than two modes
- No mode: All values appear equally
Properties:
- Only measure for categorical data
- Can have multiple values