Descriptive Statistics: Summarizing Data Distributions
**Descriptive statistics** are mathematical tools used to summarize and organize the features of a specific dataset. Rather than drawing inferences about a wider parent population, descriptive stats focus entirely on characterizing the properties of the observed sample.
Measures of Central Tendency
Central tendency summarizes the "middle" or "central" location of a dataset:
- Mean (Average): Calculated by summing all observations and dividing by the total count: Mean = Sum(x_i) / N. The mean is highly sensitive to outliers.
- Median (Q2): The exact middle value when values are ordered from smallest to largest. If the dataset size is even, the median is the average of the two middle numbers. The median is resistant to extreme values.
- Mode: The value that occurs most frequently in the dataset. A dataset can have one mode (unimodal), multiple modes (multimodal), or no mode if all values appear with equal frequency.
Measures of Dispersion (Spread)
Dispersion determines how spread out or clustered together the values are:
- Variance: The average of the squared differences from the Mean. The sample variance formula divides the sum of squared differences by $N-1$:
s² = ∑(x_i - mean)² / (N - 1)
- Standard Deviation: The square root of the Variance, representing the average distance of observations from the Mean. Because it is calculated in the same units as the original data, standard deviation is more intuitive than variance.
- Interquartile Range (IQR): The distance between the 75th and 25th percentiles ($IQR = Q_3 - Q_1$). This captures the middle 50% of the dataset and is highly resistant to extreme outliers.