1. Measures of Central Tendency
Central tendency summarizes an entire dataset with a single representative benchmark:
- Mean (Arithmetic Average): The sum of all values divided by total count:
x̄ = Σx / n. Highly sensitive to extreme outliers. - Median: The middle number when the dataset is sorted in numerical order. Robust against extreme skewness.
- Mode: The value that appears most frequently in the sample. A dataset can have one mode, multiple modes, or no mode.
2. Measures of Spread: Range, Variance & Standard Deviation
While central tendency tells you where data clusters, measures of dispersion indicate how spread out observations are:
- Range:
Max − Min. Quick to compute but ignores internal distribution. - Sample Variance (s²): Average of squared deviations from the mean:
s² = Σ(xᵢ − x̄)² / (n − 1). - Sample Standard Deviation (s): The square root of variance:
s = √s². Expressed in the exact same measurement units as raw data.
3. Worked Example with Dataset
Consider quiz scores from 10 students: 12, 15, 18, 20, 25, 18, 30, 22, 18, 27.
- Count: n = 10
- Sum: 205
- Mean: 205 / 10 = 20.50
- Sorted: 12, 15, 18, 18, 18, 20, 22, 25, 27, 30. Middle values are 18 and 20 ⟹ Median = 19.00.
- Mode: 18 occurs three times ⟹ Mode = 18.
- Sample Std Dev: 5.42