Central Tendency and Dispersion

Where is the middle, and how spread out is it?

Every dataset has a "typical" value (central tendency) and a spread (dispersion). Describing both tells you far more than a single number ever could.

Measures of center

• Mean — the arithmetic average. Sensitive to outliers.
• Median — the middle value when sorted. Robust to outliers.
• Mode — the most frequent value. Useful for categorical data.

Scores: [3, 3, 4, 5, 99]
Mean   = 22.8   (yanked up by 99)
Median = 4      (the middle value)
Mode   = 3      (appears twice)

When data is skewed, the median describes the "typical" experience far better than the mean. Income, housing prices, and web traffic are classic right-skewed datasets.

Measures of spread

• Range — max minus min. Simple but dominated by extremes.
• Variance — the average squared distance from the mean.
• Standard deviation — the square root of variance, in the same units as the data.

Variance          = Σ(xᵢ - mean)² / n
Standard deviation = √variance

Because variance squares the distances, it penalizes large deviations heavily. Standard deviation brings the result back to the original units so it's interpretable — "average temperature varies by about 3°C".

When median beats mean

Use the median whenever a few extreme values would distort the mean — salaries, net worth, city populations, response times. The mean is best for roughly symmetric data.