Cheatsheet: Summarizing data: center and spread

The one idea

Every summary answers two questions: where is the center of the data, and how spread out is it? Center without spread is half a description, and the wrong center (mean on skewed data) is a lie.

The three centers

Center	How	Uses every value?	Outlier-sensitive?	Best for
Mean	Add all, divide by count	Yes	Yes (dragged by outliers)	Symmetric, outlier-free data
Median	Middle value when sorted	No	No (robust)	Skewed data, data with outliers
Mode	Most frequent value	No	No	Categorical data

When the mean and median disagree, the data is skewed; report the median as the typical value.

The spread measures

Measure	How	Note
Range	Max minus min	Simple but fragile; only the two extremes
Variance	Average of squared distances from the mean	In squared units; hard to read directly
Standard deviation	Square root of the variance	Typical distance from the mean, in original units

Worked example (eight scores)

Data: 2, 4, 4, 4, 5, 5, 7, 9
Mean = 40 / 8 = 5      Median = (4+5)/2 = 4.5      Mode = 4
Squared distances from 5:  9,1,1,1,0,0,4,16  (sum = 32)
Variance = 32 / 8 = 4      Standard deviation = sqrt(4) = 2

Standardizing a feature (machine learning)

standardized value = (value - mean) / standard deviation

Recenters the feature at 0, rescales its spread to be comparable across features, and keeps a large-unit feature (income) from numerically dominating a small-unit one (age). The standardized value is the z-score (a later lesson).

Pitfalls to dodge

• Reporting the mean for skewed data (use the median).
• Reporting a center with no spread.
• Confusing variance (squared units) with standard deviation (original units).
• Using the range as your spread (one outlier blows it up).
• Forgetting the mode is the only center that works for categories.

Words to use precisely

• Mean: the arithmetic average; uses every value; outlier-sensitive.
• Median: the middle value when sorted; robust to outliers.
• Mode: the most frequent value; the center for categorical data.
• Variance: average squared distance from the mean.
• Standard deviation: square root of variance; typical distance from the mean, in original units.
• Skew: a few extreme values stretching one tail, pushing the mean away from the median.