Lesson 5 — Probability & Z-scores

Catching z’s

How likely is something to happen — and how do we score one data point against an entire distribution?

Probability basics

The likelihood of an event, expressed as a number from 0 to 1.

0Impossible
0.5Coin flip (heads)
0.99Rhonda being a good girl
1Certain

What is a z-score?

A z-score tells us how far a data point is from the mean, measured in standard deviations.

Z-score formula
z = (value − mean) ÷ standard deviation

Positive z = above the mean  |  Negative z = below the mean  |  z = 0 is exactly the mean

Calculating Rhonda’s z-score

Golden retrievers: mean height = 24″, SD = 2″. Rhonda is 30″.

30 − 24Subtract the mean
6 ÷ 2Divide by SD
z = 33 SD above the mean!

A z-score of 3 means Rhonda is taller than ~99.9% of golden retrievers.

Z-scores → p-values

A z-score table converts any z-score into a probability (p-value).

p < 0.001
Rhonda’s z-score of 3 corresponds to a p-value of ~0.001. Only a 0.1% chance of randomly finding a golden retriever as tall as her!

Higher z-score = rarer value = smaller p-value. Z-scores of 4+ have p-values of 0.00003 or less.

Rhonda by the numbers 🐕

Everything we now know about how special Rhonda is:

Height
30″
Z-score
3.0
P-value
0.001
Taller than
99.9%

Bonus — Other distributions

Not just z-scores

Different tests use different distributions — same idea, different shapes.

📈t-distributionSmall samples, unknown SD
F-distributionComparing variances & ANOVA
𝛘²Chi-squareCategorical data & frequencies
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