What Is Z Star (z*)?
In many textbooks — Moore's The Basic Practice of Statistics above all — the critical value in a confidence interval is written z*, "z star". It is not a new distribution or a new table: z* is simply the standard normal value that captures the stated confidence in the middle of the curve.
The definition: z* is the number of standard deviations either side of the mean that traps the confidence level's area in the centre. For 95% confidence, z* = 1.96; for 90%, 1.645; for 99%, 2.576. The margin of error is then z* × (standard error).
The standard z* values
- 90% confidence → z* = 1.645 (5% left in each tail).
- 95% confidence → z* = 1.960 (2.5% in each tail — the famous one).
- 98% confidence → z* = 2.326 (1% in each tail).
- 99% confidence → z* = 2.576 (0.5% in each tail).
To find z* for any other level — say 92% — run the reverse lookup: the tails share (1 − 0.92) = 8%, so each tail holds 4%, and z* is the value with cumulative area 0.96, namely 1.751. That is the reverse table search, and the z table for confidence intervals page tabulates the results for the levels in common use. In the zα notation of the neighbouring post, z* for confidence level 1 − α is exactly zα/2 — two naming systems, one number, chosen by which textbook raised you.
z* at work
Margin of error for a mean: m = z* × σ/√n. Estimating a population mean with σ = 12 from n = 36 at 95% confidence: m = 1.96 × (12/6) = 3.92 — report x̄ ± 3.92. Notice what choosing the level does through z*: moving from 95% to 99% confidence widens the margin by the ratio 2.576/1.96 ≈ 1.31, about 31% more width for 4 points more confidence. That trade — width for assurance, priced entirely by z* — is the honest content of every confidence interval, and the reason "how to find z star" is really the question "how sure do we need to be, and what will it cost?". Sample-size planning runs the same formula backwards: choose the margin you can tolerate, read z* for your level, solve for n.
For completeness, the lower-confidence members of the family: 80% confidence gives z* = 1.282 (10% in each tail) and 85% gives z* ≈ 1.440. They appear in pilot studies and rough planning estimates, where a narrower interval at a honestly lower confidence is the right trade. The pattern to internalise is monotonic and smooth: every step up in confidence is bought with width, and z* is the exchange rate — which is why this one value, under either of its names, sits at the centre of interval estimation.
Also asked as: how to find z star · what is z star in statistics · how to find the z star value · z star critical value.
Related questions
Frequently asked questions
Is z* the same as a z score?
No. A z score is computed from data (a value's position); z* is chosen from a table (a boundary for a confidence level). They share the z scale and nothing else — one is an observation, the other a design decision.
Why do some books write z* and others zα/2?
Different textbook traditions naming the same critical value: z* ("z star") in Moore-style texts, zα/2 in Neyman–Pearson-flavoured texts where α is the significance level. For 95% confidence both give 1.96. Recognising both notations is the entire adjustment needed when switching books.