---
title: "Critical Value (z*) — AP Stats Definition & Exam Guide"
description: "Critical value z* is the standard normal cutoff matching your confidence level. Multiply z* by standard error to get margin of error in Unit 6 z-intervals."
canonical: "https://fiveable.me/ap-stats/key-terms/critical-value-z"
type: "key-term"
subject: "AP Statistics"
unit: "Unit 3"
---

# Critical Value (z*) — AP Stats Definition & Exam Guide

## Definition

The critical value z* is the number of standard errors you go out from a point estimate to capture the middle C% of the standard normal distribution; in AP Stats Unit 6, margin of error = z* × SE, so z* = 1.645 for 90%, 1.96 for 95%, and 2.576 for 99% confidence.

## What It Is

The [critical value](/ap-stats/key-terms/critical-value "fv-autolink") z* is a cutoff from the standard normal distribution. Per the CED (6.2.D), critical values represent the [boundaries](/ap-stats/unit-2/normal-distribution-revisited/study-guide/dx4vMcx3WjSw68f1Ov66 "fv-autolink") that capture the middle C% of the standard normal curve, where C is your confidence level. Want 95% confidence? You need the z-scores that trap the middle 95% of the curve, which are -1.96 and +1.96, so z* = 1.96. For 90% it's 1.645, and for 99% it's 2.576.

Here's the intuition. A confidence interval is point estimate ± margin of error, and z* is the 'how many standard errors wide' part of that margin. The standard error tells you the size of one step; z* tells you how many steps to take. Higher confidence means you need to cast a wider net, so z* grows. That's the whole reason a 99% [interval](/ap-stats/unit-1/representing-quantitative-variable-with-graphs/study-guide/VWtyLVDvjzEgtbAi6v6j "fv-autolink") is wider than a 90% interval built from the same data. You find z* on your calculator with invNorm (for 95%, invNorm(0.975) because 2.5% sits in each tail) or by reading Table A in reverse.

## Why It Matters

z* lives in Unit 6 ([Inference for Categorical Data: Proportions](/ap-stats/unit-3 "fv-autolink")), specifically Topics 6.2 and 6.8. Learning objective 6.2.C says the [margin of error](/ap-stats/key-terms/margin-of-error "fv-autolink") for a proportion is the critical value (z*) times the standard error, giving z*√(p̂(1-p̂)/n). Then 6.2.D and 6.8.C use z* to build the full intervals, p̂ ± z*·SE for one proportion and (p̂₁ - p̂₂) ± z*·SE for two. The interval formulas aren't on the exam formula sheet, but you don't need to memorize them. The standard error formulas are on the sheet, and you just bolt z* onto them. z* also shows up when you rearrange the margin of error formula to solve for the minimum sample size n, a classic exam move. Bottom line, if you can't find z*, you can't build any z-interval, and you can't explain why confidence level changes interval width.

## Connections

### [Confidence Level (Unit 6)](/ap-stats/key-terms/confidence-level)

z* is literally the [confidence level](/ap-stats/key-terms/confidence-level "fv-autolink") translated into a z-score. C% confidence means z* marks the boundaries of the middle C% of the standard normal curve, so changing C changes z* and nothing else in the formula.

### [Confidence Interval (Unit 6)](/ap-stats/key-terms/confidence-interval)

Every z-interval has the same skeleton, [point estimate](/ap-stats/key-terms/point-estimate "fv-autolink") ± z*·SE. z* is the stretch factor that decides how wide the net is, while the standard error sets the scale of one unit of stretch.

### [Empirical Rule (Unit 1)](/ap-stats/key-terms/empirical-rule)

You've already met z* in disguise. The [Empirical Rule](/ap-stats/key-terms/empirical-rule "fv-autolink")'s 'about 95% within 2 standard deviations' is the rough version of z* = 1.96 for a 95% interval. Inference just recycles the normal curve you learned at the start of the course.

### Critical value t* for means (Unit 7)

When you do inference for means instead of proportions, the critical value comes from a t-distribution (t*) instead of the standard normal (z*), because you're estimating the standard deviation from the sample. Proportions get z*, means get t*. Mixing these up is one of the most common point-losers on inference FRQs.

## On the AP Exam

Multiple choice loves the conceptual angle. A typical stem gives you a confidence interval, then asks what happens if the confidence level changes. For example, a 95% interval for a proportion is (0.42, 0.58); switch to 99% and the interval gets wider, same center, because z* jumped from 1.96 to 2.576. The same logic gets tested with two-proportion intervals like (0.05, 0.15) at 90% confidence, where bumping to 99% widens the interval around the same point estimate. You should also expect questions where lowering the confidence level (99% to 95%) is the change that narrows an interval. On FRQs, z* shows up inside the full confidence interval procedure. You name the interval (one-sample or two-sample z-interval for proportions), check conditions, then compute p̂ ± z*·SE with the correct z* for the stated confidence level. Using the wrong z*, or using a t* when the problem is about proportions, costs you the 'correct mechanics' component of the scoring.

## critical value (z*) vs z test statistic (z)

Both are z-scores, but they answer opposite questions. The critical value z* is chosen before you look at the data; it comes straight from the confidence level (1.96 for 95%, always). The test statistic z is computed from the data; it measures how far your sample result sits from the null hypothesis value. z* sets a boundary, z is your sample's actual location. In a significance test you compare them; in a confidence interval you only use z*.

## Key Takeaways

- The critical value z* marks the boundaries of the middle C% of the standard normal distribution, where C is your confidence level.
- Memorize the big three values, z* = 1.645 for 90% confidence, 1.96 for 95%, and 2.576 for 99%.
- Margin of error equals z* times the standard error, so increasing the confidence level increases z* and widens the interval without moving its center.
- Find z* with invNorm by putting half the leftover area in each tail, so for 95% confidence you compute invNorm(0.975).
- Use z* for inference about proportions (Unit 6) and t* for inference about means (Unit 7).
- To find the minimum sample size for a desired margin of error, rearrange ME = z*√(p̂(1-p̂)/n) and solve for n.

## FAQs

### What is the critical value z* in AP Stats?

z* is the z-score that captures the middle C% of the standard normal distribution for a C% confidence level. You multiply it by the standard error to get the margin of error, so a 95% interval for a proportion is p̂ ± 1.96·SE.

### What is z* for a 95% confidence interval?

z* = 1.96 for 95% confidence. That leaves 2.5% in each tail, which is why you compute it as invNorm(0.975) on your calculator. For 90% it's 1.645 and for 99% it's 2.576.

### Is the critical value the same as the test statistic?

No. The critical value z* depends only on the confidence level and is fixed before you see any data, while the test statistic z is calculated from your sample and measures distance from the null hypothesis. They're both z-scores, but they play completely different roles.

### Do I use z* or t* for a confidence interval?

Use z* for proportions and t* for means. Proportions (Unit 6) use the standard normal distribution because the standard error formula doesn't require estimating a separate standard deviation, while means (Unit 7) use the t-distribution to account for estimating s from the sample.

### Why does a bigger z* make the confidence interval wider?

The interval is point estimate ± z*·SE, so z* directly scales the margin of error. Going from 95% to 99% confidence raises z* from 1.96 to 2.576, which widens the interval; you're casting a bigger net to be more confident it catches the true parameter.

## Related Study Guides

- [3.3 Constructing a Confidence Interval for a Population Proportion](/ap-stats/unit-3/constructing-confidence-interval-for-population-proportion/study-guide/rYCExLGlPYtMNFiJOWAR)

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