Z Table — Standard Normal Distribution Lookup

What Is a Z Table and Why Does It Matter?

A z table — also called a z-score table, standard normal table, or standard normal distribution table — is a reference grid that converts a z-score into a cumulative probability. Every value inside the table represents the area under the standard normal curve to the left of a given z-score, written as P(Z < z). Because the standard normal distribution always has a mean of 0 and a standard deviation of 1, the same table works for any normally distributed variable once it has been standardized.

Before calculators and statistical software made this instant, the z table was the only practical way to find probabilities for the normal distribution. It still shows up constantly in coursework, textbooks, and exams because it forces you to understand the relationship between a z-score and the shape of the curve — not just produce a number.

How to Read a Z Table Step by Step

Reading a z table comes down to matching a z-score to a row and column. Here is the full process using a concrete example: suppose a test has a mean (μ) of 600 and a standard deviation (σ) of 150, and a student scores 700.

• Step 1 — Standardize the score: z = (700 − 600) / 150 = 0.67
• Step 2 — Choose the table: since z = 0.67 is positive, use the positive z table
• Step 3 — Find the row: locate 0.6 in the left-hand column
• Step 4 — Find the column: locate 0.07 along the top row
• Step 5 — Read the intersection: the value is 0.7486, meaning roughly 74.86% of scores fall below 700

Negative vs. Positive Z Table: What's the Difference?

Most published z tables split the standard normal distribution into two halves. The negative z table covers every z-score below the mean (z < 0), and every value inside it is less than 0.5000, since less than half the curve lies to the left of a negative z-score. The positive z table covers z-scores at or above the mean (z ≥ 0), and every value is 0.5000 or greater.

Because the standard normal curve is perfectly symmetric around zero, you technically only need one table. The relationship P(Z < -z) = 1 − P(Z < z) lets you derive any negative-table value from the positive table, and vice versa. Most students still find it faster to use both halves directly rather than recalculating the complement every time — which is why the full negative and positive tables above are both included.

Left-Tail vs. Right-Tail Probability

Every cell in a standard z table reports the left-tail probability — the area to the left of z. To find the right-tail probability — the area to the right — subtract the table value from 1: P(Z > z) = 1 − P(Z < z). The lookup tool above calculates both automatically so you don't need to do that subtraction by hand.

Finding Critical Values for Confidence Intervals

One of the most common uses of a z table outside of basic probability problems is finding the critical z-value for a confidence interval. For a 95% confidence interval, you're looking for the z-score where 2.5% of the distribution lies in each tail — which means searching the table for 0.9750. That value lands at z = 1.96, the most frequently used critical value in introductory statistics.

The same logic applies to hypothesis testing: a one-tailed test at the 0.05 significance level uses the z-score where 95% of the area lies to the left, which is z = 1.645. The critical values table above lists the most common confidence levels so you don't need to search for them manually.

Z Table vs. T Table: When to Use Each

A z table assumes you know the population standard deviation, or that your sample size is large enough (generally n ≥ 30) for the Central Limit Theorem to make the normal approximation reliable. When the population standard deviation is unknown and the sample is small, a t-distribution table is used instead — it accounts for the extra uncertainty introduced by estimating the standard deviation from the sample itself. As sample size grows, the t-distribution converges toward the standard normal distribution, which is why large-sample t-tests often produce results nearly identical to a z-test.

In practice, if you're unsure which table applies, check whether σ is given in the problem. If it is, use the z table. If only the sample standard deviation (s) is given and the sample is small, use the t table with df = n − 1.

One-Tailed vs. Two-Tailed Z Table Lookups

The same z table works for both one-tailed and two-tailed problems — what changes is how you split up the significance level (α) before looking up the critical value.

• One-tailed test: all of α sits in a single tail. For α = 0.05, look up the z-score where the cumulative probability is 1 − 0.05 = 0.95, which gives z = 1.645.
• Two-tailed test: α is split evenly between both tails. For α = 0.05, each tail gets 0.025, so you look up 1 − 0.025 = 0.975, which gives z = 1.96.

This is exactly why the 95% confidence interval uses z = 1.96 (two-tailed) while a 95% one-tailed test uses z = 1.645 — both come from the same table, just with a different α split. The critical values table above lists both the two-tailed and one-tailed z-score for the four most common confidence levels.

Working Backward: Probability to Z-Score, and Z-Score to Percentile

Sometimes a problem gives you a percentile or probability and asks for the corresponding z-score — the reverse of a standard lookup. To do this manually, scan the body of the table for the probability closest to your target, then read off the row and column headers. For example, locating 0.9500 inside the positive table lands between 0.9495 (z = 1.64) and 0.9505 (z = 1.65), which is why 1.645 is used as the standard approximation for the 95th percentile.

This reverse process is exactly what the "Probability → Z-score" tab in the lookup tool above automates — type in the probability and it returns the closest matching z-score without manual scanning.

Z-Score to Percentile Quick Reference

A percentile is just the cumulative probability expressed as a whole number instead of a decimal — multiply P(Z < z) by 100. A few values come up often enough to memorize:

Common Z Table Mistakes to Avoid

• Using the wrong table half — always check the sign of your z-score before picking negative or positive
• Forgetting to standardize first — a raw score (like a test score) must be converted to a z-score before any table lookup applies
• Mixing up left-tail and right-tail — most z tables give left-tail area by default; right-tail requires subtracting from 1
• Rounding too early — round only the final z-score to two decimals, not the intermediate mean or standard deviation values
• Misreading rows for columns — the row gives the first decimal, the column gives the second; reversing them produces a completely different value
• Using a one-tailed critical value for a two-tailed problem — check whether α applies to one tail or is split across both before looking up z

For problems that go beyond a simple table lookup — full word problems involving binomial distributions, conditional probability, or multi-step normal distribution questions — the probability calculator walks through the entire solution with the same step-by-step approach used throughout MathGPT Free.