Statistical Reference Table
BY: Statistics Fundamentals Team
Reviewed By: Minsa A (Senior Statistics Editor)
Last Updated: August 2026

Z-Table — Standard Normal Distribution Table

The Z-table (standard normal table) gives cumulative probabilities P(Z ≤ z) for the standard normal distribution. Use it to find the probability that a value falls below, above, or between specific Z-scores — for hypothesis tests, confidence intervals, and more.

Quick Lookup Negative Z Table Positive Z Table Worked Examples FAQ
Download Z-Table: PDF Cheat Sheet

Interactive Z-Score Lookup

P(Z ≤ z) =
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Negative Z Table

Use the negative Z-table for Z-scores below 0. Values represent the cumulative area to the left of the Z-score — P(Z ≤ z) where z < 0.

Click any cell to highlight it and copy the probability value. Table shows cumulative P(Z ≤ z) for negative Z-scores.

Positive Z Table

Use the positive Z-table for Z-scores from 0.00 to 3.99. The values show P(Z ≤ z) — the proportion of the standard normal distribution falling to the left of z. For example, P(Z ≤ 1.00) = 0.8413 and P(Z ≤ 1.96) = 0.9750.

Critical Z-Score Values — Quick Reference

The table below lists the most frequently used Z-scores in statistics, including confidence interval critical values and common probability benchmarks.

Z-Score P(Z ≤ z) Left Tail P(Z ≥ z) Right Tail Two-Tail Probability Common Use
−3.000.00130.99870.002799.73% rule (3σ)
−2.5760.00500.99500.010099% CI (two-tailed)
−2.3260.01000.99000.020098% CI (two-tailed)
−1.9600.02500.97500.050095% CI (two-tailed)
−1.6450.05000.95000.100090% CI (two-tailed)
0.000.50000.50001.0000Mean of distribution
1.000.84130.15870.31741 standard deviation
1.6450.95000.05000.100090% CI (two-tailed)
1.9600.97500.02500.050095% CI ← most common
2.000.97720.02280.04562 standard deviations
2.3260.99000.01000.020098% CI (two-tailed)
2.5760.99500.00500.010099% CI (two-tailed)
3.000.99870.00130.002799.73% rule (3σ)

Key values to memorize: P(Z ≤ 1.00) = 0.8413  |  P(Z ≤ 1.96) = 0.9750  |  P(Z ≤ 0) = 0.5000

Understanding the Z-Table

What Does P(Z ≤ z) Mean?

The Z-table shows the area (probability) to the left of a Z-score under the standard normal curve. For example, P(Z ≤ 1.00) = 0.8413 means 84.13% of values fall below Z = 1.

Right-Tail Probability

For P(Z ≥ z), use: 1 − P(Z ≤ z). For example, P(Z ≥ 1.96) = 1 − 0.9750 = 0.0250, meaning only 2.5% of values lie above Z = 1.96.

Area Between Two Z-Scores

For P(z₁ ≤ Z ≤ z₂), use: P(Z ≤ z₂) − P(Z ≤ z₁). For a 95% CI: P(−1.96 ≤ Z ≤ 1.96) = 0.975 − 0.025 = 0.950.

The Z-Score Formula

Before using the Z-table, you need a Z-score. The Z-score formula converts a raw value into a standardized score that tells you how many standard deviations it is from the mean:

Z-Score Formula
z = (X − μ) / σ
X
Observed value
μ (mu)
Population mean
σ (sigma)
Standard deviation

A Z-score of 0 means the value equals the mean. A Z-score of +1 means the value is one standard deviation above the mean. A Z-score of −2 means the value is two standard deviations below the mean.

→ Learn more: Complete Guide to Z-Scores  |  Z-Score Calculator

How to Use the Z-Table — Step by Step

The Z-table is read by matching the first two digits of your Z-score against the row, and the second decimal against the column. The intersection gives P(Z ≤ z).

For a Positive Z-Score (z > 0)

Example: Find P(Z ≤ 1.96)

1Your Z-score is z = 1.96 (positive), so use the positive Z table.
2Split into row and column: row = 1.9, column = 0.06.
3Find the row 1.9 on the left-hand side of the table.
4Move across to the column 0.06 at the top.
5The intersection gives P(Z ≤ 1.96) = 0.9750. This means 97.50% of values fall below Z = 1.96.

For a Negative Z-Score (z < 0)

Example: Find P(Z ≤ −1.35)

1Your Z-score is z = −1.35 (negative), so use the negative Z table.
2Split into row and column: row = −1.3, column = 0.05.
3Find row −1.3 in the left column, then column 0.05 at the top.
4The intersection gives P(Z ≤ −1.35) = 0.0885. This means only 8.85% of values fall below Z = −1.35.

Worked Examples

The following examples cover the four most common Z-table use cases: left-tail, right-tail, area between two values, and finding a raw score from a percentile.

Example 1 — Left-Tail Probability

Exam scores are normally distributed with a mean μ = 70 and standard deviation σ = 10. What proportion of students scored less than 80?

1Calculate Z: z = (80 − 70) / 10 = 1.00
2Look up z = 1.00 in the positive Z-table. Row 1.0, column 0.00.
3P(Z ≤ 1.00) = 0.8413

Answer: 84.13% of students scored less than 80.

Example 2 — Right-Tail Probability

Using the same distribution (μ = 70, σ = 10), what proportion of students scored more than 64?

1Calculate Z: z = (64 − 70) / 10 = −0.60
2Look up z = −0.60 in the negative Z-table: P(Z ≤ −0.60) = 0.2743.
3P(Z ≥ −0.60) = 1 − 0.2743 = 0.7257

Answer: 72.57% of students scored more than 64.

Example 3 — Probability Between Two Scores

What proportion of students scored between 65 and 80? (μ = 70, σ = 10)

1Z₁ = (65 − 70) / 10 = −0.50  →  P(Z ≤ −0.50) = 0.3085
2Z₂ = (80 − 70) / 10 = 1.00  →  P(Z ≤ 1.00) = 0.8413
3P(65 < X < 80) = 0.8413 − 0.3085 = 0.5328

Answer: 53.28% of students scored between 65 and 80.

Example 4 — Find the Score at a Given Percentile

What score corresponds to the 90th percentile? (μ = 70, σ = 10)

1We need P(Z ≤ z) = 0.9000. Search the positive Z-table for the value closest to 0.9000.
2The closest value is 0.8997 at z = 1.28.
3Convert back: X = μ + z·σ = 70 + (1.28)(10) = 82.8

Answer: A score of 82.8 is approximately the 90th percentile.

→ See more Z-score examples →

Real-World Applications of the Z-Table

Hypothesis Testing

In a one-sample Z-test, the Z-table converts your test statistic into a p-value. If your Z = 2.33, the p-value = 1 − 0.9901 = 0.0099 (reject H₀ at α = 0.05).

Confidence Intervals

The Z-table gives critical values for confidence intervals. For a 95% CI, use z* = 1.96. For 99% CI, use z* = 2.576. The formula is: X̄ ± z*(σ/√n).

Standardized Testing

SAT, GRE, and IQ scores are reported as standardized scores. A score at z = 1.5 corresponds to the 93rd percentile (P(Z ≤ 1.5) = 0.9332).

Quality Control

In manufacturing, the Z-table determines defect rates. At ±3σ (the 3-sigma rule), P(−3 ≤ Z ≤ 3) = 0.9973, meaning only 0.27% of products fall outside specifications.

Frequently Asked Questions

What is a Z-table used for?
A Z-table (also called a standard normal table) is used to find cumulative probabilities for the standard normal distribution. It tells you P(Z ≤ z) — the probability that a standard normal variable is less than or equal to a given Z-score. It is essential for hypothesis testing, calculating p-values, constructing confidence intervals, and comparing values across different normal distributions.
What is the Z value for 0.8413 in the standard normal table?
The Z value for a cumulative probability of 0.8413 is Z = 1.00. This means P(Z ≤ 1.00) = 0.8413 — approximately 84.13% of values in a standard normal distribution fall below a Z-score of 1.00. This corresponds to one standard deviation above the mean.
What does 0.975 mean in the Z-table?
A cumulative probability of 0.9750 corresponds to Z = 1.96. That is, P(Z ≤ 1.96) = 0.9750. This is the most commonly used Z-score in statistics — it is the upper critical value for a 95% two-tailed confidence interval. The two-tailed critical values ±1.96 capture the middle 95% of the standard normal distribution (0.9750 − 0.0250 = 0.9500).
How do you read the Z-table for a negative Z-score?
To use the negative Z-table: (1) Find your negative Z-score, e.g. z = −1.35. (2) Locate the row for −1.3 in the leftmost column. (3) Find the column for 0.05 in the header row. (4) The cell at the intersection gives P(Z ≤ −1.35) = 0.0885. This means 8.85% of values lie below z = −1.35.
What is the Z-score for a 95% confidence interval?
For a 95% two-tailed confidence interval, the critical Z-score is z* = ±1.96. This comes from P(Z ≤ 1.96) = 0.9750 in the Z-table, which leaves α/2 = 0.025 (2.5%) in each tail for a total of 5% outside the interval. For other confidence levels: 90% CI → z* = ±1.645; 99% CI → z* = ±2.576.
What is the difference between a negative and positive Z-table?
Both negative and positive Z-tables show P(Z ≤ z) — the cumulative probability from the left. The negative Z-table covers Z-scores from −3.99 to 0.00, giving probabilities less than 0.5000 (values below the mean). The positive Z-table covers 0.00 to 3.99, giving probabilities greater than 0.5000 (values above the mean). Because the standard normal distribution is symmetric, P(Z ≤ −z) = 1 − P(Z ≤ z).
How do you find the area between two Z-scores?
To find P(z₁ ≤ Z ≤ z₂), look up both probabilities in the Z-table and subtract: P(z₁ ≤ Z ≤ z₂) = P(Z ≤ z₂) − P(Z ≤ z₁). For example, the area between Z = −1 and Z = 1 is 0.8413 − 0.1587 = 0.6827 (the 68% rule). The area between Z = −1.96 and Z = 1.96 is 0.9750 − 0.0250 = 0.9500 (the 95% rule).
How is the Z-table related to the empirical rule?
The empirical rule (68–95–99.7 rule) can be verified directly from the Z-table. P(Z ≤ 1.00) − P(Z ≤ −1.00) = 0.8413 − 0.1587 = 0.6827 ≈ 68%. P(Z ≤ 2.00) − P(Z ≤ −2.00) = 0.9772 − 0.0228 = 0.9544 ≈ 95%. P(Z ≤ 3.00) − P(Z ≤ −3.00) = 0.9987 − 0.0013 = 0.9974 ≈ 99.7%.
What is a Z-score of 0.025 (z = 0.025)?
A Z-score of 0.025 gives P(Z ≤ 0.025) ≈ 0.5100. However, you may be thinking of the critical value z for α = 0.025 (i.e., the value where the right-tail area equals 0.025). That value is z = 1.96, since P(Z ≥ 1.96) = 0.025. The notation z₀.₀₂₅ = 1.96 is commonly used in hypothesis testing for a 5% significance level (two-tailed).
When should I use a Z-table vs. a T-table?
Use the Z-table when: (1) The population standard deviation σ is known, or (2) your sample size is large (n ≥ 30). Use the T-table when: (1) σ is unknown and you are using the sample standard deviation s, or (2) your sample size is small (n < 30). As n → ∞, the t-distribution approaches the standard normal distribution, so both tables give nearly identical results for large samples.