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

Z Score Table — Standard Normal Distribution Table

The Z score 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 Score Table: PDF Cheat Sheet

Interactive Z Score Lookup

P(Z ≤ z) =
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Enter a cumulative probability P(Z ≤ z) to find the corresponding Z-score. Useful for finding critical values — e.g. enter 0.975 to get z = 1.96.

Z-Score =
—

Negative Z Table

Use the negative Z-table for Z-scores below 0. Values are the cumulative area to the left of the Z-score — P(Z ≤ z) where z < 0. Click any cell to highlight it, copy the value, and see the corresponding bell curve.

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

Positive Z Table

Use the positive Z-table for Z-scores from 0.00 to 3.99. Values show P(Z ≤ z). For example, P(Z ≤ 1.00) = 0.8413 and P(Z ≤ 1.96) = 0.9750.

Critical Z-Score Values — Quick Reference

The most frequently used Z-scores in statistics, including confidence interval critical values and common benchmarks.

Z-Score P(Z ≤ z) Left tail P(Z ≥ z) Right tail Two-tail Common use
−3.000.00130.99870.002799.73% rule (3σ)
−2.5760.00500.99500.010099% CI
−2.3260.01000.99000.020098% CI
−1.9600.02500.97500.050095% CI
−1.6450.05000.95000.100090% CI
0.000.50000.50001.0000Mean
1.000.84130.15870.31741 standard deviation
1.6450.95000.05000.100090% CI
1.9600.97500.02500.050095% CI ← most used
2.000.97720.02280.04562 standard deviations
2.3260.99000.01000.020098% CI
2.5760.99500.00500.010099% CI
3.000.99870.00130.002799.73% rule (3σ)

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

Understanding the Z Score Table

What does P(Z ≤ z) mean?

The Z score table shows the area to the left of a Z-score under the standard normal curve. 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). P(Z ≥ 1.96) = 1 − 0.9750 = 0.0250.

Area between two Z-scores

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

The Z-Score Formula

Before using the Z score table, you need a Z-score. The formula converts a raw value into a standardized score — the number of standard deviations from the mean:

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

→ Complete guide to Z-scores  |  Z-Score Calculator

How to Use the Z Score Table — Step by Step

Match the first decimal of your Z-score to the row and the second decimal to the column. The intersection gives P(Z ≤ z).

For a positive Z-score (z > 0)

Example: Find P(Z ≤ 1.96)

1z = 1.96 is positive → use the positive Z table.
2Split: row = 1.9, column = 0.06.
3Find row 1.9, move to column 0.06.
4The intersection gives P(Z ≤ 1.96) = 0.9750. 97.50% of values fall below Z = 1.96.

For a negative Z-score (z < 0)

Example: Find P(Z ≤ −1.35)

1z = −1.35 is negative → use the negative Z table.
2Split: row = −1.3, column = 0.05.
3The intersection gives P(Z ≤ −1.35) = 0.0885. Only 8.85% of values fall below z = −1.35.

Worked Examples

The four most common use cases: left-tail, right-tail, area between two Z-scores, and finding a raw score from a percentile.

Example 1 — Left-tail probability

Exam scores: μ = 70, σ = 10. What proportion scored less than 80?

1z = (80 − 70) / 10 = 1.00
2Look up z = 1.00. 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

Same distribution. What proportion scored more than 64?

1z = (64 − 70) / 10 = −0.60
2P(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 scored between 65 and 80?

1Z₁ = −0.50 → P(Z ≤ −0.50) = 0.3085
2Z₂ = 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 percentile

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

1Need P(Z ≤ z) = 0.9000. Search the positive Z table.
2Closest value: 0.8997 at z = 1.28.
3X = 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

Hypothesis testing

In a one-sample Z-test, the Z table converts a test statistic to a p-value. If Z = 2.33, p-value = 1 − 0.9901 = 0.0099.

Confidence intervals

For a 95% CI, use z* = 1.96. For 99%, use z* = 2.576. Formula: X̄ ± z*(σ/√n).

Standardized testing

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

Quality control

At ±3σ, P(−3 ≤ Z ≤ 3) = 0.9973. Only 0.27% of products fall outside ±3 standard deviations.

Frequently Asked Questions

What is a Z-table used for?
A Z-table (standard normal table) gives P(Z ≤ z) — the probability that a standard normal variable is at most z. It is essential for hypothesis testing, calculating p-values, building confidence intervals, and comparing values across normal distributions.
What is the Z value for 0.8413 in the standard normal table?
The Z value for 0.8413 is Z = 1.00. P(Z ≤ 1.00) = 0.8413 means 84.13% of values fall below Z = 1.00 — one standard deviation above the mean. Use the reverse lookup above to verify this instantly.
What does 0.975 mean in the Z-table?
P(Z ≤ z) = 0.9750 corresponds to z = 1.96. This is the critical value for 95% two-tailed confidence intervals — the most commonly used Z-score in statistics. The two critical values ±1.96 capture the middle 95%: 0.9750 − 0.0250 = 0.9500.
How do you read the Z-table for a negative Z-score?
For z = −1.35: (1) Use the negative Z table. (2) Find row −1.3. (3) Find column 0.05. (4) The intersection gives P(Z ≤ −1.35) = 0.0885. Only 8.85% of values fall below z = −1.35. Click the cell in the table above to see the bell curve automatically.
What is the Z-score for a 95% confidence interval?
For a 95% two-tailed CI, the critical Z-score is z* = ±1.96. This leaves 2.5% in each tail. For other levels: 90% CI → ±1.645; 98% CI → ±2.326; 99% CI → ±2.576. Enter 0.975 into the reverse lookup above to confirm.
What is the difference between a negative and positive Z-table?
Both show P(Z ≤ z). The negative Z-table covers −3.99 to 0.00 (probabilities < 0.5000). The positive Z-table covers 0.00 to 3.99 (probabilities > 0.5000). Because the distribution is symmetric, P(Z ≤ −z) = 1 − P(Z ≤ z).
How do you find the area between two Z-scores?
P(z₁ ≤ Z ≤ z₂) = P(Z ≤ z₂) − P(Z ≤ z₁). Area between −1 and 1: 0.8413 − 0.1587 = 0.6827 (68% rule). Area between −1.96 and 1.96: 0.9750 − 0.0250 = 0.9500 (95% rule).
How is the Z-table related to the empirical rule?
The 68–95–99.7 rule reads directly from the Z table:
0.8413 − 0.1587 = 0.6827 ≈ 68% (within ±1σ).
0.9772 − 0.0228 = 0.9544 ≈ 95% (within ±2σ).
0.9987 − 0.0013 = 0.9974 ≈ 99.7% (within ±3σ).
When should I use a Z-table vs. a T-table?
Use the Z-table when the population standard deviation σ is known, or when n ≥ 30. Use the T-table when σ is unknown and estimated from the sample, especially when n < 30. As n → ∞, the t-distribution approaches the normal distribution, so results converge for large samples.
What is a Z-score of 0.025 (z = 0.025)?
A Z-score of 0.025 gives P(Z ≤ 0.025) ≈ 0.5100. You may be thinking of the critical value where the right-tail area equals 0.025, which is z = 1.96 (P(Z ≥ 1.96) = 0.025). Enter 0.975 into the reverse lookup above to confirm.