Standard Normal Distribution (Z) Table

This interactive Z table displays probabilities (areas) under the standard normal distribution curve for specific z-scores. It helps determine the probability of values falling within particular ranges in normally distributed data. Hover over any table value to visualize the corresponding shaded area in the distribution chart along with its exact probability. You can also click on any value to pin it and maintain the shaded area visualization for closer examination.

How to Use the Z Table

Find the z-score's ones digit and first decimal place in the leftmost column
Find the second decimal place in the top row
The intersection gives you the probability
Hover over any value to see the shaded area in the distribution chart

Examples of Using the Z Table

Example 1: Z = 1.84

Locate 1.8 in the left column
Find 0.04 in the top row
The intersection gives 0.9671, meaning P(Z < 1.84) = 0.9671
P(Z > 1.84) = 1 - P(Z < 1.84) = 1 - 0.9671 = 0.0329

pnorm(1.84)

Example 2: Z = -1.25

There are two ways to calculate P(Z < z) for negative z-scores:

Simply choose the negative z-scores tab on the table above and locate the z-score
Symmetry property: P(Z < z) = 1 - P(Z < z)

Let's use the symmetry property for this example:

Use the symmetry property: P(Z < -1.25) = 1 - P(Z < 1.25)
Find P(Z < 1.25) in the table: 0.8944
Calculate: P(Z < -1.25) = 1 - 0.8944 = 0.1056

1 - pnorm(1.25)
# or
pnorm(-1.25)

Pro Tip:

For P(Z > a), calculate 1 - P(Z < a)
For P(a < Z < b), calculate P(Z < b) - P(Z < a)
Remember that the total area under the curve equals 1

Confidence Intervals and Critical Z-Scores

Confidence intervals use critical z-scores to determine the range of values that likely contains the true population parameter. Here are the most commonly used confidence levels and their corresponding z-scores:

Critical Z-Scores for Common Confidence Levels

90%

Confidence Level

±1.645

Critical Z-Score

95%

Confidence Level

±1.96

Critical Z-Score

99%

Confidence Level

±2.576

Critical Z-Score

How to Find Confidence Interval Z-Scores

For any confidence level, the critical z-score corresponds to the point where the cumulative probability equals (1 + confidence level) / 2.

95% Confidence Interval Z-Score

Alpha (α) = 1 - 0.95 = 0.05
Alpha/2 = 0.025 in each tail
Find z-score where P(Z < z) = 0.975
From the table: z = 1.96

qnorm(0.975)  # Returns 1.96

90% Confidence Interval Z-Score

Alpha (α) = 1 - 0.90 = 0.10
Alpha/2 = 0.05 in each tail
Find z-score where P(Z < z) = 0.95
From the table: z = 1.645

qnorm(0.95)   # Returns 1.645

99% Confidence Interval Z-Score

Alpha (α) = 1 - 0.99 = 0.01
Alpha/2 = 0.005 in each tail
Find z-score where P(Z < z) = 0.995
From the table: z = 2.576

qnorm(0.995)  # Returns 2.576

Confidence Interval Formula

Once you have the critical z-score, use this formula to calculate the confidence interval:

CI = x̄ ± (z × σ/√n)

Where:
- x̄ = sample mean
- z = critical z-score
- σ = population standard deviation
- n = sample size

Quick Reference:

90% CI: Use z = ±1.645
95% CI: Use z = ±1.96 (most common)
99% CI: Use z = ±2.576
Higher confidence = wider interval = larger z-score

Understanding Z Scores and the Normal Distribution

What is a Z Score?

A z-score (also called a standard score) measures how many standard deviations an observation is from the mean. The formula is:

z = (x - μ) / σ

Z Score Calculator

To calculate a z-score, you need the value (x), population mean (μ), and standard deviation (σ).