Z-Table (Standard Normal Distribution Table)

Interactive Z-Table Calculator

Standard Normal Distribution Table (Z-Table)

The z-table shows the cumulative probability P(Z ≤ z) for the standard normal distribution. The table gives the area under the curve to the LEFT of any z-value.

How to Use This Table

1. Find the z-score to one decimal place in the left column
2. Find the second decimal place in the top row
3. The intersection gives P(Z ≤ z)

Example: For z = 1.96, find row 1.9, column 0.06 → P(Z ≤ 1.96) = 0.9750

Positive Z-Values (z ≥ 0)

Negative Z-Values (z ≤ 0)

For negative z-values, use the symmetry property: P(Z ≤ -z) = 1 - P(Z ≤ z)

Common Critical Values

Using the Z-Table

Finding P(Z ≤ z)

Look up the z-value directly in the table.

Finding P(Z ≥ z)

P(Z ≥ z) = 1 - P(Z ≤ z)

Finding P(a ≤ Z ≤ b)

P(a ≤ Z ≤ b) = P(Z ≤ b) - P(Z ≤ a)

Finding z for a given probability

Reverse lookup: Find the probability in the table and read the z-value.

Step-by-Step: How to Read the Z-Table

Example 1: Find P(Z ≤ 1.96)

1. Find the row for z = 1.9
2. Find the column for 0.06
3. Read the value: 0.9750
4. This means 97.5% of values fall below z = 1.96

Example 2: Find P(Z > 2.00)

1. Find P(Z ≤ 2.00) = 0.9772
2. Calculate P(Z > 2.00) = 1 - 0.9772 = 0.0228
3. Only 2.28% of values exceed z = 2.00

Example 3: Find P(-1.50 < Z < 1.50)

1. Find P(Z ≤ 1.50) = 0.9332
2. Find P(Z ≤ -1.50) = 0.0668
3. Calculate: 0.9332 - 0.0668 = 0.8664
4. 86.64% of values fall between z = -1.50 and z = 1.50

Z-Score Formula

The z-score measures how many standard deviations a value is from the mean:

$z = \frac{x - \mu}{\sigma}$

Where:

• x = Your data value
• μ = Population mean
• σ = Population standard deviation

Frequently Asked Questions

What is a z-table?

A z-table (also called standard normal table) shows the cumulative probability P(Z ≤ z) for the standard normal distribution. It tells you what percentage of values fall below any given z-score.

How do I read a z-table?

To read a z-table: (1) Find the first decimal place of your z-score in the left column, (2) Find the second decimal place in the top row, (3) The intersection gives the probability P(Z ≤ z).

What is z-score for 95% confidence?

For a 95% confidence interval (two-tailed), the critical z-value is ±1.96. For a one-tailed test at 95%, use z = 1.645.

What is the z-score for 99% confidence?

For a 99% confidence interval (two-tailed), the critical z-value is ±2.576. For a one-tailed test at 99%, use z = 2.326.

When do I use the z-table vs t-table?

Use the z-table when: (1) The population standard deviation (σ) is known, (2) Sample size is large (n ≥ 30), or (3) Working with proportions. Use the t-table when σ is unknown and sample size is small.

What does a z-score of 0 mean?

A z-score of 0 means the value equals the mean. The probability P(Z ≤ 0) = 0.5000, meaning 50% of values fall below the mean.

What does a negative z-score mean?

A negative z-score indicates the value is below the mean. For example, z = -1.5 means the value is 1.5 standard deviations below the mean.

How do I find the z-score for a given probability?

Use reverse lookup: Find the probability closest to your target in the table body, then read the corresponding z-value from the row and column headers.

What percentage of data falls within 2 standard deviations?

For a normal distribution, approximately 95.44% of data falls within 2 standard deviations of the mean (between z = -2 and z = +2).

Common Z-Values Quick Reference

Related Resources

• Z-Score Calculator - Calculate z-scores instantly
• Normal Distribution Lesson - Learn about the normal distribution
• t-Table - For small samples when σ is unknown
• Chi-Square Table - For categorical data
• Standard Deviation Calculator - Calculate σ and s