Z-Table (Standard Normal)

What Is a Z-Table?

A Z-table (also called a standard normal table or unit normal table) displays the cumulative probabilities for the standard normal distribution. It shows the probability that a standard normal random variable Z is less than or equal to a given value z, written as P(Z ≤ z) or Φ(z).

The standard normal distribution has mean μ = 0 and standard deviation σ = 1. Any normal distribution can be converted to standard normal using the z-score transformation:

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

How to Read the Z-Table

The table is organized with:

• Rows: The first decimal place of z (e.g., 1.9)
• Columns: The second decimal place (0.00 through 0.09)

To find P(Z ≤ 1.96):

1. Find row 1.9
2. Find column 0.06
3. The intersection gives 0.9750

This means 97.5% of the standard normal distribution lies below z = 1.96.

Types of Z-Tables

Different textbooks present Z-tables in different formats:

Cumulative from Left (This Table)

Shows P(Z ≤ z) — the area from negative infinity to z. This is the most common format and what this table provides.

Cumulative from Center

Shows P(0 ≤ Z ≤ z) — the area between the mean (0) and z. To convert: subtract 0.5 from our table values for positive z.

Right Tail

Shows P(Z > z) — the area to the right of z. To convert: calculate 1 - P(Z ≤ z).

Common Applications

Hypothesis Testing

Critical values for hypothesis tests come directly from the Z-table:

• α = 0.05 (one-tailed): z = 1.645 (look up 0.9500)
• α = 0.05 (two-tailed): z = ±1.96 (look up 0.9750)
• α = 0.01 (two-tailed): z = ±2.576 (look up 0.9950)

Confidence Intervals

For a 95% confidence interval, you need z* = 1.96, because P(-1.96 < Z < 1.96) = 0.95.

Finding Percentiles

To find the 90th percentile of a normal distribution:

1. Look for 0.9000 in the table
2. Find z ≈ 1.28
3. Convert back: x = μ + 1.28σ

Quality Control

In process capability analysis, Z-values determine defect rates. A "Six Sigma" process has z = 6, meaning virtually all output falls within specifications.

Working with Negative Z-Values

For negative z-values, use the symmetry of the normal distribution:

$P(Z \leq -z) = 1 - P(Z \leq z) = P(Z > z)$

For example:

• P(Z ≤ -1.96) = 1 - P(Z ≤ 1.96) = 1 - 0.9750 = 0.0250

The 68-95-99.7 Rule

The Z-table confirms the empirical rule:

• P(-1 < Z < 1) ≈ 0.68 (68% within 1σ)
• P(-2 < Z < 2) ≈ 0.95 (95% within 2σ)
• P(-3 < Z < 3) ≈ 0.997 (99.7% within 3σ)

Interpolation

For z-values with more than two decimal places (e.g., z = 1.645):

1. Find values for z = 1.64 (0.9495) and z = 1.65 (0.9505)
2. Interpolate: 0.9495 + 0.5 × (0.9505 - 0.9495) = 0.9500

For precise calculations, use the Normal Probability Calculator or Z-Table Calculator.

Historical Note

Before computers, statisticians relied entirely on printed tables. The first comprehensive normal tables were published by Karl Pearson in 1914. Students memorized key values and interpolated for others.

Today, software computes exact probabilities instantly, but Z-tables remain valuable for:

• Learning statistical concepts
• Quick reference
• Exam situations where calculators aren't allowed
• Verifying computational results

Related Tools

• Z-Score Calculator — Convert raw scores to z-scores
• Normal Probability Calculator — Calculate exact probabilities
• Inverse Normal Calculator — Find z for a given probability
• Critical Value Z Calculator — Find critical values for tests
• P-Value from Z Calculator — Convert z to p-values

Summary

The Z-table is a fundamental reference in statistics, providing cumulative probabilities for the standard normal distribution. While modern software can compute these values precisely, understanding how to read and use a Z-table builds statistical intuition and remains essential for students and practitioners. The table connects z-scores to probabilities, enabling hypothesis testing, confidence interval construction, and probability calculations for any normal distribution.