How to Find Z Score from Probability

Last reviewed: June 2026 by the InvNorm Calculator Editorial Team. Report an issue

Finding a Z score from a probability is one of the most common tasks in statistics. Whether you are determining critical values for a hypothesis test, finding percentile cutoffs, or solving a textbook problem, the process involves using the inverse normal distribution function. This guide covers every major method: Z tables, graphing calculators, spreadsheets, and programming languages.

What Does It Mean to Find Z from Probability?

The standard normal distribution has a mean of 0 and a standard deviation of 1. Every point on the horizontal axis is a Z score. The cumulative distribution function (CDF) gives the probability that a random variable from this distribution falls at or below a given Z score:

P(Z ≤ z) = Φ(z)

Finding Z from probability means reversing this. Given a probability p, you want the Z score z such that Φ(z) = p. This reverse function is written as:

z = Φ⁻¹(p)

This is exactly what the invNorm function, NORM.S.INV, and qnorm compute.

Method 1: Using a Z Table

A standard normal Z table lists cumulative probabilities for Z scores from about −3.4 to 3.4. To use it in reverse:

1. Locate your target probability in the body of the table.
2. Find the closest value to your probability. If an exact match does not exist, choose the nearest entry.
3. Read the Z score from the row header (ones and tenths digits) and column header (hundredths digit).

Example: Find Z for p = 0.8500

1. Scan the table body for 0.8500.
2. The closest values are 0.8485 (Z = 1.03) and 0.8508 (Z = 1.04).
3. Since 0.8500 is between them, the Z score is approximately 1.04 (or more precisely, about 1.036 by interpolation).

Z tables are limited to about four decimal places. For greater precision, use a calculator or software.

Method 2: Using a TI-84 Calculator

The TI-84's invNorm function is the fastest method on a graphing calculator.

1. Press 2nd → VARS to open the DISTR menu.
2. Select 3:invNorm(.
3. Type the probability and press ENTER.

Example: Find Z for p = 0.975

invNorm(0.975)
→ 1.959963985

The Z score is approximately 1.960. This is the familiar critical value for a 95% confidence interval.

For a detailed walkthrough, see our TI-84 invNorm guide.

Method 3: Using Excel or Google Sheets

Excel provides NORM.S.INV for the standard normal distribution:

=NORM.S.INV(0.975)
→ 1.959964

Google Sheets uses the same function name, or the legacy NORMSINV:

=NORMSINV(0.975)
→ 1.959964

For a custom normal distribution (non-standard), use NORM.INV(p, mean, sd). See the full Excel and Google Sheets guide.

Method 4: Using R

R's qnorm function (quantile function for the normal distribution) is built into base R:

qnorm(0.975)
# [1] 1.959964

For a right tail probability, use the lower.tail parameter:

qnorm(0.025, lower.tail = FALSE)
# [1] 1.959964

Method 5: Using Python

In Python, use the scipy.stats module:

from scipy.stats import norm

# Left tail: find Z for cumulative probability 0.975
norm.ppf(0.975)
# 1.959963984540054

# Right tail: find Z where right tail area is 0.025
norm.isf(0.025)
# 1.9599639845400536

The ppf function (percent point function) takes the left cumulative probability. The isf function (inverse survival function) takes the right tail probability directly.

Handling Left Tail vs Right Tail

All methods above expect the left cumulative probability by default. If your problem gives you a right tail probability, convert it first:

Left cumulative = 1 − Right tail probability

Example: Right Tail Probability of 0.10

You need the Z score where 10% of the area is to the right.

1. Convert: left cumulative = 1 − 0.10 = 0.90
2. Compute: invNorm(0.90) ≈ 1.28155

This means 90% of the distribution is to the left of Z = 1.28155, and 10% is to the right.

Try this in the calculator

Example: Finding the Z Score for a Two-Tailed Test at α = 0.05

A two-tailed test splits α into two equal tails: 0.025 in each tail.

1. Lower Z: invNorm(0.025) ≈ −1.960
2. Upper Z: invNorm(0.975) ≈ 1.960

The critical values are ±1.96.

Try this in the calculator

For a deeper explanation of tail directions, see our left tail vs right tail guide.

Quick Reference: Methods Compared

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