Left Tail vs Right Tail Probability

Q: How do I convert between left tail and right tail probability?

To convert from left tail to right tail, subtract from 1: right = 1 - left. To convert from right tail to left tail, also subtract from 1: left = 1 - right. For example, a left tail probability of 0.90 corresponds to a right tail probability of 0.10, and vice versa.

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

Understanding the difference between left tail and right tail probabilities is essential for working with normal distributions. Calculators, spreadsheet functions, and statistical software all require you to know which type of probability you are providing. Getting this wrong leads to incorrect results, which is one of the most common mistakes in introductory statistics courses.

Definitions

Left Tail Probability

The left tail probability (also called the left cumulative probability or lower tail probability) is the area under the normal distribution curve to the left of a given value. It represents the probability that a random variable is less than or equal to that value:

P(X ≤ x) = left tail probability

Visually, imagine shading everything to the left of a vertical line on the bell curve. The shaded region is the left tail probability.

Right Tail Probability

The right tail probability (also called the upper tail probability or survival probability) is the area to the right of a given value. It represents the probability that a random variable is greater than or equal to that value:

P(X ≥ x) = right tail probability

This is the complement of the left tail: the unshaded region when you shade left.

The Fundamental Relationship

Since the total area under the normal curve equals 1 (representing 100% probability), the left and right tail probabilities always sum to 1:

Left tail + Right tail = 1

This gives us simple conversion formulas:

Right tail = 1 − Left tail
Left tail = 1 − Right tail

Examples of Conversion

Left Tail	Right Tail	Interpretation
0.5000	0.5000	At the mean (Z = 0), half is on each side
0.8413	0.1587	Z = 1.0: about 84% below, 16% above
0.9500	0.0500	Z ≈ 1.645: 95% below, 5% above
0.9750	0.0250	Z ≈ 1.960: 97.5% below, 2.5% above
0.9950	0.0050	Z ≈ 2.576: 99.5% below, 0.5% above
0.0500	0.9500	Z ≈ −1.645: 5% below, 95% above

Why This Matters for Calculators and Software

Most calculator functions and software commands expect the left cumulative probability as input. If you provide a right tail probability without converting, your answer will be wrong.

Tool	Expected Input	How to Handle Right Tail
TI-84 invNorm	Left cumulative	Enter `invNorm(1 - right_tail)`
Excel NORM.S.INV	Left cumulative	Enter `=NORM.S.INV(1 - right_tail)`
R qnorm	Left cumulative (default)	Use `qnorm(p, lower.tail=FALSE)`
Python norm.ppf	Left cumulative	Use `norm.isf(right_tail)`
Casio InvN	Depends on tail selector	Set tail to "Right" and enter directly

R and Python offer convenience functions (lower.tail=FALSE and norm.isf) that accept right tail probabilities directly. For TI-84 and Excel, you must always subtract from 1 yourself.

Example 1: Left Tail Problem

Problem: Heights of adult women are normally distributed with μ = 64 inches and σ = 3 inches. Find the height at the 80th percentile.

The 80th percentile is a left tail probability: 80% of the values fall below this height.

Left tail probability: p = 0.80
Find Z: Φ⁻¹(0.80) ≈ 0.84162
Convert to height: x = 64 + 0.84162 × 3 = 66.52 inches

About 80% of adult women are 66.52 inches or shorter.

Try this in the calculator

Example 2: Right Tail Problem

Problem: Using the same height distribution (μ = 64, σ = 3), find the height exceeded by only 5% of women.

This is a right tail problem: only 5% of values are above the answer.

Right tail probability: p = 0.05
Convert to left cumulative: 1 − 0.05 = 0.95
Find Z: Φ⁻¹(0.95) ≈ 1.64485
Convert to height: x = 64 + 1.64485 × 3 = 68.93 inches

Only 5% of adult women are taller than about 68.93 inches.

Try this in the calculator

Example 3: Recognizing Tail Direction from Problem Wording

The wording of a problem tells you which tail to use. Here are common patterns:

Wording	Tail	What to Enter
"at most," "no more than," "at or below"	Left	Use the probability directly
"percentile," "bottom X%"	Left	Use the probability directly
"at least," "more than," "exceeds"	Right	Subtract from 1 for most tools
"top X%," "only X% above"	Right	Subtract from 1 for most tools
"between," "middle X%"	Both tails	Use middle area mode
"significance level α," "rejection region"	Depends on one vs two-tailed	Split α for two-tailed tests

Symmetry of the Standard Normal Distribution

The standard normal distribution is symmetric around Z = 0. This symmetry creates a useful relationship:

P(Z ≤ −z) = P(Z ≥ z)

In practical terms, the left tail probability at Z = −1.96 equals the right tail probability at Z = 1.96. Both are 0.025. This symmetry is why confidence interval critical values come in ± pairs.

Connection to Confidence Intervals

Confidence intervals use both tails simultaneously. For a 95% confidence interval:

Middle area: 0.95 (95%)
Total tail area: 1 − 0.95 = 0.05
Each tail: 0.05 / 2 = 0.025
Lower critical Z: Φ⁻¹(0.025) ≈ −1.960
Upper critical Z: Φ⁻¹(0.975) ≈ 1.960

The lower critical value uses the left tail probability (0.025), and the upper critical value uses 1 minus the left tail probability (0.975). This is why understanding tail conversions is critical for hypothesis testing.

Try the 95% confidence interval in the calculator

Connection to One-Tailed and Two-Tailed Tests

In hypothesis testing, the type of test determines how you use tail probabilities:

Left-tailed test (H_a: μ < μ₀): All of α goes in the left tail. Critical Z = invNorm(α).
Right-tailed test (H_a: μ > μ₀): All of α goes in the right tail. Critical Z = invNorm(1 − α).
Two-tailed test (H_a: μ ≠ μ₀): Split α equally between both tails. Critical Z values = invNorm(α/2) and invNorm(1 − α/2).

Frequently Asked Questions

What is the difference between left tail and right tail probability?

Left tail probability is the area under the normal curve to the left of a given value, P(X ≤ x). Right tail probability is the area to the right, P(X ≥ x). They always add up to 1. For example, if the left tail probability at Z = 1.96 is 0.975, then the right tail probability is 1 − 0.975 = 0.025.

How do I convert between left tail and right tail probability?

Subtract from 1 in either direction: right = 1 − left, and left = 1 − right. For example, a left tail probability of 0.90 corresponds to a right tail probability of 0.10, and vice versa.

Does the TI-84 use left tail or right tail for invNorm?

The TI-84 invNorm function always uses left tail (left cumulative) probability. If you have a right tail probability, subtract it from 1 before entering. For a right tail probability of 0.05, enter invNorm(0.95). Some Casio calculators offer a built-in tail selector that handles this conversion for you.