---
title: "Finding Probabilities and Values from a Normal Model"
description: "Get the probability of a range of values from a normal model, and go the other way from a stated probability back to the cut-off value, sketching the region every time."
canonical: https://lightmysky.com/learn/mathematics/finding-probabilities-and-values-from-a-normal-model-mt_naXMRySvos
source: https://lightmysky.com/learn/mathematics/finding-probabilities-and-values-from-a-normal-model-mt_naXMRySvos.md
retrieved: 2026-11-12
---

> **Agent view.** This is the Markdown twin of the page, for tools and assistants.
> When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md
> API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md
> Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt
> Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json
> Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`.

# Finding Probabilities and Values from a Normal Model

Get the probability of a range of values from a normal model, and go the other way from a stated probability back to the cut-off value, sketching the region every time.

Subject: Mathematics · Area: Probability · Ages 17 to 18
Page: https://lightmysky.com/learn/mathematics/finding-probabilities-and-values-from-a-normal-model-mt_naXMRySvos

## Ready when they can

- Find left-tail, upper-tail and interval probabilities for a stated normal model
- Find normal-model cutoffs from stated cumulative or tail probabilities, including the upper 10 percent cutoff
- Find an unknown mean or standard deviation from a given probability

## Lesson: Reading the bell curve both ways

For normal X with mean μ and positive standard deviation σ, use z = (x - μ) / σ. A left-cumulative table lists Φ(z) = P(Z < z). Read its stated z row and column; do not use a mean-to-z table as if it listed left areas. The questions here include the table entries you need. With μ = 46, σ = 3 and x = 49, z = 1. Since Φ(1) = 0.8413, P(X < 49) = 0.8413. Symmetry gives Φ(0) = 0.5.

For a normal model, about 68%, 95% and 99.7% lie within one, two and three σ of μ. The central 95% leaves about 5% outside. Symmetry splits that into about 2.5% per tail. Thus an upper 2.5% cutoff is about μ + 2σ, and a lower 2.5% cutoff is about μ - 2σ. With μ = 40 and σ = 6, these approximate cutoffs are 52 and 28.

*(drawing: For the stated normal model, about 95% lies between 28 and 52, leaving about 2.5% in each tail.)*

For an upper tail, subtract the left area from 1. For a < X < b, standardise both ends and use Φ(z_b) - Φ(z_a). The area below b includes the interval and everything below a. With μ = 20 and σ = 2, boundaries 19 and 23 give z = -0.5 and 1.5. Using Φ(-0.5) = 0.3085 and Φ(1.5) = 0.9332 gives 0.6247.

To find a cutoff from a probability, first express the area to its left. Search that area among the left-cumulative table entries, then read the corresponding z. Reverse with x = μ + zσ. For the top 10%, the left area is 0.90 and z is about 1.28. With μ = 80 and σ = 5, the cutoff is about 80 + 1.28 × 5 = 86.4. Lower-tail cutoffs use negative z when their left area is below 0.5.

A given left probability supplies z through the left-cumulative table. Then solve x = μ + zσ for the unknown parameter: μ = x - zσ or σ = (x - μ) / z when z is not 0. For P(X < 35) = 0.8413 and σ = 4, z = 1 gives μ = 31. Two known quantiles also work: b - a = (z_b - z_a)σ, so divide their raw gap by their z gap to find σ.

*(drawing: Name the requested region before selecting a table entry or subtracting areas.)*

**Recap.** Convert values to z for probabilities, and work the table backwards from probabilities to marks.

*Lesson, worked questions and flashcards authored for this learning dataset.*

## Practice

14 questions on this page, each with its working shown.

## Needs first

- [The Normal Distribution and Standardised Scores](https://lightmysky.com/learn/mathematics/the-normal-distribution-and-standardised-scores-mt_Cs32DT9KnA)

## Optional connections

- [Cumulative Frequency Curves](https://lightmysky.com/learn/mathematics/cumulative-frequency-curves-mt_5U70qgbnST)

## Opens up

- [Sampling Distributions and the Central Limit Theorem](https://lightmysky.com/learn/mathematics/sampling-distributions-and-the-central-limit-theorem-mt_nugaEeUSii)
- [Confidence Intervals for Means and Proportions](https://lightmysky.com/learn/mathematics/confidence-intervals-for-means-and-proportions-mt_Pg2fPDW1sn)