---
title: "Normal: 31% under 45, 8% over 64"
subject: "Probability and Statistics"
chapter: 3
key: "semester_5/probability_and_statistics:normal-31-percent-under-45"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/normal-31-percent-under-45
site: BCT Engineering
---

# Normal: 31% under 45, 8% over 64

Probability and Statistics · Chapter 3: Continuous Probability Distributions

- Asked in: 2076 Asoj, 5 marks
- Video: [NORMAL Distribution Problem 3| 31% of items under 45 & 8% of items are over 64.Find Mean & SD | #vtu](https://www.youtube.com/watch?v=2wyFMAMQDiI) — VTU padhai
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/normal-31-percent-under-45
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-3

## Question

In a normal distribution 31% of the items are under 45 and 8% are over 64. Find the mean and standard deviation of distribution. (Given, Z0.42 = 1.4, Z0.19 = 0.5)

## Short answer

From P(X<45)=0.31, z=-0.5; P(X>64)=0.08, z=1.4. Equations: μ−0.5σ=45, μ+1.4σ=64. Solving: σ=10, μ=50.

## Answer

Let X ~ N(μ, σ²). Given: P(X < 45) = 0.31 and P(X > 64) = 0.08.

#### Z-values
- P(X < 45) = 0.31 lies below the mean, so the area between 45 and μ is 0.50 − 0.31 = 0.19. Given Z₀.₁₉ = 0.5, the z-value is **−0.5**.
- P(X > 64) = 0.08, so the area between μ and 64 is 0.50 − 0.08 = 0.42. Given Z₀.₄₂ = 1.4, the z-value is **+1.4**.

#### Equations
(45 − μ)/σ = −0.5 → μ − 0.5σ = 45 ... (1)
(64 − μ)/σ = 1.4 → μ + 1.4σ = 64 ... (2)

Subtract (1) from (2): 1.9σ = 19 → σ = 10

From (1): μ = 45 + 0.5 × 10 = 50

(Check: (64 − 50)/10 = 1.4.)

**Answer: mean = 50, standard deviation = 10**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. P(X < 45) = 0.31 gives z of:
   A. +0.5
   B. -1.4
   C. -0.5
   D. 0.31
   Answer: C. Area between 45 and mean is 0.19, Z0.19 = 0.5, and 45 is below the mean. (Key point: z for 45)

2. P(X > 64) = 0.08 gives z of:
   A. +1.4
   B. -1.4
   C. 0.08
   D. +0.42
   Answer: A. Area from mean to 64 is 0.42, Z0.42 = 1.4. (Key point: z for 64)

3. The equations are:
   A. mu + 0.5 sigma = 45 and mu - 1.4 sigma = 64
   B. mu - 0.5 sigma = 64
   C. 0.5 mu = 45
   D. mu - 0.5 sigma = 45 and mu + 1.4 sigma = 64
   Answer: D. X = mu + z sigma. (Key point: Equations)

4. The solution is:
   A. mu = 45, sigma = 5
   B. mu = 50, sigma = 10
   C. mu = 10, sigma = 50
   D. mu = 54, sigma = 19
   Answer: B. 1.9 sigma = 19. (Key point: Result)

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