---
title: "Standard normal with area property"
subject: "Probability and Statistics"
chapter: 3
key: "semester_5/probability_and_statistics:standard-normal-area-property"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/standard-normal-area-property
site: BCT Engineering
---

# Standard normal with area property

Probability and Statistics · Chapter 3: Continuous Probability Distributions

- Asked in: 2070 Asar, 6 marks
- Video: [Normal Distribution: Calculating Probabilities/Areas (z-table)](https://www.youtube.com/watch?v=p_KApjpyBHE) — Joshua Emmanuel
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/standard-normal-area-property
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-3

## Question

Define standard normal distribution with area property.

## Short answer

Standard normal distribution has mean 0 and variance 1, denoted Z ~ N(0,1), with pdf f(z) = (1/√(2π)) e^(−z²/2). Its total area under the curve is 1; the curve is symmetric about z=0. Standard normal table gives area P(0<Z<z), and P(Z<0)=0.5.

## Answer

**Standard normal distribution:** the normal distribution with mean 0 and variance 1. Z = (X − μ)/σ ~ N(0, 1), with pdf

f(z) = (1/√(2π)) e^(−z²/2), −∞ < z < ∞

#### Area property
The total area under the curve is 1 and the curve is symmetric about z = 0. The standard normal table gives the area between 0 and z, P(0 < Z < z).

```text
              .-'''-.
           .-'   |   '-.
        .-'      |      '-.
  _.--'          |          '--._
 ------+----+----+----+----+------
      -2   -1    0    1    2   z
        |<-0.3413->|<-0.3413->|
```

Key properties:
1. P(Z < 0) = P(Z > 0) = 0.5.
2. Symmetry: P(0 < Z < a) = P(−a < Z < 0), and P(Z > a) = P(Z < −a).
3. P(Z < a) = 0.5 + P(0 < Z < a) for a > 0.
4. P(a < Z < b) = P(0<Z<b) − P(0<Z<a) when both are positive; for a < 0 < b it is the sum of the two table areas.
5. Empirical rule:

| Interval | Area |
|---|---|
| −1 < Z < 1 | 0.6827 (68.27%) |
| −2 < Z < 2 | 0.9545 (95.45%) |
| −3 < Z < 3 | 0.9973 (99.73%) |
| −1.96 < Z < 1.96 | 0.9500 (95%) |

**Example:** P(−1.2 < Z < 0.8) = P(0 < Z < 1.2) + P(0 < Z < 0.8) = 0.3849 + 0.2881 = 0.6730.

## Check yourself

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

1. The standard normal table in this answer gives the area:
   A. from −∞ to z
   B. from 0 to z
   C. from z to ∞
   D. from −z to z
   Answer: B. The table lists P(0 < Z < z), so tails and other areas are built from it. (Key point: What the table gives)

2. Which relation follows from symmetry?
   A. P(Z > a) = P(Z < −a)
   B. P(Z > a) = P(Z > −a)
   C. P(Z < a) = P(Z > a) always
   D. P(Z > a) = 2P(Z < a)
   Answer: A. The curve is symmetric about 0, so the tails beyond a and beyond −a are equal. (Key point: Tail by symmetry)

3. The area between Z = −2 and Z = 2 is about:
   A. 68.27%
   B. 95.45%
   C. 99.73%
   D. 50%
   Answer: B. ±1, ±2 and ±3 standard deviations contain 68.27%, 95.45% and 99.73%. (Key point: Empirical rule)

4. P(−1.2 < Z < 0.8) is found by:
   A. 0.3849 − 0.2881
   B. 0.3849 + 0.2881 = 0.6731
   C. 0.5 − 0.3849
   D. 0.3849 × 0.2881
   Answer: B. When the interval straddles zero, the two table areas are added. (Key point: Interval across zero)

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