---
title: "Standard normal: definition, properties, importance"
subject: "Probability and Statistics"
chapter: 3
key: "semester_5/probability_and_statistics:standard-normal-definition-properties-importance"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/standard-normal-definition-properties-importance
site: BCT Engineering
---

# Standard normal: definition, properties, importance

Probability and Statistics · Chapter 3: Continuous Probability Distributions

- Asked in: 2075 Asoj, 4 marks; 2072 Kartik, 5 marks (asked 2 times)
- Video: [Z-Scores, Standardization, and the Standard Normal Distribution (5.3)](https://www.youtube.com/watch?v=2tuBREK_mgE) — Simple Learning Pro
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/standard-normal-definition-properties-importance
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-3

## Question

Define standard normal distribution. Write down its properties and importance of this distribution. (Define Standard Normal Distribution with their respective probability density function and describe its properties.)

## Short answer

Standard normal is Z=(X-mu)/sigma with mean 0, variance 1, pdf (1/sqrt(2pi))e^(-z^2/2). It is symmetric about 0, total area 1, inflection at ±1, and one table serves all normals; it underlies z-tests and CLT.

## Answer

#### Definition
A normal random variable X with mean μ and standard deviation σ is converted to the **standard normal variable** Z = (X − μ)/σ. Z has mean 0 and variance 1, written Z ~ N(0, 1). Its pdf is

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

#### Properties
1. The curve is bell shaped and symmetric about z = 0.
2. Mean = median = mode = 0; variance = standard deviation = 1.
3. The total area under the curve is 1; P(Z < 0) = P(Z > 0) = 0.5.
4. The curve is asymptotic to the z-axis; it never touches it.
5. Points of inflection are at z = ±1.
6. Areas: P(−1 < Z < 1) = 0.6827, P(−2 < Z < 2) = 0.9545, P(−3 < Z < 3) = 0.9973.
7. Φ(−z) = 1 − Φ(z). Skewness is 0 and kurtosis is 3.

```text
          .--'''--.
        .'    |    '.
      _/      |      \_
   __/        |        \__
  -3  -2  -1  0   1   2   3   z
```

#### Importance
- A single table of areas for Z serves every normal distribution, whatever μ and σ are.
- Probabilities for any N(μ, σ²) are found by standardising.
- It is the basis for the normal approximations to the binomial and Poisson distributions.
- The Central Limit Theorem makes the sample mean approximately standard normal after standardising, which is the basis of z-tests and confidence intervals.
- It allows comparison of values from different scales, using z-scores.

## Check yourself

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

1. The standard normal variable is:
   A. Z = (X - mu)/sigma with mean 0 and variance 1
   B. Z = X/sigma with mean 1
   C. Any normal with mean 0
   D. Z = (X - sigma)/mu
   Answer: A. Standardising gives N(0,1). (Key point: Definition)

2. Which is true for the standard normal?
   A. Mean 0, mode 1
   B. It is skewed right
   C. Kurtosis is 0 and skewness 3
   D. Mean = median = mode = 0
   Answer: D. It is symmetric about 0; skewness 0, kurtosis 3. (Key point: Symmetry)

3. P(-2 < Z < 2) is about:
   A. 0.6827
   B. 0.9545
   C. 0.9973
   D. 0.5
   Answer: B. 68.27%, 95.45%, 99.73% for 1, 2, 3 SD. (Key point: Area within 2 SD)

4. Why is it important?
   A. One table serves every normal distribution after standardising
   B. It has no parameters so it is exact
   C. It applies only to discrete data
   D. It removes the need for tables
   Answer: A. All normals are converted to Z. (Key point: Importance)

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