Statistics IUnit 57 min read

Normal Distribution – Key Concepts and Applications

Unit 5 of Statistics I: covers the normal distribution, its defining properties, standardisation, probability calculations, approximation to other discrete distributions, and real‑world applications with worked examples.

Padhai · Based on the official syllabus and past papers
Updated 2 Oct 2026

Key points

The normal distribution is continuous, symmetric, and fully described by its mean $ \mu $ and standard deviation $ \sigma $.
Standardisation transforms any normal variable to the standard normal $ Z \sim N(0,1) $ using $ Z = \frac{X-\mu}{\sigma} $.
Probabilities for normal variables are obtained from the standard normal table or software; the area under the curve equals 1.
The normal distribution is used to model many natural and engineered processes (e.g., measurement errors, human heights, stock returns).
Normal approximation to binomial and Poisson is valid when $ np $ and $ n(1-p) $ are large (≥ 5).

Introduction

The normal distribution, also called the Gaussian distribution, is the most frequently encountered continuous probability distribution in statistics. It arises naturally from the Central Limit Theorem: the sum (or average) of many independent, identically distributed random variables tends to a normal distribution, regardless of the original distribution.

Definition

A random variable $X$ follows a normal distribution with parameters $μ$ (mean) and $σ^{2}$ (variance) if its probability density function (pdf) is

Normal curve (left: general; right: standardised)

$f_{X} (x) = \frac{1}{σ 2 π} exp (- \frac{( x - μ ) ^{2}}{2 σ ^{2}}), - \infty < x < \infty.$

The pdf is bell‑shaped, symmetric about $μ$ , and its total area equals 1.

Key Properties

Property	Description
Mean, Median, Mode	All equal to $μ$ .
Symmetry	$f (μ + d) = f (μ - d)$ .
Tails	As $∣ x - μ ∣ \to \infty$ , $f (x) \to 0$ .
Standard Deviation	Controls spread; 68 % of data lie within $μ \pm σ$ , 95 % within $μ \pm 2 σ$ , 99.7 % within $μ \pm 3 σ$ .
Moment Generating Function	$M_{X} (t) = exp (μ t + \frac{1}{2} σ^{2} t^{2})$ .

Standardisation

To compute probabilities, we convert $X$ to the standard normal variable $Z$ :

$Z = \frac{X - μ}{σ} ⟹ Z \sim N (0, 1) .$

This transformation preserves the shape of the distribution and allows us to use the standard normal table.

Standardisation: X → Z on the standard normal curve (μ=0, σ=1)

Probability Calculations

Using the Standard Normal Table

The standard normal table lists $P (Z \leq z)$ . To find $P (a \leq X \leq b)$ :

Standardise both limits: $z_{a} = \frac{a - μ}{σ}$ , $z_{b} = \frac{b - μ}{σ}$ .
Look up $P (Z \leq z_{b})$ and $P (Z \leq z_{a})$ .
Subtract: $P (a \leq X \leq b) = P (Z \leq z_{b}) - P (Z \leq z_{a})$ .

Worked Example 1 – Resistor Tolerance

A factory produces resistors with $μ = 40 Ω$ and $σ = 2 Ω$ . What percentage of resistors lie between $38 Ω$ and $42 Ω$ ?

Solution
$z_{38} = \frac{38 - 40}{2} = - 1$
$z_{42} = \frac{42 - 40}{2} = 1$

From the standard normal table:
$P (Z \leq 1) = 0.8413$
$P (Z \leq - 1) = 0.1587$

$P (38 \leq X \leq 42) = 0.8413 - 0.1587 = 0.6826$ → 68.3 %.

Probability of a Single Value

For a continuous variable, $P (X = x) = 0$ . Probabilities are always for intervals.

Normal Approximation to Binomial

When $n$ is large and $p$ not too close to 0 or 1, the binomial $B (n, p)$ can be approximated by $N (n p, n p (1 - p))$ . Apply a continuity correction: replace $k$ by $k \pm 0.5$ .

Worked Example 2 – Binomial Approximation

A quality inspector finds that 30 % of chips are defective. In a batch of 200 chips, what is the probability that at most 70 are defective?

$n = 200, p = 0.3$
Mean $μ = n p = 60$ , SD $σ = n p (1 - p) = 200 \times 0.3 \times 0.7 \approx 6.48$ .

Standardise with continuity correction:
$P (X \leq 70) \approx P (Z \leq \frac{70 + 0.5 - 60}{6.48}) = P (Z \leq 1.64)$ .

From the table: $P (Z \leq 1.64) \approx 0.9495$ .
So 94.95 % chance that at most 70 chips are defective.

Applications in the Real World

In the Real World

Product / Company	Idea Used	How It Works
eSewa	Transaction amount distribution	User payment amounts are roughly normal; risk models use mean and SD to flag outliers.
Daraz	Order processing time	Average delivery time $μ = 30$  min, $σ = 5$  min; 95 % of orders delivered within 20–40 min.
Ncell	Signal strength	Received signal strength $μ = - 80$  dBm, $σ = 3$  dBm; 68 % of users experience between –83 and –77 dBm.
NEPSE	Daily stock return	Daily return $μ = 0.2%$ , $σ = 1.5%$ ; probability of a > 3 % gain is $P (Z > 1.67) \approx 4.8%$ .
Google Search Latency	Query response time	Mean latency $μ = 120$  ms, $σ = 15$  ms; 99.7 % of queries finish within 75–165 ms.

Worked Example – Bank Loan Interest

A bank offers loans with an average annual interest rate of $8%$ and a standard deviation of $2%$ . What is the probability that a randomly chosen loan has an interest rate exceeding $10%$ ?

Solution
$z = \frac{10 - 8}{2} = 1.0$ .
$P (Z > 1.0) = 1 - P (Z \leq 1.0) = 1 - 0.8413 = 0.1587$ .
So 15.9 % of loans have rates above 10 %.

Comparison with Other Distributions

Distribution	Shape	Parameters	When to Use
Normal	Bell‑shaped, symmetric	$μ, σ$	Continuous data, measurement errors, CLT
Binomial	Discrete, skewed	$n, p$	Number of successes in fixed trials
Poisson	Discrete, right‑skewed	$λ$	Rare events over fixed interval
Exponential	Continuous, right‑skewed	$λ$	Time between events

Advantages of Normal

Closed‑form pdf and cdf (via error function).
Simple to standardise and use tables or software.
Central Limit Theorem guarantees normality for sums of many variables.

Disadvantages

Assumes infinite support; real data may have natural bounds.
Sensitive to outliers; heavy‑tailed data may be poorly modeled.

Visualising Normal Data

Histogram of Sample Data

A sample of 50 measurements of a manufactured part’s length (mm) is shown below. The histogram approximates a normal shape, confirming the assumption.

Standard Normal Table Snippet

A compact view of the standard normal cumulative probabilities for $z$ from 0.00 to 3.00.

Summary

The normal distribution is fully characterised by its mean and standard deviation.
Standardisation to $Z$ allows universal probability calculations.
Normal approximation is a powerful tool for discrete problems.
Many everyday systems (payments, logistics, telecommunications, finance) rely on normal models for decision making.

Exam Tip

Know the formulae: pdf, standardisation, probability bounds (68‑95‑99.7 rule).
Practice table look‑ups: be comfortable reading $P (Z \leq z)$ and applying continuity corrections.
Work through approximation problems: identify $n$ and $p$ , compute $μ$ and $σ$ , apply continuity correction.
Draw the curve: sketching the normal curve with shaded areas helps visualise the problem.
Check units: when dealing with percentages or physical units, convert to the same scale before standardising.

Based on the TU BSc CSIT syllabus for Statistics I (STA169), unit 5.

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