Statistics 2nd Year Chapter 10: Normal Distribution Notes (PDF Download)

The normal distribution is the single most important probability distribution in all of statistics. It shows up everywhere — heights and weights of individuals, IQ scores, measurement errors — because it is also the limiting form that many other probability distributions approach, and it underlies the central limit theorem that makes so much of inferential statistics possible. This chapter opens the second half of the ICS Statistics course: everything from Chapter 11 onward (sampling distributions, estimation, hypothesis testing) leans on the normal distribution introduced here.

The chapter builds the normal probability density function and its key properties, then introduces the standard normal random variable Z — a single reference distribution (mean 0, variance 1) that lets us answer probability questions for ANY normal distribution using one shared set of tables, through the simple transformation Z = (X – mu)/sigma.

Learning Objectives

• State the normal probability density function and identify its two parameters, mu and sigma
• List the key properties of the normal distribution: symmetry, mode, special areas, moments
• Standardize any normal random variable X into the standard normal variable Z
• Use the standard normal cumulative distribution function Phi(z) to calculate probabilities
• Find quantiles and percentiles of the standard normal distribution and de-standardize them back to X
• Solve applied problems involving normally distributed variables: scores, heights, dimensions, times
• Work backward to find mu or sigma (or both) given probability information

Key Concepts

10.1 The Normal Probability Density Function

A continuous random variable X is normally distributed if its probability density function is f(x) = [1/(sigma.sqrt(2.pi))] . e^[-(1/2)((x-mu)/sigma)^2], for -infinity < x < infinity, where mu can be any real number and sigma must be positive. A normal distribution is fully characterized by just these two parameters — mu (its mean) and sigma (its standard deviation) — and is commonly written X ~ N(mu, sigma-squared).

The graph of this function is called a normal curve: unimodal, symmetrical, and bell-shaped. The parameter sigma controls how flat or peaked the curve is — decreasing sigma (with mu fixed) makes the curve more sharply peaked (higher probability of X being close to mu); increasing sigma flattens it. Changing mu (with sigma fixed) simply slides the same-shaped curve left or right along the x-axis.

10.1 Properties of the Normal Distribution

The normal distribution has several defining properties worth memorizing. It is a continuous distribution ranging from -infinity to +infinity, with total area under the curve equal to 1 (P(-infinity < X < infinity) = 1). It is unimodal with its single peak (mode) at x = mu, where the maximum ordinate is f(mu) = 1/(sigma.sqrt(2.pi)). It is perfectly symmetrical, so mean = median = mode = mu, the lower and upper quartiles are equidistant from mu, and all odd-order moments about the mean are zero.

Three 'special areas' recur constantly in problems: P(mu-sigma < X < mu+sigma) = 0.6827 (about 68%), P(mu-2sigma < X < mu+2sigma) = 0.9545 (about 95%), and P(mu-3sigma < X < mu+3sigma) = 0.9973 (about 99.7%) — together known as the empirical rule. The quartile deviation is Q.D.(X) = 0.6745.sigma, the mean deviation is M.D.(X) = 0.7979.sigma, and the variance/SD are simply sigma-squared and sigma. The curve is asymptotic to the x-axis (never touches it) and has points of inflexion at x = mu – sigma and x = mu + sigma. A useful reproductive property: if X1 ~ N(mu1, sigma1^2) and X2 ~ N(mu2, sigma2^2) are independent, then X1+X2 ~ N(mu1+mu2, sigma1^2+sigma2^2).

10.2 The Standard Normal Random Variable

Because the normal cumulative distribution function cannot be integrated in closed form, and because there are infinitely many possible (mu, sigma) combinations, statisticians instead standardize: any normal random variable X can be transformed into the standard normal random variable Z = (X-mu)/sigma, which always has mean 0 and variance 1, regardless of X's original mu and sigma. This lets a single set of tables answer probability questions for every normal distribution.

The standard normal density function is written phi(z) = [1/sqrt(2.pi)].e^(-z^2/2), and its cumulative distribution function is written Phi(z) = P(Z <= z) — the area under the standard normal curve up to z. Because the curve is symmetric, phi(-z) = phi(z). Four handy results follow directly from the definition: P(Z<=a) = Phi(a); P(Z>=a) = 1-Phi(a); P(a<=Z<=b) = Phi(b)-Phi(a); and using symmetry, Phi(-a) = 1-Phi(a), so P(|Z|<a) = 2.Phi(a)-1 and P(|Z|>a) = 2.Phi(-a). Phi(z) and its inverse Phi-inverse(p) are both read from standard tables.

10.2 Quantiles, Percentiles, and De-standardizing

The p-th quantile (or 100p-th percentile) of Z is the value z_p such that Phi(z_p) = p, i.e., z_p = Phi-inverse(p) — found by looking up p in the cumulative table and reading off the corresponding z. Once a quantile z_p of the standard normal distribution is known, it can be de-standardized back into the original scale of any normal variable X using x_p = mu + sigma.z_p, obtained by rearranging Z = (X-mu)/sigma into X = mu + sigma.Z.

For applied problems on any normal variable X ~ N(mu, sigma-squared), the same standardization logic applies directly to probability statements: P(X<=a) = Phi[(a-mu)/sigma]; P(X>=a) = 1-Phi[(a-mu)/sigma]; P(a<=X<=b) = Phi[(b-mu)/sigma] – Phi[(a-mu)/sigma]. This single technique — standardize, look up in the Z table, and de-standardize when needed — solves essentially every problem type in this chapter, from finding probabilities given X-values to finding X-values (or even mu or sigma) given probabilities.

Important Definitions

Key Facts and Relations

Topic	Key Fact / Relation
Normal PDF	f(x) = [1/(sigma.sqrt(2.pi))] e^[-(1/2)((x-mu)/sigma)^2]
Standardization	Z = (X – mu) / sigma
De-standardizing	X = mu + sigma.Z
Standard normal PDF	phi(z) = [1/sqrt(2.pi)] e^(-z^2/2)
Standard normal CDF	Phi(z) = P(Z <= z)
Symmetry results	Phi(-a) = 1-Phi(a); P(|Z|<a) = 2Phi(a)-1; P(|Z|>a) = 2Phi(-a)
Quartile deviation	Q.D.(X) = 0.6745 sigma
Mean deviation	M.D.(X) = 0.7979 sigma
Special areas (empirical rule)	mu+/-sigma: 68.27% | mu+/-2sigma: 95.45% | mu+/-3sigma: 99.73%
Quantile of X	x_p = mu + sigma . z_p, where z_p = Phi-inverse(p)
Moments about mean	mu1=0, mu2=sigma^2, mu3=0, mu4=3sigma^4; Beta1=0, Beta2=3

Diagrams

Normal Distribution Shapes for Different Sigma: Three normal curves with the same mean but different standard deviations, showing how a smaller sigma produces a sharper, more peaked curve while a larger sigma produces a flatter, more spread-out curve

Empirical Rule: 68-95-99.7: The standard normal curve with the areas within 1, 2, and 3 standard deviations of the mean shaded and labelled 68.27%, 95.45%, 99.73%, illustrating the special areas under the normal curve

Standardization: Raw Scores to Z-scores: Two aligned normal curves — the raw test-score distribution (mean 500, SD 100) on top and the standard normal Z distribution below — with the region X<400 (Z<-1) shaded on both, illustrating the standardization technique from Example 10.7

Short Questions & Answers

Long Questions & Answers

Explain the normal probability density function and its key properties, and explain why the normal distribution is considered the most important distribution in statistics.

What is the normal probability density function, and what do its two parameters represent?

A continuous random variable X is normally distributed if its density function is f(x) = (1/(sigma*sqrt(2*pi))) * e^(-1/2 * ((x-mu)/sigma)^2), defined for all real x. The curve's entire shape and location are governed by just two parameters: mu, which can be any real number and equals the mean, and sigma, which must be strictly positive and equals the standard deviation. Because these two numbers completely determine the distribution, it is written compactly as X ~ N(mu, sigma-squared). This economy of parameters is part of why the normal distribution is so easy to work with.

What does the normal curve look like, and how do mu and sigma affect its shape?

The normal curve is unimodal (one peak), perfectly symmetrical about that peak, and forms the familiar bell shape. Holding mu fixed, decreasing sigma makes the curve narrower and more sharply peaked, concentrating probability tightly around the mean; increasing sigma flattens and widens the curve, spreading probability more evenly. Holding sigma fixed while changing mu leaves the shape completely unchanged and simply slides the whole curve left or right to centre on the new mean value.

What formal properties follow from the normal curve's symmetry, and what is the empirical rule?

Because the normal distribution is a valid continuous probability distribution, the total area under the curve equals exactly 1, and the probability of X falling in an interval equals the area above that interval. Its symmetry means the mean, median, and mode are always identical, all equal to mu, the quartiles sit equidistant from mu, and every odd moment about the mean equals zero. Most usefully, the empirical rule holds for every normal distribution: about 68.27% of its area lies within one standard deviation of the mean, 95.45% within two, and 99.73% within three.

Why is the normal distribution considered the most important distribution in statistics?

Three reinforcing reasons stand out. Empirically, many real variables genuinely follow something close to a normal distribution, including heights, weights, IQ scores, and small measurement errors. Theoretically, the normal distribution often emerges as the limiting form that many other distributions converge toward, making it a useful approximation even when the true underlying process isn't normal. Most importantly, it is the limiting distribution described by the Central Limit Theorem, which explains why sample means tend to be approximately normal even from non-normal populations — the property underlying most statistical inference, from confidence intervals to hypothesis tests.

Explain the concept of the standard normal random variable, describe the process of standardization and de-standardization, and explain how the standard normal table is used to solve problems involving probabilities, quantiles, and unknown parameters.

Why is standardization necessary, and how is the standard normal variable Z defined?

Since infinitely many normal distributions exist, one for every combination of mu and sigma, it is impractical to build a separate probability table for each one. Standardization solves this by converting any normal variable X into a single common reference variable: Z = (X – mu) / sigma. Whatever mu and sigma the original X had, the resulting Z is guaranteed to have mean 0 and variance 1. This shared distribution is called the standard normal distribution, written Z ~ N(0,1), and it is why one single table can serve every normal distribution.

What are phi(z) and Phi(z), and why is Phi(z) so important?

The standard normal density function, written phi(z), gives the height of the curve at a point z: phi(z) = (1/sqrt(2*pi)) * e^(-z^2/2). The cumulative distribution function, written Phi(z), instead gives the total area under the curve up to that point: Phi(z) = P(Z <= z). Phi(z) is the single most frequently used tool in this chapter, because it directly answers the recurring question of what proportion of the standard normal distribution's area lies at or below a given z-value.

What symmetry-based shortcut formulas apply to the standard normal distribution?

Because the standard normal curve is perfectly symmetric about zero, several shortcuts follow directly: P(Z > a) = 1 – Phi(a); P(a < Z < b) = Phi(b) – Phi(a); and Phi(-a) = 1 – Phi(a). These lead to two further useful formulas for absolute-value statements: P(|Z| < a) = 2*Phi(a) – 1, and P(|Z| > a) = 2*Phi(-a). All of these let probabilities be found quickly from a single table without extra calculation.

What is a quantile of the standard normal distribution, and how is it found?

A quantile (or percentile) is the reverse of a probability lookup: given a target probability p, the task is to find the z-value that produces it. The p-th quantile, z_p, is defined by Phi(z_p) = p, written using inverse-function notation as z_p = Phi inverse of p. In practice it is found by locating the probability p within the body of the standard normal table and reading across to the corresponding z-value.

How is a Z quantile converted back to the original X scale, and what is the overall three-step method?

Once a quantile z_p has been found from the table, it can be converted back to the original X scale by de-standardizing: rearranging Z = (X – mu)/sigma into X = mu + sigma*Z, so the p-th quantile on the X scale is x_p = mu + sigma*z_p. Overall, nearly every problem in this chapter follows the same three-step method: standardize into Z, use the standard normal table to find a probability or a quantile, and de-standardize back to X's own scale when required.

Multiple Choice Questions (MCQs)

Quick Revision Summary

• Normal PDF: f(x) = [1/(sigma.sqrt(2pi))] e^[-(1/2)((x-mu)/sigma)^2], parameters mu and sigma
• Symmetric, unimodal, bell-shaped; mean=median=mode=mu; total area = 1
• Empirical rule: mu+/-sigma = 68.27% | mu+/-2sigma = 95.45% | mu+/-3sigma = 99.73%
• Standardization: Z = (X-mu)/sigma; standard normal Z~N(0,1)
• Phi(z) = P(Z<=z); P(Z>a)=1-Phi(a); P(a<Z<b)=Phi(b)-Phi(a)
• Symmetry: Phi(-a)=1-Phi(a); P(|Z|<a)=2Phi(a)-1; P(|Z|>a)=2Phi(-a)
• Quantile: z_p = Phi-inverse(p); de-standardize: x_p = mu + sigma.z_p
• Q.D.(X) = 0.6745 sigma; M.D.(X) = 0.7979 sigma; Var(X)=sigma^2

Exam Tips

• Always sketch a quick bell curve and shade the region asked for — it prevents sign errors with Phi(a) vs 1-Phi(a)
• For 'P(X between a and b)' problems, standardize BOTH endpoints before subtracting Phi values
• For 'find X given a probability' problems, always look up the z-value FIRST, then de-standardize with x = mu + sigma.z
• Remember Phi(-z) = 1 – Phi(z) — most tables only list positive z, so use this to handle negative z-values
• When solving for unknown mu or sigma, set up the standardization equation and solve algebraically — don't guess
• Watch for continuity correction language ('less than 60mm' on discrete/measured data may mean 'less than 59.5' in a problem's intended interpretation)