Normal distribution is a probability distribution for a continuous random variable. It produces a symmetrical, bell-shaped curve where the mean, median, and mode are equal at the centre. By transforming any normal variable \(X \sim \text{N}(\mu, \sigma^2)\) into the standard normal variable \(Z \sim \text{N}(0, 1)\), we can calculate probabilities and determine values of parameters from standard normal tables.
Explore the key properties of the normal curve \(X \sim \text{N}(\mu, \sigma^2)\), the standardisation process to \(Z \sim \text{N}(0, 1)\), and the standard tail areas.
The curve is symmetrical about the vertical line \(X = \mu\). The mean, median and mode are equal at the centre. The total area under the graph is \(1\text{ unit}^2\), with \(50\%\) of the data on each side of the mean.
Using \(Z = \dfrac{X - \mu}{\sigma}\), any normal variable is transformed into a standard normal variable with mean \(0\) and standard deviation \(1\). The standard normal table gives upper-tail probabilities \(P(Z > z)\).
Standard score (z-score)
Converts \(X \sim \text{N}(\mu, \sigma^2)\) to \(Z \sim \text{N}(0, 1)\).
Symmetry property
Upper-tail and lower-tail probabilities are equal by symmetry.
Understand the normal curve, standardisation, and probability calculations.
When sample size \(n\) is large (e.g., \(n > 30\)) and binomial table values cannot be obtained, normal distribution estimates the answer if \(np \ge 10\) and \(n(1 - p) \ge 10\).
Defined for a continuous random variable \(X\), denoted as \(X \sim \text{N}(\mu, \sigma^2)\). The curve is symmetrical, bell-shaped, and asymptotic. Total area under the graph is \(1\text{ unit}^2\).
Changing \(\mu\) shifts the axis of symmetry horizontally without changing shape. Changing \(\sigma\) affects height and width: larger \(\sigma\) gives a broader, flatter curve; smaller \(\sigma\) gives a narrower, taller peak.
Approximately \(68\%\) of data lies within \(\mu \pm 1\sigma\), \(95\%\) within \(\mu \pm 2\sigma\), and \(99.8\%\) within \(\mu \pm 3\sigma\).
As the number of experiments or sample size increases, the experimental mean gets closer to the theoretical mean value of the population. The larger the sample size, the smaller the random variation.
A standard normal distribution has mean \(\mu = 0\) and standard deviation \(\sigma = 1\). The standard score is \(Z = \dfrac{X - \mu}{\sigma}\), with \(E(Z) = 0\) and \(\text{Var}(Z) = 1\).
The standard normal table gives upper-tail probabilities \(P(Z > z)\). Use symmetry and complement rules: \(P(Z < a) = 1 - P(Z > a)\), \(P(a < Z < b) = P(Z > a) - P(Z > b)\).
When \(P(Z > a) > 0.5\), \(a\) must be negative. Equate standard scores to solve for unknown parameters: \(z = \dfrac{x - \mu}{\sigma} \implies x = \mu + z\sigma\).
Identify the mean \(\mu\) and standard deviation \(\sigma\).
Convert \(X\) to \(Z\) using \(Z = \dfrac{X - \mu}{\sigma}\).
Use the standard normal table to find \(P(Z > z)\).
Apply symmetry and complement rules to find the required probability.
Answer in your mind, then press "Check Answer".
A continuous random variable \(X\) is normally distributed with a mean of \(24\) and a standard deviation of \(6\). Find the \(z\)-score when \(X = 19.5\).
A continuous random variable \(Z\) has a standard normal distribution. Find \(P(Z > 2.153)\) using the standard normal distribution table.
The masses of bread produced by a bakery are normally distributed with a mean of \(350\text{ g}\) and a standard deviation of \(45\text{ g}\). Convert \(P(280 < X < 375)\) into a standard normal probability statement in terms of \(Z\).
Changing the standard deviation shifts the position of the curve along the horizontal axis.
Changing \(\sigma\) only affects the width and height (dispersion) of the curve. The position is determined solely by the mean \(\mu\).
The tails of the normal curve touch the horizontal axis when \(z > 3\) or \(z < -3\).
The curve is asymptotic; both ends extend indefinitely without touching the \(x\)-axis.
The area under the curve can exceed \(1\) if the values of \(X\) are very large.
The total area under the probability density curve represents the sum of all probabilities and is strictly equal to \(1\text{ unit}^2\), regardless of the scale of \(X\).
Objective: Test your understanding of normal distribution properties, standardisation, and probability rules.
1 What happens to the graph of a normal distribution if the standard deviation \(\sigma\) decreases while the mean \(\mu\) remains constant?
2 Given a continuous random variable \(X \sim \text{N}(45, \sigma^2)\) such that \(P(X > 51) = 0.2888\). If the corresponding \(z\)-score obtained from the standard normal table is \(0.557\), what is the standard deviation \(\sigma\)?
3 For a continuous random variable \(X\) with a normal distribution, the percentage of data that lies within the standard deviation \(\pm 1\) from the mean is ________%.
4 The formula used to convert a continuous random variable \(X \sim \text{N}(\mu, \sigma^2)\) into a standard normal score \(Z\) is \(Z = \dfrac{X - \mu}{________}\).
Drag each concept to the correct description. On a phone, tap a concept first, then tap the matching box.
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