5.3 Normal Distribution
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.
Learning Objectives
- Investigate and explain the characteristics and properties of a normal distribution graph.
- Understand the effects of varying the mean (\(\mu\)) and standard deviation (\(\sigma\)) on the shape and position of the normal distribution curve.
- Understand the law of large numbers and its significance in random variations.
- Standardise a continuous random variable \(X \sim \text{N}(\mu, \sigma^2)\) to obtain standard normal score (z-score), \(Z \sim \text{N}(0, 1)\).
- Determine and interpret the probability of events using the standard normal distribution table or scientific calculator.
- Solve real-world mathematical problems involving normal distributions.
Anatomical Features and Transformations of the Normal Curve
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.
Symmetrical Bell-Shaped Curve
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.
Standardisation to \(Z \sim \text{N}(0, 1)\)
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 \(P(Z > z)\).
| Region | Percentage of data | Interpretation |
|---|---|---|
| \(\mu \pm 1\sigma\) | Approximately \(68\%\) | \(34\%\) on each side of the mean. |
| \(\mu \pm 2\sigma\) | Approximately \(95\%\) | \(13.5\%\) in each region between \(1\sigma\) and \(2\sigma\). |
| \(\mu \pm 3\sigma\) | Approximately \(99.8\%\) | \(2.4\%\) between \(2\sigma\) and \(3\sigma\), \(0.1\%\) beyond \(3\sigma\). |
Standard score (z-score)
\(Z = \dfrac{X - \mu}{\sigma}\)Converts \(X \sim \text{N}(\mu, \sigma^2)\) to \(Z \sim \text{N}(0, 1)\).
Symmetry property
\(P(Z > a) = P(Z < -a)\)Upper-tail and lower-tail probabilities are equal by symmetry.
Short Explanation
Understand the normal curve, standardisation, and probability calculations.
1. Conditions for Estimating Binomial Distribution
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 \(n(1 - p) \ge 10\).
2. Properties of the Normal Distribution Graph
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 \(1\text{ unit}^2\).
3. Effects of Changing \(\mu\) and \(\sigma\)
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.
4. Percentage Distribution of Data
Approximately \(68\%\) of data lies within \(\mu \pm 1\sigma\), \(95\%\) within \(\mu \pm 2\sigma\), and \(99.8\%\) within \(\mu \pm 3\sigma\).
5. Random Variation and the Law of Large Numbers
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.
6. Standard Normal Distribution
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 \(\text{Var}(Z) = 1\).
7. Determining Probabilities
The standard normal table gives upper-tail probabilities \(P(Z > z)\). Use symmetry and complement rules: \(P(Z < a) = 1 - P(Z > a)\), \(P(a < Z < b) = P(Z > a) - P(Z > b)\).
8. Finding Values Given Probabilities
When \(P(Z > a) > 0.5\), \(a\) must be negative. Equate standard scores to solve for unknown parameters: \(z = \dfrac{x - \mu}{\sigma} \implies x = \mu + z\sigma\).
| Probability statement | Equivalent form | Condition |
|---|---|---|
| \(P(Z < a)\) | \(1 - P(Z > a)\) | Total area equals \(1\). |
| \(P(a < Z < b)\) | \(P(Z > a) - P(Z > b)\) | For \(a < b\). |
| \(P(|Z| > a)\) | \(2P(Z > a)\) | For \(a > 0\). |
| \(P(|Z| < a)\) | \(1 - 2P(Z > a)\) | For \(a > 0\). |
Steps for Standardising and Finding Probabilities
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.
Try to Answer First
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 \(Z\).
Common Mistakes
- Forgetting to square root the variance: when given \(X \sim \text{N}(500, 169)\), students often use \(\sigma = 169\) instead of \(\sigma = \sqrt{169} = 13\) in the denominator of \(Z = \dfrac{X - \mu}{\sigma}\).
- Assigning the wrong sign to \(z\)-scores: when a value lies to the left of the mean (\(X < \mu\)), \(z\) must be negative. Students frequently forget the negative sign.
- Adding instead of subtracting the value in the subtraction column: the small digits on the right side of the standard normal table must be subtracted from the probability, not added.
- Treating continuous boundaries strictly: students worry about whether the inequality contains \(\le\) or \(<\). In continuous normal distribution, \(P(a < X < b) = P(a \le X \le b)\) because \(P(X = a) = 0\).
Concept Misunderstandings
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 \(X\).
Summary
- Normal distribution models continuous variables with a symmetrical bell-shaped curve where \(\text{mean} = \text{median} = \text{mode} = \mu\).
- The total area under the curve is \(1\), with \(50\%\) lying on each side of \(\mu\).
- Increasing \(\mu\) shifts the curve to the right, whereas increasing \(\sigma\) widens and flattens the curve.
- Any normal distribution \(X \sim \text{N}(\mu, \sigma^2)\) can be converted to the standard normal distribution \(Z \sim \text{N}(0, 1)\) via \(Z = \dfrac{X - \mu}{\sigma}\).
- Standard normal distribution tables allow calculating upper-tail probabilities \(P(Z > z)\), which, combined with symmetry and complement rules, allow evaluating any probability interval or solving for unknown parameters.
Short Activity
Objective: Test your understanding of normal distribution properties, standardisation, and probability rules.
A. Objective Quiz
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 \(\sigma\)?
B. Fill in the Blanks
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 \(Z = \dfrac{X - \mu}{________}\).
C. Matching / Drag and Drop
Drag each concept to the correct description. On a phone, tap a concept first, then tap the matching box.
