Normal Distribution and Its Applications
FoundationThe normal distribution is the commonest bell-shaped model for continuous data: symmetric about the mean, with σ controlling the spread; standardise X into Z = (X−μ)/σ, read probabilities from the standard normal table, and combine symmetry with the 68–95–99.7 rule to solve quality-control and applied problems.
- Normal distribution X ~ N(μ, σ²): bell-shaped, symmetric about μ, mean equals mode equals median, σ controls flatness, total area one.
- Standardisation Z = (X−μ)/σ reduces any normal problem to the standard normal Z ~ N(0, 1) and its table.
- The table gives P(Z ≤ z); 'greater than' subtracts from one, 'between' subtracts two entries, then restore the context to X.
文體小知識：數學概念說明文——由連續隨機變量引入正態分佈的性質與記法，講解標準化與標準正態分佈表的用法，配合對稱性與經驗法則解決概率應用題。
§Text Passage
1Random variables in the compulsory part are mostly discrete, with probabilities concentrated on individual values; but quantities such as height, lifetime and volume can take any value in an interval — continuous random variables, with continuous probability distributions. For a continuous distribution, probability is only meaningful over an interval, represented by area under a curve: the probability that a person's height is exactly 170 centimetres is actually zero; what matters is the probability of falling between two values.
2The normal distribution is the most important continuous probability distribution, written X ~ N(μ, σ²). Its curve is bell-shaped and symmetric about the mean; the mean, mode and median are all equal, lying on the axis of symmetry; the flatness of the curve depends on the value of σ — the larger the σ, the wider and flatter the curve; and the total area under the curve is one, representing all probability. The two parameters have distinct roles: μ fixes the position (changing μ slides the whole curve), while σ fixes the spread.
3Each pair of μ and σ gives a different curve, so tabulating them all is impossible; instead we standardise: Z equals X minus μ, divided by σ, counting how many standard deviations x lies from the mean. The standardised Z follows the standard normal distribution with mean 0 and variance 1, and every normal calculation in the world reduces to this single curve, read from the standard normal table.
4Finding probabilities follows three fixed steps: standardise, look up the table, restore the context. The standard normal table lists the probability that Z is less than or equal to a value; for the probability between two values, subtract two table entries; for the probability above a value, subtract the entry from one. Symmetry is the second key: the curve is symmetric about zero, with half the area on each side, and the left-tail probability at −z equals one minus the left-tail probability at z.
5Applications add one memorised rule: about 68% of data lie within one standard deviation of the mean, about 95% within two, and about 99.7% within three. Quality control on production lines, setting warranty periods and judging outliers all rely on this 68–95–99.7 rule; applied questions on the normal distribution are a staple of the M1 paper.
1Summary 重點摘要
The normal distribution is the commonest bell-shaped model for continuous data: symmetric about the mean, with σ controlling the spread; standardise X into Z = (X−μ)/σ, read probabilities from the standard normal table, and combine symmetry with the 68–95–99.7 rule to solve quality-control and applied problems.
2Vocabulary cards Vocabulary
A random variable taking any value in an interval, such as height or lifetime; probabilities are areas under a curve.
The bell-shaped continuous probability distribution symmetric about its mean, written X ~ N(μ, σ²).
The probability curve of a normal distribution: bell-shaped, symmetric, with total area one underneath.
The location parameter of the normal distribution; the mean, mode and median are all equal, on the axis of symmetry.
The spread parameter controlling the flatness of the curve; the larger the σ, the wider and flatter the curve.
Converting X into Z = (X−μ)/σ, counting how many standard deviations x lies from the mean.
The normal distribution with mean 0 and variance 1, Z ~ N(0, 1), read from its own table.
A table listing P(Z ≤ z); combined with symmetry it yields all interval probabilities.
The standard normal curve is symmetric about zero: P(Z < 0) = 0.5 and P(Z < −a) = P(Z > a).
About 68%, 95% and 99.7% of data lie within 1, 2 and 3 standard deviations of the mean.
3Step by step Steps
- Step 1: From discrete to continuousRandom variables in the compulsory part are mostly discrete, with probability concentrated on individual values; but quantities like height, lifetime and volume can take any value in an interval — continuous random variables — and probability is only meaningful over an interval, as area under a curve. For example, in a continuous model the probability that height is exactly 170 centimetres is zero; what is meaningful is the probability of height between 169.5 and 170.5 centimetres — the area under the curve over that interval. This area-based thinking underpins every calculation in the lesson.
- Step 2: The normal distribution and its propertiesThe normal distribution is the most important continuous distribution: the curve is bell-shaped and symmetric about the mean; the mean, mode and median are all equal; the flatness depends on σ; and the total area under the curve is one. For example, with the same mean of 170 centimetres, σ = 5 gives a tall narrow curve while σ = 10 gives a low wide one — yet both enclose area one, the whole probability. In the notation X ~ N(μ, σ²), remember the second parameter is the variance, not the standard deviation: a favourite exam trap.
- Step 3: The roles of the two parametersμ fixes position: changing μ slides the whole bell sideways. σ fixes spread: the larger the σ, the more scattered the data and the wider, flatter the curve. For example, two production lines both produce screws averaging 50 millimetres, but with σ of 0.2 and 1 millimetre respectively: the first is consistent, with a tall narrow curve; the second is erratic, with a low wide one. When comparing data sets, the mean answers 'what is typical' and the standard deviation answers 'how spread out' — both are needed.
- Step 4: Standardisation and the standard normal distributionEvery pair of μ and σ gives its own curve, so tabulating all is impossible; instead convert once and for all: Z = (X−μ)/σ, the number of standard deviations x lies from the mean. After conversion, Z ~ N(0, 1), the unique standard normal distribution. For example, if X ~ N(50, 8²), then at x = 66, Z = (66−50)÷8 = 2 — two standard deviations above the mean; at x = 34, Z = −2, two below. Standardisation is the first step of every normal calculation, and the formula must be remembered exactly.
- Step 5: Reading the standard normal tableThe standard normal table lists P(Z ≤ z). To find P(X < x): standardise to get z, read the table, then restore the context. For example, X ~ N(50, 8²), find P(X < 66): z = 2, and the table gives P(Z ≤ 2) ≈ 0.9772. For an interval, subtract: P(42 < X < 66) = P(Z < 2) − P(Z < −1) ≈ 0.9772 − 0.1587 = 0.8185, where P(Z < −1) = 1 − P(Z < 1) uses symmetry. Look up, subtract, restore — the standard format for calculation questions.
- Step 6: Symmetry and solving for xThe standard normal curve is symmetric about zero: P(Z < 0) = 0.5, and P(Z < −a) = P(Z > a) = 1 − P(Z < a). To find x from a given probability, reverse-read the table for z, then solve Z = (X−μ)/σ for X = μ + zσ. For example, X ~ N(170, 7²), find h such that P(X > h) = 0.95: h lies to the left, P(Z < z) = 0.05 reverse-reads to z ≈ −1.645, so h = 170 + (−1.645)(7) ≈ 158.5 centimetres. The marks in reverse questions hinge on the reverse lookup from probability to z.
- Step 7: The 68–95–99.7 ruleThe normal distribution has an elegant memory rule: about 68% of data lie within μ±1σ, about 95% within μ±2σ, and about 99.7% within μ±3σ. For example, light bulbs with lifetime X ~ N(2000, 100²): about 68% last between 1900 and 2100 hours, about 95% between 1800 and 2200 hours, so only about 2.5% outlast 2200 hours. Quality tolerance, warranty estimates and outlier checks all use this mental arithmetic; convert the interval endpoints into numbers of standard deviations from the mean, apply the rule, then split the two tails equally by symmetry.
5Worked example Example
Example: the lifetime X (in hours) of a model of light bulb follows X ~ N(2000, 100²). Using the 68–95–99.7 rule, find (a) P(1900 < X < 2100); (b) P(X > 2200); (c) P(2100 < X < 2200).
- Locate the intervals: μ = 2000 and σ = 100; 1900 and 2100 are μ−σ and μ+σ, while 1800 and 2200 are μ−2σ and μ+2σ
- (a) The probability within μ±1σ is about 0.68, so P(1900 < X < 2100) ≈ 0.68
- (b) The probability within μ±2σ is about 0.95 and the two tails are equal, so P(X > 2200) ≈ (1 − 0.95)÷2 = 0.025
- (c) P(2100 < X < 2200) = P(μ+σ < X < μ+2σ) = (0.95 − 0.68)÷2 = 0.135
- Answer: (a) about 0.68; (b) about 0.025; (c) about 0.135. The key is converting endpoints into numbers of standard deviations from the mean, then using the rule and symmetry to split the tails equally; state the multiple of σ for every endpoint in the solution.
6Common mistakes Mistakes
7Quick check Quick check
Q1In a normal distribution, what is the relationship among the mean, mode and median?
Q2Let X ~ N(60, 5²). Find the Z value corresponding to x = 47.5.
Q3In the standard normal distribution, what is P(Z < 0) and why?
Q4Given P(Z < 1.5) = 0.9332, find P(−1.5 < Z < 1.5).
Q5Let X ~ N(2000, 100²). Use the 68–95–99.7 rule to find P(X < 1800).
8Recap Recap
- Normal distribution X ~ N(μ, σ²): bell-shaped, symmetric about μ, mean equals mode equals median, σ controls flatness, total area one.
- Standardisation Z = (X−μ)/σ reduces any normal problem to the standard normal Z ~ N(0, 1) and its table.
- The table gives P(Z ≤ z); 'greater than' subtracts from one, 'between' subtracts two entries, then restore the context to X.
- To find x: reverse-read z from the probability, then use X = μ + zσ, watching the sign of z.
- The 68–95–99.7 rule estimates the proportions within 1, 2 and 3 standard deviations, with the two tails split equally.