📚 Variance of a Normal Distribution | 正态分布的方差
In Edexcel A Level Mathematics, the normal distribution is written as X ~ N(μ, σ²), where σ² is the variance. Understanding variance is essential because it controls the spread, shape, and probability calculations for any normal model.
在 Edexcel A Level 数学中，正态分布记作 X ~ N(μ, σ²)，其中 σ² 就是方差。理解方差非常重要，因为它决定了正态模型的离散程度、曲线形状以及概率计算。
1. What the Variance Parameter Means | 方差参数的含义
The variance σ² measures the average squared distance of a normal random variable from its mean μ. It is always positive, and its square root σ is the standard deviation.
方差 σ² 衡量正态随机变量与其均值 μ 的平均平方距离。方差始终为正，其平方根 σ 就是标准差。
A larger variance means observations are more spread out; a smaller variance means they are concentrated near the mean.
方差越大，观测值越分散；方差越小，观测值越集中在均值附近。
In normal distribution problems, the variance is not just an extra detail: it determines how likely extreme values are. Two normal distributions with the same mean but different variances can give very different probabilities for the same interval.
在正态分布问题中，方差并非只是附加信息：它决定了极端值出现的可能性。均值相同但方差不同的两个正态分布，对于同一区间可能给出截然不同的概率。
2. Notation and Parameters | 符号与参数
In Edexcel questions, the normal distribution is always specified by two parameters: the mean μ and the variance σ². The notation X ~ N(μ, σ²) is standard, and the second parameter is the variance, not the standard deviation.
在 Edexcel 考题中，正态分布总是由两个参数确定：均值 μ 和方差 σ²。标准记法为 X ~ N(μ, σ²)，第二个参数是方差，而不是标准差。
For example, X ~ N(50, 16) means μ = 50 and σ² = 16, so σ = 4.
例如，X ~ N(50, 16) 表示 μ = 50，σ² = 16，因此 σ = 4。
| Parameter | Symbol | Meaning | Example: X ~ N(50, 16) |
|---|---|---|---|
| Mean | μ | Centre of the distribution | 50 |
| Variance | σ² | Spread of the distribution | 16 |
| Standard deviation | σ | Square root of variance | 4 |
Always check whether a question gives the variance or the standard deviation. Many exam questions deliberately write N(μ, σ²) with a square value, such as 36 or 49, to test whether you remember to take the square root.
始终要检查题目给出的是方差还是标准差。许多考题故意用平方数来书写 N(μ, σ²)，例如 36 或 49，以检验你是否记得开平方。
3. Effect of Variance on the Curve | 方差对曲线形状的影响
The probability density function of a normal distribution reaches its maximum at x = μ, and the variance controls how quickly the curve falls away from that peak.
正态分布的概率密度函数在 x = μ 处达到最大值，方差控制曲线从峰值向两侧下降的速度。
If σ² is small, the bell curve is tall and narrow. If σ² is large, the curve is low and flat, with more probability in the tails.
如果 σ² 较小，钟形曲线又高又窄；如果 σ² 较大，曲线变得低平，尾部概率更大。
The points of inflection occur exactly at μ ± σ, which gives a useful visual check when sketching normal curves.
拐点恰好出现在 μ ± σ 处，这为绘制正态曲线提供了一个实用的视觉检查方法。
This means that the variance is not just a number to plug into a formula; it tells you how concentrated the data are. In context, a small variance might represent precise manufacturing, while a large variance might represent more unpredictable natural measurements.
这意味着方差不仅仅是一个带入公式的数字；它还告诉你数据的集中程度。在实际情境中，小方差可能代表精确的制造过程，而大方差可能代表更不可预测的自然测量值。
4. Standard Deviation as the Natural Scale | 标准差作为自然尺度
Although normal distributions are defined by variance, most probability calculations use the standard deviation σ because it has the same units as the data.
虽然正态分布由方差定义，但大多数概率计算使用标准差 σ，因为它与数据具有相同的单位。
You should convert the problem’s variance σ² into σ before standardising, as the formula needs σ in the denominator.
在标准化之前，应先将题目给出的方差 σ² 转换为标准差 σ，因为公式的分母需要 σ。
σ = √σ²
For example, if X ~ N(80, 25), then the standard deviation is σ = 5, not 25. Using σ = 25 would place most of the data impossibly far from the mean.
例如，如果 X ~ N(80, 25)，那么标准差是 σ = 5，而不是 25。如果误用 σ = 25，会使大部分数据位于离均值不可能远的位置。
5. Standardising and the Role of σ | 标准化与 σ 的作用
Standardising changes X ~ N(μ, σ²) into the standard normal Z ~ N(0, 1). Dividing by σ rescales the spread so that the variance becomes 1.
标准化将 X ~ N(μ, σ²) 转化为标准正态分布 Z ~ N(0, 1)。除以 σ 重新调整了离散程度，使方差变为 1。
Z = (X − μ) / σ
For a probability such as P(X < a), write P(Z < (a − μ)/σ) and then use the standard normal table or calculator.
对于 P(X < a) 这类概率，先写成 P(Z < (a − μ)/σ)，然后查标准正态分布表或使用计算器。
P(X < a) = P(Z < (a − μ)/σ)
This step works because subtracting μ shifts the mean to 0, and dividing by σ rescales the units so that the new variance is exactly 1.
这一步之所以有效，是因为减去 μ 会将均值移到 0，而除以 σ 会重新调整单位，使新的方差恰好为 1。
6. Finding Probabilities with Given Variance | 给定方差求概率
When μ and σ² are known, every normal probability can be reduced to a standard normal problem. First find σ = √σ², then calculate the z-value.
当 μ 和 σ² 已知时，任何正态概率都可以转化为标准正态问题。先求 σ = √σ²，再计算 z 值。
Example: X ~ N(100, 25). Find P(X < 110). Here σ = 5, so z = (110 − 100)/5 = 2. Then P(Z < 2) ≈ 0.9772.
例：X ~ N(100, 25)，求 P(X < 110)。这里 σ = 5，所以 z = (110 − 100)/5 = 2，因此 P(Z < 2) ≈ 0.9772。
For interval probabilities, convert both endpoints: P(a < X < b) = P((a − μ)/σ < Z < (b − μ)/σ).
对于区间概率，需同时转换两个端点：P(a < X < b) = P((a − μ)/σ < Z < (b − μ)/σ)。
This endpoint conversion is essential because the standard normal table only applies to the Z distribution. You cannot look up probabilities for X ~ N(μ, σ²) directly unless your calculator uses the original parameters.
这种端点转换至关重要，因为标准正态分布表只适用于 Z 分布。除非计算器使用原始参数，否则不能直接查 X ~ N(μ, σ²) 的概率。
7. Calculating Unknown Variance from Probabilities | 由概率求未知方差
Many Edexcel questions give a probability and ask you to find σ or σ². Use the inverse normal function on the standard normal distribution to find z, then rearrange the standardising equation.
许多 Edexcel 题目会给出一个概率，要求求 σ 或 σ²。先利用标准正态分布的逆正态函数求出 z，然后重新整理标准化方程。
If P(X < k) = p, then z = Φ⁻¹(p) and z = (k − μ)/σ, so σ = (k − μ)/z. Finally square to get σ².
如果 P(X < k) = p，则 z = Φ⁻¹(p)，且 z = (k − μ)/σ，因此 σ = (k − μ)/z。最后平方得到 σ²。
σ = (k − μ) / z
Remember that σ must be positive, so discard any negative square root.
记住 σ 必须为正，因此舍去任何负的平方根。
If the probability is given as a right-tail such as P(X > k) = p, convert it to a left-tail first: P(X < k) = 1 − p. This is the most reliable way to use inverse normal functions on standard calculators.
如果给出的是右尾概率，如 P(X > k) = p，请先将其转换为左尾概率：P(X < k) = 1 − p。这是在标准计算器上使用逆正态函数最可靠的方法。
8. Sample Variance vs Population Variance | 样本方差与总体方差
In statistical modelling, the population variance σ² is a fixed parameter of the true normal distribution. In sampling, the sample variance s² estimates σ² from data.
在统计建模中，总体方差 σ² 是真实正态分布的一个固定参数。在抽样中，样本方差 s² 用于从数据中估计 σ²。
The formula for sample variance uses n − 1 in the denominator to give an unbiased estimate: s² = Σ(xᵢ − x̄)² / (n − 1).
样本方差公式使用 n − 1 作分母以得到无偏估计：s² = Σ(xᵢ − x̄)² / (n − 1)。
In normal distribution probability questions, you are usually given σ² directly or asked to find it; do not confuse this with sample variance unless the question explicitly says a sample was taken.
在正态分布概率题中，通常会直接给出 σ² 或要求求出 σ²；除非题目明确说明抽取了样本，否则不要将其与样本方差混淆。
Understanding the difference helps you choose the correct method: known population variance leads to z-procedures, while estimated sample variance can lead to t-procedures in further study.
理解这一区别有助于选择正确的方法：已知总体方差时使用 z 方法，而在进一步学习中，用估计的样本方差则可能使用 t 方法。
9. Common Mistakes | 常见错误
Using σ² in place of σ is the most
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