📚 Investigation 2 - Standard Deviation | 探究2：标准差
Standard deviation is one of the most important measures of spread in statistics. In this investigation, you will explore how it is calculated, how it behaves when data sets change, and why it matters when making inferences from samples. Whether you are working with raw data or grouped frequencies, a solid understanding of standard deviation will help you describe variability reliably.
标准差是统计学中最重要的离散程度度量之一。在本探究中，你将探讨它的计算方法、当数据集发生变化时它的表现，以及为什么在从样本推断总体时它如此重要。无论你处理的是原始数据还是分组频率数据，扎实理解标准差都能帮助你可靠地描述变异性。
1. Introduction to Standard Deviation | 标准差入门
Standard deviation quantifies how far individual data points tend to be from the mean. A small standard deviation means most values cluster closely around the average, while a large standard deviation indicates the data are widely spread.
标准差量化了各个数据点远离均值的典型距离。标准差小意味着大多数数值紧密聚集在平均值附近，而标准差大则表明数据分布非常分散。
The symbol σ (sigma) is used for population standard deviation, and the letter s is used for sample standard deviation. Both are always non‑negative; a standard deviation of zero occurs only when all values are identical.
总体标准差的符号是 σ（西格玛），样本标准差的符号是 s。两者都非负；只有当所有数值完全相同时，标准差才为零。
2. Calculating Standard Deviation by Hand | 手动计算标准差
For a population of N values, the steps are: find the mean μ, compute each deviation (x − μ), square the deviations, average them, and finally take the square root.
对于容量为 N 的总体，计算步骤是：求均值 μ，计算每个离差 (x − μ)，将离差平方，取平均值，最后开平方根。
σ = √[ Σ (xᵢ − μ)² / N ]
For a sample with n values, you replace μ by the sample mean x̄ and divide by (n − 1) instead of n. We will explore the reason behind (n − 1) in a later section.
对于样本容量为 n 的样本，用样本均值 x̄ 代替 μ，并且除以 (n − 1) 而不是 n。我们将在后一个小节探讨除以 (n − 1) 的原因。
s = √[ Σ (xᵢ − x̄)² / (n − 1) ]
3. Population vs Sample Standard Deviation | 总体标准差与样本标准差
The population standard deviation σ describes every member of a defined group. In the IB course, you may be given the whole population or asked to treat a data set as a population. In that case, use N in the denominator.
总体标准差 σ 描述的是某个确定群体中的每个成员。在 IB 课程中，你可能会被给予整个总体，或被要求把一个数据集当作总体处理。此时分母使用 N。
The sample standard deviation s is used when you only have a subset of the population and wish to estimate the variability in the whole population. Dividing by (n − 1) gives an unbiased estimate of σ², which is why s is often called the “estimated standard deviation”.
当你只拥有总体的一个子集并希望估计总体的变异性时，就要使用样本标准差 s。除以 (n − 1) 可以得到 σ² 的无偏估计，因此 s 常被称为“估计标准差”。
Your calculator typically offers both σ and s. For IB investigations, always check whether your data represent a population or a sample before choosing the appropriate measure.
你的计算器通常同时提供 σ 和 s。在 IB 探究中，选择适当的度量前，一定要先确认你的数据代表的是总体还是样本。
4. Why Divide by n−1? Bessel's Correction | 为何除以 n−1？贝塞尔校正
Using n in the denominator when estimating from a sample tends to underestimate the true population variance. This happens because the sample mean x̄ is the value that minimises the sum of squared deviations from that sample, making the spread appear smaller than it really is.
当从样本估计时，分母使用 n 往往会低估真实的总体方差。这是因为样本均值 x̄ 是使该样本离差平方和最小的值，使得离散程度看起来比实际的小。
By dividing by (n − 1), we inflate the variance slightly to correct this bias. This adjustment is called Bessel’s correction. As n grows larger, the difference between n and n − 1 becomes negligible, so for large samples the two versions almost coincide.
通过除以 (n − 1)，我们将方差略微放大以校正这种偏差。这种调整称为贝塞尔校正。当 n 越来越大时，n 和 n − 1 的差异变得可以忽略，因此对于大样本，两个版本几乎重合。
This correction is only needed for variance and standard deviation calculated from a sample. If you have the whole population, always divide by N.
这种校正仅对通过样本计算的方差和标准差是必要的。如果你拥有整个总体，始终除以 N。
5. Standard Deviation for Grouped Data | 分组数据的标准差
When data are presented in a frequency table, use the midpoint of each class interval as the representative value. Multiply each squared deviation by the class frequency, sum these weighted values, then divide by the total frequency.
当数据以频数表呈现时，使用每个组区间的组中值作为代表值。将每个平方离差乘以组频数，求这些加权值的和，再除以总频数。
σ = √[ Σ f (m − μ)² / Σ f ]
For a sample, replace μ by the estimated mean x̄ and the denominator Σ f by (Σ f − 1). Remember that grouped data give an approximation; the exact standard deviation can only be computed from the original raw values.
对于样本，用估计均值 x̄ 代替 μ，并将分母 Σ f 替换为 (Σ f − 1)。请记住，分组数据只能给出近似值；精确的标准差只能通过原始数值计算。
6. Properties of Standard Deviation | 标准差的性质
Standard deviation is always measured in the same units as the original data. If your data are in metres, the standard deviation is also in metres. This makes it easier to interpret than the variance, which is in squared units.
标准差的单位始终与原始数据的单位相同。如果你的数据以米为单位，标准差也以米为单位。这使它比方差更易于解释，因为方差的单位是平方单位。
Standard deviation is sensitive to outliers. A single extreme value can increase the standard deviation substantially, so it is often wise to report the interquartile range alongside it.
标准差对异常值敏感。一个极端值就能使标准差大幅增加，因此通常明智的做法是同时报告四分位距。
When all data are shifted by a constant, the standard deviation remains unchanged. However, multiplying every data point by a constant multiplies the standard deviation by the absolute value of that constant.
当所有数据都加上一个常数时，标准差保持不变。但是，如果将每个数据点乘以一个常数，标准差将乘以该常数的绝对值。
7. Effect of Linear Transformations | 线性变换的影响
aX + b applied to every data point. The mean becomes aμ + b, but the variability only depends on a.
考虑对每个数据点施加的线性变换 Y = aX + b。均值变为 aμ + b，但变异性仅取决于 a。
σᵧ = |a| σₓ
This property is useful when converting units, such as from kilometres to metres. Multiplying by 1000 multiplies the standard deviation by 1000, while adding a constant offset does not affect the spread at all.
这个性质在转换单位时非常有用，例如从千米转换为米。乘以 1000 会使标准差乘以 1000，而加上一个常数偏移量则完全不会影响离散度。
8. Standard Deviation and Outliers | 标准差与异常值
Although the interquartile range is often preferred for identifying outliers, standard deviation can also offer a rule of thumb. In an approximately symmetric, bell‑shaped distribution, data points more than two or three standard deviations from the mean are considered unusual.
虽然四分位距更常用于识别异常值，但标准差也可以提供一种经验法则。在一个近似对称的钟形分布中，距离均值超过两三个标准差的数据点被视为不寻常。
In a small sample, one outlier can dramatically inflate the standard deviation. Therefore, always examine a box plot or a dot plot alongside the numerical summary to understand the shape of the data.
在小样本中，一个异常值就可能极大地夸大标准差。因此，在查看数值摘要时，一定要同时检查箱形图或点图，以了解数据的形状。
9. Using Standard Deviation with the Normal Distribution | 标准差与正态分布
key model in the IB syllabus, relies entirely on the mean and standard deviation. About 68 % of data lie within ±1σ of the mean, roughly 95 % within ±2σ, and about 99.7 % within ±3σ.
正态分布是 IB 课程大纲中的一个关键模型，完全依赖于均值和标准差。大约 68% 的数据落在均值 ±1σ 范围内，大约 95% 落在 ±2σ 内，约 99.7% 落在 ±3σ 内。
This empirical rule can only be applied if the distribution is genuinely bell‑shaped. When it is, the standard deviation provides a powerful tool for calculating probabilities and setting confidence intervals.
这一经验法则只有在分布确实呈钟形时才能应用。当条件满足时，标准差便成为计算概率和设定置信区间的有力工具。
10. Investigation: Comparing Data Sets | 探究：比较数据集
In your investigation, you will likely be asked to compare two or more data sets using both their means and their standard deviations. A higher mean indicates a typical value larger on average; a higher standard deviation indicates greater inconsistency or spread.
在你的探究中，你很可能需要同时使用均值和标准差来比较两个或多个数据集。较大的均值表明典型值平均更大；较大的标准差则表明更大的不一致性或离散程度。
For example, if two IB classes sit the same test, Class A might have a mean of 72 with a standard deviation of 8, while Class B has a mean of 72 with a standard deviation of 15. Although their average performance is the same, Class B’s scores are far more varied, suggesting a wider range of abilities or preparation levels.
例如，如果两个 IB 班级参加相同的测试，A 班均值为 72、标准差为 8，而 B 班均值为 72、标准差为 15。尽管他们的平均表现相同，但 B 班的分数差异大得多，这表明能力或准备水平的范围更广。
Always support your comparison with numbers and context. Saying “the standard deviation is larger” is not enough; explain what that larger spread means in the real‑world situation.
在比较时始终要用数字和背景信息支撑。只说“标准差更大”是不够的，要解释这种更大的离散度在实际情况中意味着什么。
11. Common Mistakes and Misconceptions | 常见错误与误解
One frequent mistake is confusing the formulas for σ and s. When you are given a data set without any mention of it being a sample, IB questions often expect you to treat it as a population. Read the wording carefully.
一个常见错误是混淆 σ 和 s 的公式。当题目给出的数据集未提及是样本时，IB 题目通常希望你将其视为总体。请仔细阅读题目措辞。
Another misconception is assuming that a small standard deviation always means the data are “good” or “accurate”. A small spread could simply imply low variability, which may be desirable or not depending on the context.
另一个误解是认为小的标准差总意味着数据“好”或“精确”。小的离散度可能仅仅意味着变异性低，它是否理想取决于具体情境。
Students also sometimes forget that standard deviation carries units. Omitting units or treating standard deviation as unit‑free can cost marks in an investigation write‑up.
学生有时还会忘记标准差带有单位。在探究报告中省略单位或将标准差当作无量纲量可能导致失分。
12. Summary and Key Formulas | 总结与关键公式
Standard deviation is the square root of the average squared distance from the mean. Its calculation differs slightly between a population (σ) and a sample (s). For grouped data, midpoints and frequencies are used.
标准差是离差平方的平均值的平方根。在总体 (σ) 和样本 (s) 中，它的计算略有不同。对于分组数据，使用组中值和频数。
| Measure | Formula | Denominator |
|---|---|---|
| Population σ | √[ Σ(x−μ)² / N ] | N |
| Sample s | √[ Σ(x−x̄)² / (n−1) ] | n−1 |
| Grouped σ | √[ Σ f(m−μ)² / Σ f ] | Σ f |
Understanding the properties of standard deviation — its units, its response to transformations, and its link to the normal distribution — will equip you to write a thorough and well‑reasoned investigation.
理解标准差的性质——它的单位、对变换的响应以及与正态分布的联系——将使你能够写出一篇详尽且推理充分的探究报告。
Published by TutorHao | IB Mathematics Revision Series | aleveler.com
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