📚 Investigation 3: Standard Deviation | 探究三：标准差
Standard deviation is one of the most fundamental concepts in statistics, providing a measure of how spread out a set of data is around its mean. In IB Mathematics, understanding standard deviation is essential for analysing data distributions, comparing datasets, and interpreting real-world phenomena. This investigation explores the definition, calculation, properties, and applications of standard deviation through a series of guided explorations.
标准差是统计学中最基本的概念之一，用于衡量一组数据围绕其均值的分散程度。在IB数学中，理解标准差对于分析数据分布、比较数据集以及解释现实世界现象至关重要。本次探究通过一系列引导性探索，深入探讨标准差的定义、计算、性质及其应用。
1. Understanding Variability in Data | 理解数据的变异性
Whenever we collect data, values are rarely identical. Consider the heights of students in a classroom: they range from shorter to taller, and simply knowing the average height does not capture how varied the heights are. Measures of variability—such as range, interquartile range, and standard deviation—help us quantify this spread. Standard deviation is particularly useful because it takes every data point into account and is expressed in the same units as the original data.
当我们收集数据时，数值很少完全相同。以教室里的学生身高为例：身高有高有矮，仅仅知道平均身高并不能反映身高的差异程度。变异性度量——例如极差、四分位距和标准差——帮助我们量化这种分散程度。标准差特别有用，因为它考虑了每个数据点，并且以与原始数据相同的单位表示。
, but the values in B are much more spread out. Standard deviation quantifies this difference: for A, the standard deviation is 0; for B, it is approximately 2.58. The larger the standard deviation, the greater the dispersion.
考虑两个数据集：A = {4, 4, 4, 4} 和 B = {1, 3, 5, 7}。两者的均值都是4，但B中的数值更加分散。标准差量化了这种差异：对于A，标准差为0；对于B，标准差约为2.58。标准差越大，离散程度越大。
2. Defining Standard Deviation | 标准差的定义
Standard deviation is defined as the square root of the variance. Variance measures the average squared deviation of each data point from the mean. By taking the square root, we return to the original units, making interpretation more intuitive. For a population, the standard deviation is denoted by the Greek letter σ (sigma).
标准差的定义为方差的平方根。方差衡量每个数据点与均值的偏差平方的平均值。通过取平方根，我们回到了原始单位，使解释更加直观。对于总体，标准差用希腊字母 σ（西格玛）表示。
The formula for population standard deviation is:
总体标准差公式为：
σ = √( Σ(xᵢ - μ)² / N )
Where μ is the population mean, xᵢ represents each data value, and N is the number of data points.
其中 μ 为总体均值，xᵢ 代表每个数据值，N 为数据点的个数。
3. Population vs Sample Standard Deviation | 总体标准差与样本标准差
In practice, we often work with samples rather than entire populations. When calculating the standard deviation of a sample, we use a slightly different formula to obtain an unbiased estimate of the population standard deviation. The divisor becomes n - 1 instead of n, where n is the sample size. This correction, known as Bessel's correction, accounts for the fact that a sample tends to underestimate the true variability.
在实践中，我们通常处理的是样本而非整个总体。在计算样本的标准差时，我们使用一个略有不同的公式，以获得对总体标准差的无偏估计。此时除数变为 n - 1 而非 n，其中 n 为样本大小。这一修正被称为贝塞尔校正，它解释了样本倾向于低估真实变异性的情况。
The sample standard deviation, denoted by s, is given by:
样本标准差记作 s，公式为：
s = √( Σ(xᵢ - x̄)² / (n - 1) )
Understanding which formula to use is crucial in IB investigations, especially when conducting statistical tests or creating confidence intervals.
在IB探究中，理解使用哪个公式至关重要，特别是在进行统计检验或创建置信区间时。
4. Step-by-Step Calculation of Standard Deviation | 标准差的计算步骤
Let us compute the sample standard deviation for the dataset: 5, 7, 8, 10, 12. Follow these steps:
让我们计算数据集 5, 7, 8, 10, 12 的样本标准差。按照以下步骤：
- Step 1: Calculate the mean x̄ = (5+7+8+10+12)/5 = 42/5 = 8.4.
- 步骤1：计算均值 x̄ = (5+7+8+10+12)/5 = 42/5 = 8.4。
- Step 2: Find the deviations from the mean: -3.4, -1.4, -0.4, 1.6, 3.6.
- 步骤2：求各值与均值的偏差：-3.4, -1.4, -0.4, 1.6, 3.6。
- Step 3: Square each deviation: 11.56, 1.96, 0.16, 2.56, 12.96.
- 步骤3：将各偏差平方：11.56, 1.96, 0.16, 2.56, 12.96。
- Step 4: Sum the squared deviations: 11.56+1.96+0.16+2.56+12.96 = 29.2.
- 步骤4：求平方和：11.56+1.96+0.16+2.56+12.96 = 29.2。
- Step 5: Divide by n-1 (since sample): 29.2 / 4 = 7.3.
- 步骤5：除以 n-1（因为是样本）：29.2 / 4 = 7.3。
- Step 6: Take the square root: √7.3 ≈ 2.70.
- 步骤6：取平方根：√7.3 ≈ 2.70。
Thus, the sample standard deviation is approximately 2.70. This process can be efficiently performed using a calculator or spreadsheet, but understanding the manual method deepens conceptual understanding.
因此，样本标准差大约为2.70。这一过程可以通过计算器或电子表格高效完成，但理解手动计算方法可以加深概念理解。
5. Properties of Standard Deviation | 标准差的性质
Standard deviation has several important properties that are helpful in data analysis:
标准差具有几个重要的性质，在数据分析中很有帮助：
- It is always non-negative. A standard deviation of zero indicates no variation; all data values are identical.
- 始终为非负值。标准差为零表示没有变异；所有数据值都相同。
- It is sensitive to outliers, as extreme values increase the squared deviations significantly.
- 对异常值敏感，因为极端值会显著增加偏差平方。
- Adding a constant to all data values does not change the standard deviation (shift invariance).
- 给所有数据值加上一个常数不会改变标准差（位移不变性）。
- Multiplying all data values by a constant multiplies the standard deviation by the absolute value of that constant (scaling property).
- 将所有数据值乘以一个常数，会使标准差乘以该常数的绝对值（缩放性质）。
These properties are particularly useful when standardising data or performing transformations, and they form the basis for many statistical techniques.
这些性质在标准化数据或进行变换时特别有用，它们构成了许多统计技术的基础。
6. Standard Deviation and the Normal Distribution | 标准差与正态分布
In a normal distribution, approximately 68% of the data lies within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. This empirical rule allows us to make quick estimates about data spread and identify unusual observations.
在正态分布中，大约68%的数据位于均值的一个标准差范围内，95%位于两个标准差内，99.7%位于三个标准差内。这一经验法则使我们能够快速估计数据分布并识别异常观测值。
For example, if the heights of adult males are normally distributed with a mean of 175 cm and a standard deviation of 7 cm, we can predict that about 95% of heights fall between 161 cm and 189 cm. This concept is foundational in inferential statistics and hypothesis testing.
例如，如果成年男性身高呈正态分布，均值为175 cm，标准差为7 cm，我们可以预测大约95%的身高位于161 cm到189 cm之间。这一概念是推论统计和假设检验的基础。
7. Comparing Data Sets: The Coefficient of Variation | 比较数据集：变异系数
ratio of the standard deviation to the mean, often expressed as a percentage: CV = (s / x̄) × 100%.
在比较具有不同均值或单位的数据集的变异性时，仅靠标准差可能会产生误导。变异系数（CV）是一个无量纲度量，定义为标准差与均值的比率，通常以百分比表示：CV = (s / x̄) × 100%。
For instance, suppose we are comparing the consistency of two machines: Machine X produces bolts with mean length 50 mm and standard deviation 2 mm; Machine Y produces nails with mean length 20 mm and standard deviation 1.5 mm. The CV for X is 4%, while for Y it is 7.5%. Thus, Machine X is more consistent relative to its mean, despite having a larger absolute standard deviation.
例如，假设我们要
Published by TutorHao | IB Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图