📚 Measuring the Spread of Data | 衡量数据的离散程度
When analysing a data set, knowing the central tendency (mean, median, mode) is not enough. The spread, or dispersion, tells us how much the data values vary from each other and from the centre. Two data sets can have the same mean but very different spreads. Measuring spread is essential in statistics to understand consistency, risk, and variability.
在分析数据集时，仅仅知道集中趋势（均值、中位数、众数）是不够的。离散程度告诉我们数据值彼此之间以及与中心值的差异有多大。两个数据集可能有相同的均值，但离散程度却截然不同。衡量离散程度对于理解一致性、风险和变异性至关重要。
1. Range | 极差
pread. It is the difference between the maximum and minimum values. Written as Range = max – ve to outliers and does not give information about the distribution of the remaining data.
极差是最简单的离散度量，即最大值与最小值的差：极差 = 最大值 – 最小值。由于它只用到两个数据点，对异常值极为敏感，并且无法反映其余数据的分布情况。
2. Interquartile Range (IQR) | 四分位距
The interquartile range focuses on the middle 50% It is the data. It is the difference between the third quartile (Q3) and the first quartile (Q1): IQR = Q3 – use it ignores the extremes. It is often used alongside the median as a measure of spread.
四分位距关注数据的中间50%。它是第三四分位数(Q3)与第一四分位数(Q1)之差：IQR = Q3 – Q1。四分位数将有序数据集分成四等份，中位数即Q2。由于忽略极值，IQR不受异常值影响，常与中位数一起使用作为离散度的度量。
3. Box Plots and Spread | 箱线图与离散度
A box plot (box-and-whisker plot) displays the five-number summary: minimum, Q1, median, Q3, and maximum. The length of the box represents the IQR, giving a visual impression of spread. The whiskers extend to the smallest and largest values within 1.5 IQR from the quartiles. Points beyond the whiskers are potential outliers. Box plots are excellent for comparing spreads across multiple groups.
箱线图（盒须图）展示五数概括：最小值、Q1、中位数、Q3和最大值。箱体的长度代表IQR，直观呈现离散度。胡须延伸至距四分位数1.5倍IQR范围内的最小和最大值，胡须之外的点为潜在异常值。箱线图非常适合比较多组数据的离散程度。
4. Variance and Standard Deviation | 方差与标准差
Variance measures the average squared deviation of each data point from the mean. The standard deviation is the square root of the variance and has the same units as the original data. For a population, variance is denoted by σ² and standard deviation by σ. For a sample, we use s² and s. Standard deviation is the most common measure of spread for roughly symmetric distributions, and it is key to many statistical analyses.
方差衡量每个数据点与均值之差的平方的平均值。标准差是方差的平方根，单位与原始数据相同。对于总体，方差记为σ²，标准差为σ；对于样本，用s²和s。标准差是近似对称分布最常用的离散度度量，也是许多统计分析的核心。
5. Population vs Sample Formulas | 总体与样本公式
The formulas differ in the denominator. Population variance uses N, the size of the entire population. Sample variance uses n – 1 to give an unbiased estimate of the population parameter. This adjustment is called Bessel's correction.
公式的分母不同。总体方差使用N，即整个总体的容量。样本方差使用n – 1，以给出总体参数的无偏估计，这种调整称为贝塞尔校正。
σ² = Σ(xᵢ – μ)² / N and s² = Σ(xᵢ – x̄)² / (n – 1)
In IB exams, you will be expected to know which formula to apply based on whether the data represents a population or a sample. Always check the context: population data gives σ and σ²; sample data gives s and s². Many GDC outputs label both, so you must pick the correct one.
在IB考试中，需要根据数据代表总体还是样本选择正确的公式。务必检查上下文：总体数据使用σ和σ²；样本数据使用s和s²。许多图形计算器会同时输出两者，你必须选择正确的结果。
6. Calculating Standard Deviation by Hand | 手动计算标准差
To calculate the sample standard deviation by hand, follow these steps:
1. Find the mean of the data set, x̄.
2. Subtract the mean from each data value to obtain deviations (xᵢ – x̄).
3. Square each deviation.
4. Sum all the squared deviations.
5. Divide by (n – 1) to get the sample variance s².
6. Take the square root to find the sample standard deviation s.
手动计算样本标准差的步骤如下：
1. 计算数据集的均值 x̄。
2. 每个数据值减去均值，得到离差 (xᵢ – x̄)。
3. 将每个离差平方。
4. 将所有平方离差相加。
5. 除以 (n – 1) 得到样本方差 s²。
6. 取平方根求得样本标准差 s。
Consider the data set {5, 7, 8, 9, 11}. The mean x̄ 7+8+9+11)/5 = 8. The table below shows the deviation and squared deviation for each value.
考虑数据集 {5, 7, 8, 9, 11}。均值 x̄ = (5+7+8+9+11)/5 = 8。下表展示了每个值的离差和平方离差。
| xᵢ | xᵢ – x̄ | (xᵢ – x̄)² |
|---|---|---|
| 5 | -3 | 9 |
| 7 | -1 | 1 |
| 8 | 0 | 0 |
| 9 | 1 | 1 |
| 11 | 3 | 9 |
sum of squared deviations is 9+1+0+1+9 = 20. With n = 5, the sample variance is s² = 20/(5–1) = 5. The sample standard deviation is s = √5 ≈ 2.24. This value of 2.24 represents the typical distance of a data point from the mean.
平方离差之和为 9+1+0+1+9 = 20。n = 5，样本方差 s² = 20/(5–1) = 5。样本标准差 s = √5 ≈ 2.24。这个值 2.24 表示数据点与均值的典型距离。
7. Using Technology (GDC) | 使用图形计算器
In IB Mathematics, you are permitted to use a graphic display calculator (GDC) to compute summary statistics. Enter the data into a list and run the 1-Variable Statistics function. The GDC will output the mean (x̄), the sum of squares, the sample standard deviation (often labelled sx or Sx), and the population standard deviation (σx). Always double-check which standard deviation you need for the problem. On the TI-Nspire, use Stat Calculations; on Casio, use the STAT mode.
在IB数学中，你可以使用图形计算器(GDC)来计算汇总统计量。将数据输入列表，运行单变量统计功能。GDC会输出均值(x̄)、平方和、样本标准差（通常标记为sx或Sx）以及总体标准差(σx)。务必根据题目要求确认应使用哪一个标准差。在TI-Nspire上使用统计计算功能，在Casio上使用STAT模式。
8. Comparing Data Sets Using Spread | 利用离散度比较数据集
When two data sets have similar means, the one with a smaller standard deviation is more consistent. If the means differ greatly, comparing standard deviations directly may be misleading. In such cases, consider the spreads relative to their means. Box plots allow immediate visual comparison of IQRs and medians. In IB exam questions, you may be asked to identify which group shows greater variability or to justify your choice using appropriate statistics.
当两个数据集的均值相近时，标准差较小的那一个更加稳定。如果均值差异很大，直接比较标准差可能会产生误导，这时应考虑离散度与均值的相对关系。箱线图可以直观地比较IQR和中位数。在IB考题中，你可能会被要求判断哪一组数据的变异性更大，并用恰当的统计量进行论证。
9. Outliers and Spread | 异常值与离散度
abnormal distance from other values. A common rule using the IQR states: a value is a potential outlier if it is less than Q1 – Outliers can heavily inflate the range and the standard deviation, making the data appear more spread out than it truly is. When outliers are present, the IQR and median are more resistant measures. Identifying outliers helps you decide whether to use resistant or non-resistant measures of spread.
异常值是指与其他数值距离异常远的数据点。常用的IQR规则为：若某数值小于 Q1 – 1.5×IQR 或大于 Q3 + 1.5×IQR，则被视为潜在异常值。异常值会大幅扩大极差和标准差，使数据看起来比实际更分散。存在异常值时，IQR和中位数是更具抗性的度量。识别异常值有助于决定应该使用抗性还是非抗性的离散度度量。
10. Coefficient of Variation (Optional Extension) | 变异系数（可选拓展）
f variation (CV) expresses the standard deviation as a percentage of the mean: CV = (s / x̄) × 100% hile not always required in IB, it can deepen your understanding of how spread relates to the centre
Published by TutorHao | IB Mathematics Revision Series | aleveler.com
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