Measures of Central Tendency and Dispersion | IB 数学：描述性统计中的集中与离散趋势

📚 Measures of Central Tendency and Dispersion | IB 数学：描述性统计中的集中与离散趋势

Descriptive statistics help us turn a list of numbers into a compact, meaningful summary. In IB Mathematics, you are expected to choose appropriate measures of centre and spread, interpret them in context, and justify your choices when a data set is skewed or contains outliers.

描述性统计帮助我们化繁为简，用几个关键数字概括整组数据。在 IB 数学中，你需要选择合适的集中量数与离散量数，结合背景解释这些统计量，并在数据偏态或存在异常值时说明选择理由。

1. Two Families of Measures | 两类核心统计量

A summary statistic is only useful if it captures both the location and the variability of the data. The first family, measures of central tendency, describes the typical or central value. The second family, measures of dispersion, describes how spread out the observations are around that centre.

一个统计量若要有用，就必须同时刻画数据的“位置”与“波动”。第一类统计量称为集中趋势量数，它描述数据的一般水平；第二类称为离散趋势量数，它描述数据围绕中心值的分散程度。

The table below shows the most common measures you will meet in the IB syllabus.

下表列出了 IB 课程中最常见的集中趋势与离散趋势量数。

Central tendency	Mean, median, mode
Dispersion	Range, interquartile range, variance, standard deviation

2. Mean | 均值

The mean is the arithmetic average. For a raw data set with n values, the sample mean is found by adding all values and dividing by n.

均值是算术平均数。对于包含 n 个数据的原始数据组，样本均值等于所有数据之和除以 n。

x̄ = (Σxᵢ) / n

For a population, the mean is denoted by μ. The mean uses every data value, so it is the only measure of centre that must be affected by every observation. However, this also means it can be pulled towards an extreme outlier.

总体均值通常记为 μ。均值利用了每一个数据，因此它是唯一必然受到每个观测值影响的集中量数。但这也意味着它容易被极端值“拉向”异常值一侧。

When the same value appears with a frequency f, you multiply each value by its frequency before summing.

当同一数值出现频数为 f 时，应先将每个数值与其频数相乘再求和。

x̄ = Σfᵢxᵢ / Σfᵢ

3. Median | 中位数

The median is the middle value when the data are arranged in ascending order. It divides the ordered data set into two equal halves.

中位数是将数据按升序排列后位于正中间的数，它把有序数据组分成个数相等的两半。

If n is odd, the median is the value in position (n + 1) / 2. If n is even, the median is the mean of the two middle values.

当 n 为奇数时，中位数是第 (n + 1) / 2 个数据；当 n 为偶数时，中位数是中间两个数据的平均数。

odd n: median = x₍ₙ₊₁₎/₂ | even n: median = (xₙ/₂ + xₙ/₂₊₁) / 2

The median is resistant to extreme values. If a data set contains a huge outlier, the median may still describe the “typical” value much better than the mean.

中位数不受极端值影响。当数据中含有极大异常值时，用中位数描述“典型水平”通常要比均值更合适。

4. Mode | 众数

The mode is the value that occurs most frequently. A data set can have one mode, more than one mode, or no mode if all values appear with equal frequency.

众数是出现次数最多的数值。一组数据可以有一个众数、多个众数，也可能没有众数，即所有数值出现次数相同时。

For grouped data, the mode is given as the modal class, which is the class interval with the highest frequency. You do not use the formula for the mean here; instead you state the class interval itself.

对于分组数据，众数通常表述为“众数类”，即频数最高的组区间。此时不需要使用均值公式，而是直接给出该组区间。

Mode is suitable for categorical or discrete data.

众数适用于分类数据或离散数据。
It is often the weakest measure of central tendency because it ignores most of the data.

众数常被认为是最弱的集中量数，因为它忽略了大部分数据。

5. Range and Percentiles | 极差与百分位数

The range is the simplest measure of dispersion: it is the difference between the largest and smallest values.

极差是最简单的离散量数：它等于最大值与最小值之差。

Range = x_max − x_min

The range is very sensitive to outliers because it uses only two data values. One unusually large value can make the whole data set appear more variable than it really is.

极差只用两个数据，因此对异常值非常敏感。一个异常大的数值可能使整组数据看起来波动比实际情况更大。

A percentile is a measure of position. The k-th percentile is the value below which k% of the data lie. The median is the 50th percentile; the lower quartile Q₁ is the 25th percentile; the upper quartile Q₃ is the 75th percentile.

百分位数是一种位置量数。第 k 百分位数是指有 k% 的数据小于该数值。中位数就是第 50 百分位数；下四分位数 Q₁ 是第 25 百分位数；上四分位数 Q₃ 是第 75 百分位数。

On a cumulative frequency graph, you can read the 25th, 50th and 75th percentiles by locating the values corresponding to n/4, n/2 and 3n/4 on the vertical axis.

在累积频率图上，可以在纵轴分别找到 n/4、n/2 和 3n/4，再向横轴读取对应的数值，即可得到下四分位数、中位数和上四分位数。

6. Interquartile Range and Box Plots | 四分位距与箱线图

The interquartile range measures the spread of the middle 50% of the data by calculating the distance between the upper and lower quartiles.

四分位距通过上四分位数与下四分位数之差，衡量中间 50% 数据的离散程度。

IQR = Q₃ − Q₁

The IQR is more robust than the range because it ignores the lowest 25% and the highest 25% of the data. It is an excellent choice when the distribution is skewed or outliers are present.

由于四分位距忽略最低的 25% 和最高的 25% 数据，它比极差更稳健。当分布偏态明显或存在异常值时，四分位距是很好的选择。

A box-and-whisker diagram is drawn using five key numbers: minimum value, Q₁, median, Q₃, and maximum value. Values beyond 1.5 × IQR below Q₁ or above Q₃ are often flagged as outliers.

箱线图由五个关键数构成：最小值、Q₁、中位数、Q₃ 和最大值。小于 Q₁ − 1.5 × IQR 或大于 Q₃ + 1.5 × IQR 的数值通常会被判定为异常值。

Outlier boundaries: Q₁ − 1.5 × IQR and Q₃ + 1.5 × IQR

7. Variance and Standard Deviation | 方差与标准差

Variance and standard deviation are the most important measures of dispersion because they use every data value. They measure how far observations are from the mean.

方差与标准差是最重要的离散量数，因为它们使用到全部数据，度量观测值偏离均值的平均程度。

For a population with N values and mean μ, the variance is the mean of the squared deviations.

对于含有 N 个数据、均值为 μ 的总体，方差等于所有离差平方的平均数。

σ² = Σ(xᵢ − μ)² / N

For a sample of n values with sample mean x̄, IB Mathematics uses the sample variance with denominator n − 1 because it gives an unbiased estimate of the population variance.

对于样本量 n、样本均值为 x̄ 的样本，IB 数学使用分母 n − 1 计算样本方差，这样可以更无偏地估计总体方差。

s² = Σ(xᵢ − x̄)² / (n − 1)

The standard deviation is the square root of the variance. It is especially useful because it has the same units as the original data.

标准差等于方差的算术平方根。它最重要的优点是单位与原数据一致，更容易解释。

σ = √σ² and s = √s²

In your GDC, the symbol σx or xσn refers to the population standard deviation, while sx or xσn−1 refers to the sample standard deviation.

在图形计算器中，σx 或 xσn 表示总体标准差，而 sx 或 xσn−1 表示样本标准差，使用前要确认题目问的是总体还是样本。

8. Grouped Data | 分组数据

When raw data are organised into class intervals, the original values are lost. We estimate the mean by assuming each observation in a class is equal to the midpoint of that class.

当原始数据被整理成组区间后，原本的数据值已经无法逐一得知。我们通常假设每个组内的观测值都等于该组的中点值，从而估计均值。

midpoint xᵢ = (lower bound + upper bound) / 2

Then multiply each midpoint by its class frequency fᵢ, sum these products, and divide by the total frequency.

然后将每个组中点值与对应频数 fᵢ 相乘，求和后再除以总频数。

x̄ ≈ Σfᵢxᵢ / Σfᵢ

The sample standard deviation for grouped data follows the same idea: replace individual values by the class midpoints.

分组数据的样本标准差也使用相同思路，即将原始数值替换为各组中点值。

s² ≈ Σfᵢ(xᵢ − x̄)² / (Σfᵢ − 1)

Remember that these are estimates, not exact values. The more information you preserve, the more accurate the estimate will be.

请记住这些结果是估计值而非精确值。原始信息保留得越多，估计就越准确。

9. Effect of Linear Transformations | 线性变换对统计量的影响

If every value in a data set is transformed by the rule y = ax + b, the measures of location and spread change in predictable ways.

如果数据组中的每一个值都按规则 y = ax + b 进行线性变换，那么位置量数和离散量数都会按可预测的方式变化。

The mean, median, quartiles, and mode are shifted by the same linear transformation. The standard deviation and IQR are multiplied by |a|, while the variance is multiplied by a².

均值、中位数、四分位数和众数都经过相同的线性变换；标准差与四分位距则乘以 |a|，方差乘以 a²。

ȳ = ax̄ + b, s_y = |a| s_x, s_y² = a² s_x²

This idea is frequently tested in IB questions involving coded data. If you know the mean and standard deviation of coded data, you can recover the original mean and standard deviation by solving these equations.

IB 考试经常通过“编码数据”考查这一概念。若已知编码后数据的均值与标准差，就可以利用上述等式反推原始数据的均值与标准差。

10. Choosing the Correct Measures: Skewness | 选择合适的统计量：偏态

For a roughly symmetric distribution, the mean together with the standard deviation is the most informative pair. For a skewed distribution, the median and the interquartile range are usually preferable.

当分布大致对称时，“均值 + 标准差”是最具信息量的组合；当分布偏态明显时，通常应优先选择“中位数 + 四分位距”。

Mean is pulled towards the long tail, so in a positively skewed distribution mean > median.

均值会偏向长尾一侧，因此右偏分布中通常有均值 > 中位数。
In a negatively skewed distribution mean < median.

在左偏分布中通常有均值 < 中位数。
The standard deviation is also affected by outliers, so the IQR is more robust when extreme values exist.

标准差同样受异常值影响，因此当存在极端值时，四分位距更加稳健。
Mode is best for categorical data, where calculating a mean is not meaningful.

对于分类数据，计算均值没有意义，众数是最合适的集中量数。

If the mean is greater than the median, the distribution is positively skewed. If the mean is less than the median, the distribution is negatively skewed. If the mean and median are nearly equal, the distribution is approximately symmetric.

若均值大于中位数，说明分布右偏；若均值小于中位数，说明分布左偏；若两者几乎相等，则分布大致对称。

11. Exam Strategies | 考试实战策略

In the IB exam, descriptive statistics questions often require both GDC use and written reasoning. A correct answer may still lose marks if you do not communicate the right notation.

在 IB 考试中，描述性统计题通常既需要计算器操作，也需要书面推理。即使答案正确，若统计符号或计算步骤表达不清，仍可能被扣分。

State whether you are using a population or a sample before selecting the standard deviation symbol.

先判断题目涉及总体还是样本，再选择相应的标准差符号。
When entering grouped data, input midpoints into one list and frequencies into another; do not enter each raw value separately.

输入分组数据时，应把组中值存入一个列表，把频数存入另一个列表，不要逐一输入原始数据。
Always include units in your final answer, for example “the standard deviation is 3.4 kg”.

最终答案要写出单位，例如“标准差为 3.4 kg”。
Compare measures in context: “Because the mean is greater than the median, the data are positively skewed, so the median and IQR are more appropriate.”

结合背景进行比较：“因为均值大于中位数，说明数据右偏，所以中位数和四分位距更合适。”
When drawing a box plot, clearly show the positions of the median and quartiles on a labelled scale.

绘制箱线图时，应在标注了刻度的坐标轴上清晰标明中位数和四分位数的位置。

Mastering these ideas will help you move from simply pressing calculator buttons to genuinely understanding what the numbers say about a data set.

掌握这些概念后，你不仅能熟练操作计算器，更能真正理解这些数字背后的数据含义。

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Measures of Central Tendency and Dispersion | IB 数学：描述性统计中的集中与离散趋势