IB Mathematics: Descriptive Statistics Key Points | IB数学：描述性统计要点梳理

📚 IB Mathematics: Descriptive Statistics Key Points | IB数学：描述性统计要点梳理

Before you begin any statistical investigation, you must know which type of data you are working with. Descriptive statistics helps us summarise, organise and present data in a clear way, so that meaningful conclusions can be drawn without examining every single observation.

在开始任何统计调查之前，必须先分清所处理的数据类型。描述性统计帮助我们以清晰的方式对数据进行概括、整理和呈现，从而无需逐一检查每一个观测值就能得出有意义的结论。

1. Types of Data | 数据类型

Data can be classified by its nature. Categorical data is also called qualitative data and consists of labels or names such as favourite colour or country of birth.

数据可以按照其性质进行分类。分类数据又称定性数据，由标签或名称组成，例如最喜欢的颜色或出生国家。

Numerical data is quantitative and can be split into continuous and discrete data. Discrete data takes separate, countable values, while continuous data can take any value within a given interval.

数值数据是定量数据，可进一步分为连续型数据和离散型数据。离散型数据取的是可数的独立数值，而连续型数据可在给定区间内取任意值。

Continuous data: height, time, temperature.

连续型数据：身高、时间、温度。
Discrete data: number of children, shoe size, number of cars.

离散型数据：孩子数量、鞋码、汽车数量。

In IB Mathematics, you also need to know the difference between univariate and bivariate data. Descriptive statistics usually focuses on univariate data, where only one variable is measured.

在 IB 数学中，还需要知道单变量数据与双变量数据之间的区别。描述性统计通常关注单变量数据，即只测量一个变量。

2. Measures of Central Tendency | 集中趋势的度量

Central tendency describes the typical value of a data set. The three most common measures are the mean, median and mode.

集中趋势描述一组数据的典型取值。最常用的三个指标是均值、中位数和众数。

The mean is the sum of all observations divided by the number of observations. It is written as x̄, which is read as x-bar.

均值是所有观测值之和除以观测值的个数，记作 x̄，读作 x 杠。

x̄ = Σx / n

The median is the middle value when data is ordered in ascending or descending order. If there are n values, the median is the (n + 1)/2-th value for an odd n, or the average of the two middle values for an even n.

中位数是将数据按升序或降序排列后的中间值。若 n 为奇数，中位数是第 (n + 1)/2 个值；若 n 为偶数，则取中间两个值的平均数。

The mode is the value that occurs most frequently. A data set may have no mode, one mode, or more than one mode.

众数是出现次数最多的值。一组数据可能没有众数、只有一个众数，也可能有多个众数。

3. Measures of Dispersion | 离散程度的度量

Measures of dispersion describe how spread out the data is. Range is the simplest measure: it is the difference between the largest and smallest values.

离散程度的度量描述数据的散布范围。极差是最简单的度量，它是最大值与最小值之差。

The interquartile range, or IQR, is the difference between the upper quartile Q₃ and the lower quartile Q₁. It is a robust measure because it is not affected by extreme values.

四分位距，即 IQR，是上四分位数 Q₃ 与下四分位数 Q₁ 之差。它是一个稳健度量，不受极端值影响。

IQR = Q₃ – Q₁

When comparing two data sets, always compare both centre and spread. A data set with the same mean but a larger range or IQR has greater variability.

比较两组数据时，必须同时比较中心与散布。如果两组数据均值相同，但其中一组的极差或四分位距更大，说明该组数据的变异性更大。

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

Variance measures the average squared deviation from the mean. The standard deviation is the square root of the variance, and it has the same unit as the original data.

方差衡量各观测值相对于均值的平均平方偏差。标准差是方差的平方根，其单位与原始数据一致。

For a sample, the variance is denoted by s² and is calculated by dividing the sum of squared deviations by n – 1. This is called Bessel’s correction and is used because n – 1 gives an unbiased estimate of the population variance.

对于样本，方差记为 s²，计算方法是将平方偏差之和除以 n – 1。这称为贝塞尔修正，因为使用 n – 1 可以得到总体方差的无偏估计。

s² = Σ(x – x̄)² / (n – 1)

The sample standard deviation is the positive square root of this value: s = √(s²). If you are working with an entire population, use N and the symbol σ.

样本标准差是该值的正平方根：s = √(s²)。如果处理的是整个总体，则应使用 N 和符号 σ。

σ = √(Σ(x – μ)² / N)

On your GDC, the button labelled s is the sample standard deviation, while the button labelled σ is the population standard deviation. Choosing the wrong one is a common IB exam error.

在图形计算器（GDC）上，标有 s 的按键是样本标准差，标有 σ 的按键是总体标准差。选错按键是 IB 考试中非常常见的错误。

5. Quartiles and Interquartile Range | 四分位数与四分位距

Quartiles divide an ordered data set into four equal parts. The lower quartile Q₁ is the median of the lower half of the data, and the upper quartile Q₃ is the median of the upper half.

四分位数将有序数据划分为四个等量部分。下四分位数 Q₁ 是数据下半部分的中位数，上四分位数 Q₃ 是数据上半部分的中位数。

The median Q₂ splits the data set in half. To find quartiles, first put the data in order, then locate the position using the method your exam board expects. In IB, the GDC is often used for this calculation.

中位数 Q₂ 将数据集一分为二。求四分位数时，首先要将数据排序，然后按考试局要求的方法确定位置。在 IB 考试中，通常可使用 GDC 完成计算。

Lower quartile Q₁: 25% of the data lie below this value.

下四分位数 Q₁：25% 的数据小于该值。
Median Q₂: 50% of the data lie below this value.

中位数 Q₂：50% 的数据小于该值。
Upper quartile Q₃: 75% of the data lie below this value.

上四分位数 Q₃：75% 的数据小于该值。

The IQR is often used alongside the median to describe typical data. Since the IQR ignores the lowest 25% and highest 25% of values, it remains stable even when outliers are present.

四分位距常和中位数一起使用来描述典型数据。由于四分位距忽略了最低的 25% 和最高的 25% 的值，即使存在异常值时它仍然比较稳定。

6. Box-and-Whisker Plots | 箱线图

A box-and-whisker plot is a visual summary of the minimum, Q₁, median, Q₃ and maximum. It is also called a boxplot.

箱线图是对最小值、Q₁、中位数、Q₃ 和最大值的图形化概括，也简称箱形图。

A box is drawn from Q₁ to Q₃, with a vertical line at the median. Whiskers extend from the box to the smallest and largest non-outlier values.

箱体从 Q₁ 绘制到 Q₃，并在中位数处画一条竖线。须线从箱体延伸到最小和最大的非异常值。

Min → Q₁ → Median → Q₃ → Max

Boxplots are powerful for comparing two or more data sets. By drawing them on the same scale, you can compare the typical value, spread and skewness at a glance.

箱线图在比较两组或多组数据时非常有力。将它们绘制在同一尺度上，可以一眼比较典型值、散布程度和偏态。

Boxplots can also show outliers as separate points beyond the whiskers. A common rule in IB is that a value is an outlier if it is less than Q₁ – 1.5 × IQR or greater than Q₃ + 1.5 × IQR.

箱线图还可以将异常值显示为须线之外的点。IB 中常用规则是：如果某值小于 Q₁ – 1.5 × IQR 或大于 Q₃ + 1.5 × IQR，则该值为异常值。

7. Frequency Tables and Grouped Data | 频数表与分组数据

When data contains many repeated values, it is convenient to present it in a frequency table. The mean can then be calculated using the formula with frequencies f.

当数据包含大量重复值时，用频数表呈现更为方便。此时均值可通过包含频数 f 的公式计算。

For ungrouped frequency data, the total sum is Σfx and the total frequency is Σf. Therefore the mean is:

对于未分组的频数数据，总和为 Σfx，总频数为 Σf。因此均值为：

x̄ = Σfx / Σf

For grouped data, each class interval is represented by its midpoint x, also called the class mark. The midpoint is calculated as (lower bound + upper bound) ÷ 2.

对于分组数据，每个组区间以其组中值 x 代表，组中值按（下界 + 上界）÷ 2 计算。

Because the original individual values are lost when data is grouped, the mean and standard deviation calculated from midpoints are approximations.

由于分组后原始个体值会丢失，因此根据组中值计算的均值和标准差只是近似值。

8. Cumulative Frequency Graphs | 累积频数图

A cumulative frequency graph is plotted with the upper boundary of each class on the horizontal axis and the cumulative frequency on the vertical axis.

累积频数图以各组上界为横轴、累积频数为纵轴绘制。

You can use a cumulative frequency graph to estimate the median, quartiles and percentiles. The median is the value corresponding to a cumulative frequency of n/2. Q₁ corresponds to n/4, and Q₃ corresponds to 3n/4.

可以利用累积频数图估计中位数、四分位数和百分位数。中位数对应累积频数为 n/2 处的值；Q₁ 对应 n/4 处；Q₃ 对应 3n/4 处。

Median: cumulative frequency = n/2

Q₁: cumulative frequency = n/4

Q₃: cumulative frequency = 3n/4

When drawing the graph, remember to start the cumulative frequency at zero on the x-axis. Use a smooth curve through the cumulative points rather than a jagged line.

画图时，请记住在横轴上从零开始累积频数。应使用平滑曲线穿过累积点，而不是折线。

9. Mean and Standard Deviation for Grouped Data | 分组数据的均值与标准差

For grouped data, the standard formula for the sample variance is adapted so that each midpoint contributes with its frequency.

对于分组数据，样本方差的标准公式会用频数对每个组中值进行加权。

The sample variance for grouped data is:

分组数据的样本方差为：

s² = Σf(x – x̄)² / (Σf – 1)

Alternatively, a computational formula can be used to avoid subtracting the mean from every midpoint.

也可以使用计算型公式，避免将每个组中值都减去均值。

s² = [Σfx² – (Σfx)² / Σf] / (Σf – 1)

This second formula is especially useful for written solutions because it is easier to input into a non-programmable calculator step by step.

第二个公式尤其适合笔答过程，因为它可以逐步输入到普通计算器中。

10. Outliers and Skewness | 异常值与偏态

An outlier is an observation that lies unusually far from the rest of the data. Outliers can strongly inflate the mean and standard deviation, but they have little effect on the median and IQR.

异常值是指距离数据主体异常远的观测值。异常值会严重拉高均值和标准差，但对中位数和四分位距影响很小。

Using the boxplot rule, a value is a potential outlier if it lies beyond 1.5 × IQR from Q₁ or Q₃.

根据箱线图规则，若某值偏离 Q₁ 或 Q₃ 超过 1.5 × IQR，则为潜在异常值。

Outlier below: x < Q₁ - 1.5 × IQR

Outlier above: x > Q₃ + 1.5 × IQR

Skewness describes the symmetry of a distribution. If the mean is greater than the median, the distribution is positively skewed, with a long tail to the right.

偏态描述分布的对称性。如果均值大于中位数，则分布呈正偏态，右尾较长。

If the mean is less than the median, the distribution is negatively skewed, with a long tail to the left. When mean and median are approximately equal, the distribution is symmetric.

如果均值小于中位数，则分布呈负偏态，左尾较长。当均值和中位数大致相等时，分布接近对称。

11. Comparing Distributions | 比较分布

When comparing two distributions, you should always describe both central tendency and dispersion. Use units and compare values directly.

比较两个分布时，应同时描述集中趋势和离散程度。请使用单位，并直接比较数值。

Aspect	What to compare
Centre	Mean or median; state which is higher
Spread	Range, IQR or standard deviation; state which is more consistent
Shape	Symmetry, skewness, presence of outliers

For example, if one class has a higher median but also a larger IQR, it means that class tends to score higher on average but has less consistent performance.

例如，如果一个班级中位数更高，但四分位距也更大，说明该班级平均成绩更好，但表现不够稳定。

Always mention units in your comparison, such as marks or measurements, and refer to the context of the question.

在比较中始终要提及单位，比如分数或测量值，并联系题目所给的情境进行说明。

12. Common Errors and Exam Tips | 常见错误与考试技巧

A common mistake is confusing sample and population standard deviation. Read the question carefully: if the data is from a sample, use n – 1; if it is the whole population, use N.

常见错误是混淆样本标准差与总体标准差。请仔细审题：如果数据来自样本，使用 n – 1；如果是整个总体，则使用 N。

Another common error is finding the median before sorting the data. You must order the data first. For grouped data, remember to use midpoints, not the original boundaries.

另一个常见错误是排序前就求中位数。必须先对数据排序。对于分组数据，请使用组中值，而不是原来的边界值。

Use your GDC wisely: become familiar with entering raw data, frequency tables and grouped data lists.

善用图形计算器：熟练输入原始数据、频数表和分组数据列表。
Write down formulas: show enough working to earn method marks, even when using a GDC.

写出公式：即使使用图形计算器，也要写出足够步骤以获得方法分。
Interprete in context: after calculating a statistic, state what it means in the context of the question.

结合情境解释：计算出统计量后，请说明它在题目情境中的含义。

Finally, check that your answers have sensible units and that the mean lies between the minimum and maximum values. This simple sanity check can help you catch careless errors.

最后，检查答案是否有合理的单位，并确认均值应处在最小值和最大值之间。这个简单的合理性检查可以帮助你发现粗心错误。

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IB Mathematics: Descriptive Statistics Key Points | IB数学：描述性统计要点梳理