📚 Statistical Measures: Mean, Median, Mode and Range | 统计学指标：均值、中位数、众数与极差
In IGCSE Mathematics, statistical measures help us summarise a large set of data using a few key numbers. The three main averages are the mean, median and mode, while the range describes how spread out the data are.
在 IGCSE 数学中，统计学指标帮助我们用几个关键数值概括一组大量数据。三种主要的平均数是均值、中位数和众数，而极差则描述数据的离散程度。
1. Why Statistical Measures Matter | 为什么统计指标很重要
When you are given a raw list of 30 or 40 exam scores, it is difficult to see the overall picture. A single representative value, called an average, tells you the typical score; the range tells you whether the scores are clustered together or widely separated.
当你面对一份包含三四十个考试成绩的原始列表时，很难看清整体情况。一个具有代表性的数值（称为平均数）告诉你典型成绩是多少；极差则告诉你成绩是集中在一起还是分散得很开。
Different averages answer different questions. The mean is the arithmetic balance point, the median is the middle value after sorting, and the mode is the most common value. Each has strengths and weaknesses, so you must know which one to use in a given context.
不同的平均数回答不同的问题。均值是算数平衡点，中位数是排序后的中间值，众数是最常见的数值。每一种都有优点和缺点，因此你必须知道在给定情境中该选用哪一个。
2. The Mean | 均值
The mean is found by adding all data values together and dividing by the total number of values. It is the most widely used average and works best when there are no extreme outliers.
均值的计算方法是把所有数据值相加，再除以数值的总个数。它是最常用的平均数，当数据中没有极端异常值时最为适用。
Mean = (sum of all values) ÷ (number of values) = Σx ÷ n
For example, the scores of five students are 6, 8, 5, 9 and 7. The mean is calculated as follows.
例如，五名学生的成绩是 6、8、5、9 和 7。均值的计算过程如下。
Mean = (6 + 8 + 5 + 9 + 7) ÷ 5 = 35 ÷ 5 = 7
Always remember to divide by the correct total frequency. If a question provides a frequency table, multiply each value by its frequency before summing.
始终记住要除以正确的总频数。如果题目给出频数表，必须先将每个数值乘以其频数，然后再求和。
3. The Median | 中位数
The median is the middle value when all data are arranged in ascending order. It is not affected by extremely large or small values, which makes it a robust measure of centre.
中位数是将所有数据按升序排列后位于中间位置的数值。它不受极端大值或极端小值的影响，因此是一种稳健的中心度量。
If there are n values, the median is the value in position (n + 1) ÷ 2. For an odd count, this is a single value; for an even count, you take the average of the two middle values.
如果有 n 个数值，中位数是位于第 (n + 1) ÷ 2 位的数值。当 n 为奇数时，这是一个单独的值；当 n 为偶数时，取中间两个值的平均数。
Consider the sorted data 5, 6, 7, 8, 9. Since n = 5, the median is the 3rd value, which is 7. For the even data set 4, 7, 8, 8, 10, 12, 15, 21, the two middle values are 8 and 10, so the median is 9.
考虑排序数据 5、6、7、8、9。因为 n = 5，中位数是第 3 个数，即 7。对于偶数个数据的集合 4、7、8、8、10、12、15、21，中间两个值是 8 和 10，所以中位数是 9。
4. The Mode | 众数
The mode is the value that appears most frequently in a data set. Unlike the mean and median, the mode can be used with non-numerical data, such as favourite colours or transport types.
众数是数据集中出现频率最高的数值。与均值和中位数不同，众数可以用于非数值型数据，例如最喜欢的颜色或交通工具类型。
For the data set 4, 7, 8, 8, 10, 12, 15, 21, the mode is 8 because it occurs twice. A data set may have two modes (bimodal), as in 2, 3, 3, 5, 5, 8 where the modes are 3 and 5.
对于数据集合 4、7、8、8、10、12、15、21，众数是 8，因为它出现了两次。一个数据集可能有两个众数（双峰），例如 2、3、3、5、5、8 中众数是 3 和 5。
If every value appears exactly once, the data set has no mode. If every value appears the same number of times, statisticians say there is no single mode and simply report all the candidate values.
如果每个数值都恰好出现一次，则该数据集合没有众数。如果每个数值出现的次数相同，统计学家认为没有唯一众数，并如实报告所有候选值。
5. The Range | 极差
The range is the simplest measure of spread. It is calculated by subtracting the smallest value from the largest value in the data set.
极差是最简单的离散程度度量。它的计算方法是用数据集中的最大值减去最小值。
Range = largest value − smallest value
For the data set 4, 7, 8, 8, 10, 12, 15, 21, the range is 21 − 4 = 17. This tells us that the data span 17 units from the lowest to the highest score.
对于数据集合 4、7、8、8、10、12、15、21，极差为 21 − 4 = 17。这告诉我们数据从最低分到最高分跨越了 17 个单位。
The main weakness of the range is that it depends only on two values, so a single outlier can make the range misleadingly large. Despite this, it is quick to calculate and useful for a rough comparison.
极差的主要缺点在于它只依赖两个数值，因此一个异常值就可能使极差大得具有误导性。尽管如此，它计算快速，适合进行粗略比较。
6. Grouped Data: Estimating the Mean | 分组数据：估计均值
When data are grouped into intervals, the original individual values are lost. We estimate the mean by using the midpoint of each class interval as a representative value for that class.
当数据被分组为区间时，原始的个体数值就丢失了。我们通过使用每个组区间的中点作为该组的代表值来估计均值。
Estimated mean = Σ(fx) ÷ Σf
Here, x is the midpoint of each class and f is the frequency of that class. The following table shows the heights of 40 students.
其中 x 是每个组的中点，f 是该组的频数。下表显示了 40 名学生的身高。
| Height (cm) | Midpoint x | Frequency f | fx |
|---|---|---|---|
| 150–154 | 152 | 5 | 760 |
| 155–159 | 157 | 8 | 1256 |
| 160–164 | 162 | 12 | 1944 |
| 165–169 | 167 | 9 | 1503 |
| 170–174 | 172 | 6 | 1032 |
| Total | 40 | 6495 |
The estimated mean is therefore 6495 ÷ 40 = 162.375 cm. Because we used midpoints, this is an estimate, not an exact value.
因此估计均值为 6495 ÷ 40 = 162.375 cm。由于我们使用了中点，这是一个估计值，而不是精确值。
7. Modal Class and Median for Grouped Data | 分组数据的模态组与中位数
For grouped data, the modal class is the class interval with the highest frequency. In the height table above, the interval 160–164 has frequency 12, so it is the modal class.
对于分组数据，模态组是频数最高的组区间。在上面的身高表中，160–164 区间的频数为 12，因此它是模态组。
To find the median class, we first calculate cumulative frequencies. Then we locate the position n ÷ 2 and identify which class interval contains that position.
为了找到中位数所在组，我们首先计算累积频数。然后定位第 n ÷ 2 位，并判断该位置落在哪个组区间内。
For the height data, the cumulative frequencies are 5, 13, 25, 34, 40. Since n ÷ 2 = 20, the 20th value lies in the interval 160–164. This interval is called the median class.
对于身高数据，累积频数为 5、13、25、34、40。因为 n ÷ 2 = 20，第 20 个数值位于 160–164 区间内。该区间被称为中位数所在组。
Remember that the modal class and median class are intervals, not single numbers. State the full interval in your final answer.
请记住，模态组和中位数所在组是区间，而不是单个数值。在最终答案中必须写出完整的区间。
8. Choosing the Right Average | 选择合适的平均数
Each average suits a different situation. The mean is preferred when data are symmetric and contain no extreme values, because it uses every data point.
每种平均数适用于不同情况。当数据对称且没有极端值时，均值是首选，因为它使用了每一个数据点。
The median is better when there are outliers, such as house prices or salaries, because the median ignores extreme values. For example, one very high salary should not pull the typical salary upwards.
当存在异常值时，中位数更合适，例如房价或工资数据，因为中位数忽略极端值。例如，一个极高的工资不应该拉高典型工资的水平。
The mode is most useful for categorical data or for businesses that need to know the most popular product. For example, a shoe shop needs the modal shoe size to keep enough stock.
众数最适合类别数据，或需要了解最受欢迎产品的企业。例如，鞋店需要知道众数鞋码以保证充足的库存。
In exam questions, read the context carefully. Terms such as “average salary” or “typical age” hint that you should use the median; “most common” or “most popular” point to the mode.
在考试题目中，请仔细阅读情境。像”平均工资”或”典型年龄”这样的表述暗示你应使用中位数；”最常见”或”最受欢迎”则指向众数。
9. Cumulative Frequency and the Median | 累积频数与中位数
A cumulative frequency graph, or ogive, is a curve that shows the running total of frequencies. It is drawn by plotting cumulative frequency against the upper boundary of each class interval.
累积频数图（又称 ogive）是显示频数累计总数的曲线。绘制方法是以每个组区间的上边界为横坐标、累积频数为纵坐标进行描点连线。
Once the ogive is drawn, the median can be read from the graph. Draw a horizontal line from n ÷ 2 on the cumulative frequency axis to the curve, then drop a vertical line to the data axis.
绘制好累积频数图后，可以从图上读出中位数。从累积频数轴上的 n ÷ 2 处画水平线与曲线相交，再向数据轴画垂线。
For a frequency table, you can also locate the median position directly. Just remember to use cumulative frequencies and check whether n is odd or even.
对于频数表，你也可以直接定位中位数的位置。只需记住使用累积频数，并检查 n 是奇数还是偶数。
10. 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 value below which 25% of the data lie, and the upper quartile Q₃ is the value below which 75% of the data lie.
四分位数将有序数据集分成四个相等的部分。下四分位数 Q₁ 是 25% 数据位于其下的数值，上四分位数 Q₃ 是 75% 数据位于其下的数值。
The interquartile range (IQR) is the difference between Q₃ and Q₁. It measures the spread of the middle 50% of the data, so it is not affected by outliers.
四分位距（IQR）是 Q₃ 与 Q₁ 之差。它衡量中间 50% 数据的离散程度，因此不受异常值影响。
IQR = Q₃ − Q₁
For the data set 4, 7, 8, 8, 10, 12, 15, 21, the median is 9. The lower half is 4, 7, 8, 8, so Q₁ = (7 + 8) ÷ 2 = 7.5. The upper half is 10, 12, 15, 21, so Q₃ = (12 + 15) ÷ 2 = 13.5. Therefore IQR = 13.5 − 7.5 = 6.
对于数据集合 4、7、8、8、10、12、15、21，中位数是 9。下半部分为 4、7、8、8，所以 Q₁ = (7 + 8) ÷ 2 = 7.5。上半部分为 10、12、15、21，所以 Q₃ = (12 + 15) ÷ 2 = 13.5。因此 IQR = 13.5 − 7.5 = 6。
11. Box-and-Whisker Plots | 箱线图
A box-and-whisker plot is a visual display that shows the minimum, lower quartile, median, upper quartile and maximum of a data set. It is drawn as a box between Q₁ and Q₃ with a line at the median, and whiskers extending to the minimum and maximum.
箱线图是一种可视化展示，显示数据集的最小值、下四分位数、中位数、上四分位数和最大值。它表现为一个介于 Q₁ 和 Q₃ 之间的箱子，在中位数处画一条线，并由须线延伸至最小值和最大值。
Using the data above, the five-number summary is: min = 4, Q₁ = 7.5, median = 9, Q₃ = 13.5, max = 21. These five numbers fully describe the distribution’s centre and spread.
使用上述数据，五数概括为：最小值 = 4，Q₁ = 7.5，中位数 = 9，Q₃ = 13.5，最大值 = 21。这五个数值完整描述了分布的中心和离散程度。
In IGCSE questions, you may be asked to compare two box plots. Compare the medians for average level and compare the IQR or range for consistency. State clearly which group is higher and which is more consistent.
在 IGCSE 题目中，你可能会被要求比较两个箱线图。比较中位数以判断平均水平，比较 IQR 或极差以判断一致性。明确说明哪一组更高、哪一组更稳定。
12. Exam Tips and Common Mistakes | 考试技巧与常见错误
Always sort data before finding the median. A very common mistake is to identify the middle number in the original unsorted order, which gives a wrong answer.
在求中位数之前一定要先对数据排序。一个非常常见的错误是在未排序的原始顺序中直接找中间数，这样会得到错误答案。
Read the scale carefully when using a cumulative frequency graph. Misreading the axes is a frequent source of lost marks in statistics questions.
使用累积频数图时仔细阅读刻度。误读坐标轴是统计题中常见的失分原因。
When estimating the mean from grouped data, always use the class midpoint, not the lower or upper boundary. Check your arithmetic when computing Σ(fx).
从分组数据估计均值时，务必使用组中点，而不是下边界或上边界。计算 Σ(fx) 时要检查算术是否正确。
Finally, show all your working. Many IGCSE statistics questions award method marks, so even if your final answer is wrong, a correct calculation step can still earn credit.
最后，展示所有计算过程。许多 IGCSE 统计题按步骤给分，因此即使最终答案错误，正确的计算步骤仍然可以获得分数。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导Cancel reply