Comparing Data: Averages, Range and Statistical Diagrams | 数据比较：平均数、极差与统计图

📚 Comparing Data: Averages, Range and Statistical Diagrams | 数据比较：平均数、极差与统计图

In KS3 Cambridge Mathematics, handling data is about much more than drawing a chart. You need to summarise a set of numbers, choose the most useful average, measure how spread out the data are, and justify your conclusions clearly. This article revises the key ideas of averages, range, frequency tables, bar charts, pie charts and scatter graphs, with worked examples and exam-style tips.

在KS3剑桥数学中，数据处理远不止画一张统计图。你需要总结一组数据，选择最有用的平均数，衡量数据的分散程度，并清楚地说明你的结论。本文复习平均数、极差、频数表、条形图、饼图和散点图的核心概念，并配有详细例题和考试技巧。

1. Mean, Median, Mode and Range | 平均数、中位数、众数和极差

The mean is found by adding all the values and dividing by the number of values. The median is the middle value when the data are written in order. The mode is the value that appears most often. The range is the difference between the largest value and the smallest value.

平均数是将所有数值相加后除以数值的个数。中位数是将数据按顺序排列后位于中间位置的数。众数是出现次数最多的数。极差是最大值与最小值之间的差。

Mean = (Sum of values) ÷ (Number of values)

For the set 4, 7, 7, 9, 13, the mean is 8, the median is 7, the mode is 7, and the range is 9. Each measure tells you something different about the data.

对于数据集 4、7、7、9、13，平均数是 8，中位数是 7，众数是 7，极差是 9。每个统计量都能从不同角度描述数据。

Measure	Value for 4, 7, 7, 9, 13	What it tells you
Mean	8	The balanced centre of the data
Median	7	The middle value when ordered
Mode	7	The most frequent value
Range	9	The spread of the data

Remember that the range is always a single number, never a pair of numbers. You calculate it as largest minus smallest, not as ’13 to 4′.

请记住，极差始终是一个数字，而不是一对数字。计算方法是用最大值减去最小值，而不是写成 13 到 4。

---

2. Calculating the Mean from a Frequency Table | 从频数表计算平均数

When data are grouped in a frequency table, you cannot simply add the values in the table and divide by the number of rows. You must multiply each value by its frequency to find the total sum of all data.

当数据用频数表整理后，你不能只把表中的数值相加再除以行数。你必须将每个数值乘以它出现的频数，才能得到所有数据的总和。

Suppose a survey asks students how many books they read in a week. The results are: value 2 appears 3 times, value 3 appears 5 times, value 4 appears 2 times and value 5 appears 6 times. To find the mean, first calculate total frequency = 3 + 5 + 2 + 6 = 16. Then calculate total sum = 2 × 3 + 3 × 5 + 4 × 2 + 5 × 6 = 6 + 15 + 8 + 30 = 59.

假设一项调查询问学生一周阅读了多少本书。结果是：数值 2 出现 3 次，数值 3 出现 5 次，数值 4 出现 2 次，数值 5 出现 6 次。要求平均数，先计算总频数 = 3 + 5 + 2 + 6 = 16。再计算总和 = 2 × 3 + 3 × 5 + 4 × 2 + 5 × 6 = 6 + 15 + 8 + 30 = 59。

Mean = 59 ÷ 16 = 3.6875 ≈ 3.69

Always check that your mean is between the smallest and largest data values. Here 3.69 lies between 2 and 5, which is a useful check.

始终检查你的平均数是否在最小值和最大值之间。这里 3.69 介于 2 和 5 之间，这是一个有用的检查方法。

---

3. Using the Range to Compare Spread | 用极差比较离散程度

The range gives a quick measure of how spread out the values are. A small range means the data are tightly packed together. A large range means the data are widely spread.

极差可以快速衡量数据的分散程度。极差小意味着数据紧密地聚集在一起。极差大意味着数据分布范围很广。

For example, class A scores 55, 56, 58, 60, 62 on a test. Class B scores 40, 52, 61, 75, 88 on the same test. Class A has range 62 − 55 = 7, while class B has range 88 − 40 = 48. This shows class A is much more consistent, but the range alone does not tell you which class performed better.

例如，A班的测试成绩为 55、56、58、60、62。B班在同一测试中的成绩为 40、52、61、75、88。A班的极差是 62 − 55 = 7，而B班的极差是 88 − 40 = 48。这说明A班的成绩稳定得多，但仅凭极差无法判断哪个班表现更好。

In an exam answer, always quote the range values and explain what they mean in context. Do not just write ‘the range is bigger’ without saying why that matters.

在考试回答中，一定要写出极差的具体数值，并说明它们在题目情境中的含义。不要只写极差更大而不解释为什么重要。

---

4. Choosing the Best Average | 选择最合适的平均数

Sometimes one average is more useful than the others. The mode is best for non-numerical data, such as favourite colour or type of transport. The median is best when there are extreme values or outliers, because it is not affected by them. The mean is best when every value matters and the data are fairly symmetrical.

有时某个平均数比其他平均数更有用。众数最适合非数值数据，比如最喜欢的颜色或交通方式。中位数最适合存在极端值或异常值的场合，因为它不受其影响。平均数最适合每个数值都很重要且数据较为对称的情况。

Imagine the weekly pocket money of five students: £4, £5, £5, £6 and £100. The mean is 120 ÷ 5 = £24, which is not representative of most students because one very high value pulls it up. The median is £5, which is more typical.

假设五名学生每周的零花钱为：4英镑、5英镑、5英镑、6英镑和100英镑。平均数是 120 ÷ 5 = 24英镑，这不能代表大多数学生，因为一个非常高的数值拉高了平均数。中位数是 5英镑，这更典型。

In this situation, the median is the better average to use because it is robust against outliers. Examiners often ask you to justify this choice.

在这种情况下，中位数是更好的平均数，因为它对异常值具有抗干扰性。考官经常要求你说明选择它的理由。

---

5. Bar Charts and Dual Bar Charts | 条形图与双条形图

A bar chart is used to show frequencies of categories. The bars must have equal width and there must be gaps between them. A dual bar chart compares two related sets of data side by side, such as boys and girls or two year groups.

条形图用于显示各类别的频数。条形必须等宽，条形之间必须有间隔。双条形图将两组相关的数据并排比较，例如男生和女生，或两个年级。

When drawing a bar chart, always label both axes, give the chart a title, use a sensible scale, and leave equal gaps between bars. If you are asked to read a bar chart, check the frequency axis carefully rather than estimating from the height of a bar alone.

画条形图时，务必标注两个坐标轴，给图表加标题，使用合理的刻度，并在各条形之间留出相等的间隔。如果要求你读取条形图，请仔细查看频数轴，而不要仅凭条形的高度进行估计。

• Equal width bars | 条形等宽
• Even gaps between bars | 条形之间间隔均匀
• Clear axis labels and title | 清晰的坐标轴标签和标题
• Accurate scale | 精确的刻度

Dual bar charts are particularly useful for making visual comparisons, but you must use a key to show what each bar represents.

双条形图特别适合进行直观比较，但你必须使用图例来说明每个条形代表什么。

---

6. Pie Charts and Proportions | 饼图与比例

A pie chart shows how a total is divided into parts. The size of each sector is proportional to the frequency of that category. To find the angle for a sector, multiply the fraction of the total by 360°.

饼图显示整体如何被分成各个部分。每个扇形的大小与该类别的频数成正比。要计算扇形的角度，用该类别占总数的比例乘以 360°。

Angle = (Frequency ÷ Total frequency) × 360°

For example, 20 students choose their favourite colour. If 8 students choose red, the red sector has angle 8 ÷ 20 × 360° = 144°. If 5 students choose blue, the blue sector has angle 5 ÷ 20 × 360° = 90°.

例如，20 名学生选择他们最喜欢的颜色。如果 8 名学生选择红色，那么红色扇形的角度是 8 ÷ 20 × 360° = 144°。如果 5 名学生选择蓝色，那么蓝色扇形的角度是 5 ÷ 20 × 360° = 90°。

Colour	Frequency	Angle
Red	8	144°
Blue	5	90°
Green	4	72°
Other	3	54°

Always check that the angles add up to 360°. This is a quick way to find arithmetic mistakes.

始终检查所有角度之和是否为 360°。这是发现计算错误的快速方法。

---

7. Scatter Graphs and Correlation | 散点图与相关性

A scatter graph shows the relationship between two sets of numerical data. Each point represents one item with two measurements, such as height and arm span, or hours of revision and test score.

散点图显示两组数值数据之间的关系。每个点代表一个具有两个测量值的项目，例如身高和臂展，或复习时间与测试成绩。

Correlation describes the trend in the points. Positive correlation means both quantities increase together. Negative correlation means one quantity increases while the other decreases. No correlation means there is no clear pattern.

相关性描述点的总体趋势。正相关意味着两个量一起增加。负相关意味着一个量增加而另一个量减少。无相关意味着没有明显的模式。

If the points mostly follow a straight trend, you can draw a line of best fit. This line should have roughly the same number of points above and below it and should pass through the middle of the data. You can then use the line to make predictions.

如果点大致沿直线分布，你可以画出最佳拟合线。这条线上下两边的点数应大致相等，并且穿过数据的中心区域。然后你可以利用这条线进行预测。

Be careful: correlation does not always mean causation. Two quantities may show a strong correlation simply because of coincidence or a third factor.

注意：相关性并不总是意味着因果关系。两个量可能只是由于巧合或第三个因素而表现出很强的相关性。

---

8. Drawing Conclusions from Data | 从数据得出结论

A strong conclusion compares a measure of centre, such as the median or mean, and a measure of spread, such as the range. It also refers back to the context of the question rather than making a vague statement.

一个有说服力的结论需要比较中心位置指标（如中位数或平均数）和离散程度指标（如极差），并且要回到题目情境，而不是笼统地叙述。

For example, two runners record their training times over ten days. Runner A has median time 24.5 seconds and range 3 seconds. Runner B has median time 25.8 seconds and range 9 seconds. You could write: Runner A is generally faster because her median time is lower, and she is more consistent because her range is smaller.

例如，两名跑步运动员记录了十天的训练时间。A运动员

Published by TutorHao | KS3 Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）