Handling Data 6 | 数据处理 6

📚 Handling Data 6 | 数据处理 6

Welcome to Handling Data 6, the final instalment in our IGCSE Edexcel Mathematics data-handling series. In this lesson, we consolidate the key concepts of scatter graphs, correlation, lines of best fit, time series, and cumulative frequency – with a special focus on interpreting data and avoiding common exam pitfalls.

欢迎来到数据处理 6，这是我们 IGCSE Edexcel 数学数据处理系列的最后一讲。在本课中，我们将系统梳理散点图、相关性、最佳拟合线、时间序列和累积频率图的核心概念，并特别关注数据解读和常见考试陷阱。

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

A scatter graph is a plot of paired data points on a coordinate grid, used to investigate whether two variables are related. Each point represents one observation with its x- and y-coordinates.

散点图是将成对的数据点绘制在坐标网格上，用来研究两个变量之间是否存在关系。每个点代表一个观测值，具有对应的 x 坐标和 y 坐标。

Correlation describes the strength and direction of a linear relationship. Positive correlation means that as one variable increases, the other tends to increase; negative correlation means they tend to move in opposite directions. If there is no apparent pattern, we say there is no correlation.

相关性描述的是线性关系的强度和方向。正相关意味着当一个变量增大时，另一个变量也趋于增大；负相关则意味着它们趋于反向变化。如果数据没有明显规律，我们就称之为无相关。

Types of correlation include:

Strong positive (r close to +1) – points cluster tightly along an upward-sloping line.
Strong negative (r close to −1) – points cluster tightly along a downward-sloping line.
Weak positive / weak negative (r near 0) – points are loosely scattered but show a slight tendency.
No correlation (r ≈ 0) – points form a random cloud.

相关类型包括：

强正相关（r 接近 +1） – 数据点紧密聚集在上升直线附近。
强负相关（r 接近 −1） – 数据点紧密聚集在下降直线附近。
弱正相关 / 弱负相关（r 接近 0） – 数据点较分散，但有轻微趋势。
无相关（r ≈ 0） – 数据点呈现随机分布。

Pearson’s product-moment correlation coefficient: r = ∑(x − x̄)(y − ȳ) / √[∑(x − x̄)² ∑(y − ȳ)²]

You are not required to calculate r by hand in the IGCSE exam – calculators or given values are used – but you must be able to estimate its value from a scatter diagram and to interpret it in context.

在 IGCSE 考试中，你不需要手动计算 r 的值——通常使用计算器或题目给出的数值——但你必须能够从散点图估计 r 的大小，并结合实际问题解释其含义。

2. The Line of Best Fit | 最佳拟合线

A line of best fit is a straight line drawn through a scatter graph so that the points are as close as possible to the line, with roughly the same number of points above and below it. The line does not have to pass through any particular data point, but it should reflect the general trend.

最佳拟合线是一条穿过散点图的直线，使得所有数据点尽可能靠近这条线，并且线上下两侧的点数大致相等。这条线不必经过任何特定的数据点，但应当反映数据的总体趋势。

To draw a line of best fit by eye: first locate the mean point (x̄, ȳ), which is the centre of mass of the data. Then draw a straight line through this point, balancing the number of points above and below the line.

徒手绘制最佳拟合线的方法是：先确定均值点 (x̄, ȳ)，即数据的中心位置。然后过该点画一条直线，使直线上方的点数和下方的点数大致平衡。

Once the line is drawn, it can be used to make predictions. To interpolate within the data range, read off the y-value for a given x-value (or vice versa). To extrapolate beyond the data range, extend the line carefully – but remember that extrapolation is less reliable because the trend may not continue.

画出最佳拟合线后，可以用它进行预测。若要在数据范围内进行内插，只需为给定的 x 值读出对应的 y 值（或反之）。若要在数据范围之外进行外推，则需要小心延长这条线——但要注意，外推的可靠性较低，因为数据趋势可能不会继续延伸。

3. Interpolation vs Extrapolation | 内插与外推

Interpolation is the process of estimating a value that lies within the known range of data. Because we are using the line within the region where we have actual observations, interpolation is generally reliable.

内插是指估计位于已知数据范围之内的值。因为我们在有实际观测数据的区域内使用拟合线，所以内插通常是可靠的。

Extrapolation is the process of estimating a value that lies outside the known data range. This relies on the assumption that the trend continues beyond the observed data, which is often risky. For example, if you are predicting the height of a child at age 20 from data collected ages 2–12, the line may suggest a value that is biologically unrealistic.

外推是指估计位于已知数据范围之外的值。这依赖于“趋势在观测数据之外仍然继续”的假设，而这往往是有风险的。例如，用 2–12 岁的身高数据预测一个孩子 20 岁时的身高，拟合线可能给出一个生物学上不合理的数值。

Key exam point: Always state that extrapolation assumes the trend continues and may be unreliable. This simple sentence earns marks in examiners’ reports year after year.

考试要点：始终说明外推的前提是“趋势继续延伸”，并且可能不可靠。这句话在考官报告中年复一年地帮学生得分。

4. Time Series and Moving Averages | 时间序列与移动平均

A time series is a set of data points recorded at successive times, usually at equal intervals. Examples include monthly temperatures, annual sales figures, or daily rainfall totals. A time series graph plots the data against time on the horizontal axis.

时间序列是一组按时间顺序记录的数据点，时间间隔通常相等。例如每月气温、年度销售额或每日降雨量。时间序列图以水平轴表示时间，将数据点依次绘出。

The main purpose of analysing a time series is to identify a trend – the long-term direction in which the data are moving. One common method to smooth out short-term fluctuations is to calculate moving averages.

分析时间序列的主要目的是识别趋势——即数据长期移动的方向。平滑短期波动的一个常用方法是计算移动平均。

How to calculate a moving average:

For an odd-order moving average (e.g., 3-point), the average of the first, second and third data points is placed against the middle time (the second time).
For an even-order moving average (e.g., 4-point), the average of points 1–4 is placed between time 2 and time 3; then a second stage of averaging (centring) is needed.

如何计算移动平均：

对于奇数阶移动平均（例如 3 点移动平均），将第 1、2、3 个数据点的平均值放在中间时间（即第 2 个时间点）。
对于偶数阶移动平均（例如 4 点移动平均），将第 1–4 个点的平均值放在时间 2 和时间 3 之间；然后需要进行第二次平均（居中化）。

Example of a 3-point moving average: For data 4, 6, 5, 8, 9, the first moving average is (4+6+5) ÷ 3 = 5, placed at time 2; the second is (6+5+8) ÷ 3 = 6.33, placed at time 3; the third is (5+8+9) ÷ 3 = 7.33, placed at time 4.

3 点移动平均示例：对于数据 4, 6, 5, 8, 9，第一个移动平均值为 (4+6+5) ÷ 3 = 5，放在时间 2；第二个为 (6+5+8) ÷ 3 = 6.33，放在时间 3；第三个为 (5+8+9) ÷ 3 = 7.33，放在时间 4。

Centring a 4-point moving average:

Step 1: Calculate the 4-point moving totals (T₁ = d₁+d₂+d₃+d₄, T₂ = d₂+d₃+d₄+d₅, etc.).
Step 2: Calculate the moving averages M₁ = T₁ ÷ 4, M₂ = T₂ ÷ 4, etc.
Step 3: Centre each pair: C = (M₁ + M₂) ÷ 2, placed at time 3.

4 点移动平均的居中化：

步骤 1：计算 4 点移动总和（T₁ = d₁+d₂+d₃+d₄，T₂ = d₂+d₃+d₄+d₅，等等）。
步骤 2：计算移动平均值 M₁ = T₁ ÷ 4，M₂ = T₂ ÷ 4，等等。
步骤 3：对每一对求居中值：C = (M₁ + M₂) ÷ 2，放在时间 3。

5. Cumulative Frequency Tables | 累积频数表

A cumulative frequency is the running total of frequencies up to a given value. The cumulative frequency table shows, for each class boundary, the total number of data points less than or equal to that boundary.

累积频数是指截止到某个值为止的所有频数之和。累积频数表显示的是，在每个组界处，小于或等于该组界的数据点总数。

Example: the heights of 30 students are grouped as follows:

Height (cm)	Frequency	Cumulative Frequency
120 < x ≤ 130	4	4
130 < x ≤ 140	8	12
140 < x ≤ 150	10	22
150 < x ≤ 160	5	27
160 < x ≤ 170	3	30

To plot a cumulative frequency graph, plot the upper class boundary on the x-axis against the cumulative frequency on the y-axis. Join the points with a smooth curve (or straight-line segments as instructed).

绘制累积频率图时，在 x 轴上标出各组的上界，在 y 轴上标出累积频数。用平滑曲线（或按题目要求用直连线段）连接各点。

6. Median, Quartiles and Interquartile Range | 中位数、四分位数与四分位距

From a cumulative frequency graph you can read the median and quartiles directly:

Median (Q₂): the value at the 50th percentile, i.e. at cumulative frequency n/2.
Lower quartile (Q₁): the value at the 25th percentile, i.e. at cumulative frequency n/4.
Upper quartile (Q₃): the value at the 75th percentile, i.e. at cumulative frequency 3n/4.
Interquartile range (IQR): IQR = Q₃ − Q₁, a measure of the spread of the middle 50% of the data.

从累积频率图中可以直接读出中位数和四分位数：

中位数（Q₂）：第 50 百分位数，即累积频数 n/2 处对应的值。
下四分位数（Q₁）：第 25 百分位数，即累积频数 n/4 处对应的值。
上四分位数（Q₃）：第 75 百分位数，即累积频数 3n/4 处对应的值。
四分位距（IQR）：IQR = Q₃ − Q₁，用于衡量中部 50% 数据的离散程度。

For the example above with n = 30: the median is read at 15, Q₁ at 7.5, and Q₃ at 22.5 on the vertical axis. Drawing horizontal lines from these values to the curve and then reading down to the horizontal axis gives the required values.

对于上述 n = 30 的例子：中位数在纵轴 15 处读取，Q₁ 在 7.5 处读取，Q₃ 在 22.5 处读取。从这些值画水平线与曲线相交，再垂直向下读横轴，即可得到所需数值。

7. Box-and-Whisker Plots | 箱线图

A box-and-whisker plot (box plot) is a visual display that summarises the five-number summary: minimum, Q₁, median (Q₂), Q₃, and maximum.

箱线图是一种可视化展示，用于概括五数概括：最小值、Q₁、中位数（Q₂）、Q₃ 和最大值。

To construct a box plot:

Draw a horizontal (or vertical) scale covering the range of the data.
Draw a box extending from Q₁ to Q₃.
Draw a vertical line inside the box at the median.
Draw whiskers from the box to the minimum and maximum values.

构造箱线图的步骤：

绘制一个覆盖数据范围的水平（或垂直）数轴。
画一个从 Q₁ 延伸到 Q₃ 的矩形盒子。
在盒子内部的中位数处画一条竖线。
从盒子两端画触须延伸到最小值和最大值。

Box plots are excellent for comparing two or more data sets side by side, as they immediately show differences in median, spread, and skewness. A longer whisker on one side indicates skewness in that direction.

箱线图非常适合并排比较两个或多个数据集，因为它能直观显示中位数、离散程度和偏态的差异。某一侧较长的触须表示数据在该方向上存在偏态。

8. Skewness | 偏态

Skewness describes the asymmetry of a distribution. In a symmetric distribution, the mean, median and mode are all equal. When the data are not symmetric, we say the distribution is skewed.

偏态描述的是分布的不对称性。在对称分布中，均值、中位数和众数都相等。当数据不对称时，我们称之为偏态分布。

For a distribution skewed positively (right-skewed), the tail extends towards the higher values. In this case, the mean is typically greater than the median: mean > median > mode.

对于正偏态（右偏）分布，尾部向较高值方向延伸。此时，均值通常大于中位数：均值 > 中位数 > 众数。

For a distribution skewed negatively (left-skewed), the tail extends towards the lower values, and the mean is typically less than the median: mean < median < mode.

对于负偏态（左偏）分布，尾部向较低值方向延伸，均值通常小于中位数：均值 < 中位数 < 众数。

Exam tip: Use the relationship between the mean and median to decide skewness quickly. If mean > median, positive skew; if mean < median, negative skew.

考试技巧：利用均值和中位数的大小关系快速判断偏态。若均值 > 中位数，则为正偏；若均值 < 中位数，则为负偏。

9. Choosing the Correct Average | 选择合适的平均数

The three main averages – mean, median and mode – each have strengths and weaknesses. The choice of which to use depends on the data set and the purpose of the analysis.

三种主要平均数——均值、中位数和众数——各有优缺点。选择使用哪一种取决于数据集本身和分析目的。

Mean: uses all data values, ideal for further calculation; but it is heavily influenced by outliers.
Median: unaffected by outliers, good for skewed distributions; but it uses only the middle value(s).
Mode: useful for categorical data or for identifying the most common value; but it may not exist or may not be unique.

均值：使用所有数据值，便于进一步计算；但受极端值（离群值）影响很大。
中位数：不受极端值影响，适合偏态分布；但仅使用中间的一个或两个值。
众数：适用于分类数据或确定最常见值；但可能不存在或可能不唯一。

In an exam, you may be asked which average is most appropriate for a given data set. Justify your answer by referring to the presence or absence of outliers and the skewness of the data.

在考试中，你可能会被问到哪个平均数最适合给定的数据集。请通过分析是否存在极端值以及数据的偏态来证明你的答案。

10. Probability and Relative Frequency | 概率与相对频率

Probability is a measure of the likelihood of an event occurring, expressed as a number between 0 and 1. The theoretical probability of an event A is P(A) = number of favourable outcomes ÷ total number of equally likely outcomes.

概率是衡量某个事件发生可能性的指标，用 0 到 1 之间的数字表示。事件 A 的理论概率是 P(A) = 有利结果数 ÷ 所有等可能结果总数。

Relative frequency is an experimental estimate of probability based on trials: Relative frequency = frequency of the event ÷ total number of trials.

相对频率是基于试验的概率实验估计值：相对频率 = 事件发生的频数 ÷ 总试验次数。

As the number of trials increases, the relative frequency tends to stabilise towards the theoretical probability. This is known as the law of large numbers. In an experiment with a biased coin, for example, the relative frequency of “heads” approaches the true probability of “heads” as more tosses are made.

随着试验次数的增加，相对频率趋于稳定并接近理论概率。这被称为大数定律。例如，对于一枚有偏的硬币，随着投掷次数的增加，“正面”的相对频率会接近“正面”的真实概率。

P(A or B) for mutually exclusive events: P(A ∪ B) = P(A) + P(B)

P(A and B) for independent events: P(A ∩ B) = P(A) × P(B)

互斥事件的加法法则：P(A ∪ B) = P(A) + P(B)

独立事件的乘法法则：P(A ∩ B) = P(A) × P(B)

11. Common Mistakes and How to Avoid Them | 常见错误及避免方法

Students often lose marks in Handling Data questions due to avoidable errors. Here are the most common pitfalls:

学生在数据处理题中经常因为可以避免的错误而失分。以下是最常见的陷阱：

Using the wrong boundary on cumulative frequency graphs. Always plot the upper class boundary, not the midpoint or the lower boundary.
Drawing the line of best fit through the origin. Only do this if the data actually support it; otherwise the line should simply balance the points.
Confusing interpolation and extrapolation. Interpolation occurs within the data range; extrapolation occurs outside it.
Forgetting the units in your answer. Always include the units given in the question (cm, kg, £, etc.).
Rounding at an intermediate stage. Keep full precision in your calculations and round only your final answer.

在累积频率图上用错边界。务必使用各组的上界，而不是中点或下界。
将最佳拟合线强行画过原点。只有当数据真实支持时才这样做；否则直线只需平衡各点即可。
混淆内插与外推。内插发生在数据范围内；外推发生在数据范围之外。
答案忘记写单位。始终包含题目中给出的单位（cm、kg、£ 等）。
在中间步骤过早四舍五入。计算过程中保持完整精度，只对最终答案进行四舍五入。

Moreover, always read questions carefully: some questions require a written explanation, such as “Explain why the median is more suitable than the mean for this data set.” These explanation questions are worth 2–3 marks and are often left blank by hurried students.

此外，务必仔细审题：有些题目要求文字解释，例如“解释为什么对于该数据集使用中位数比使用均值更合适”。这类解释题通常值 2–3 分，但常常被匆忙作答的学生留空。

12. Revision Checklist for Handling Data 6 | 数据处理 6 复习清单

Before the exam, make sure you can confidently:

考试之前，请确保你能自信地做到以下各项：

Draw and interpret scatter graphs, and estimate the correlation coefficient from the diagram.
Draw a line of best fit through the mean point, and use it for interpolation and extrapolation.
Calculate and plot moving averages for time series data, and identify the trend.
Construct a cumulative frequency table and draw the cumulative frequency curve.
Read the median, quartiles, and interquartile range from a cumulative frequency graph.
Draw and compare box-and-whisker plots.
Identify skewness from a distribution or a box plot.
Choose and justify the most appropriate average for a given data set.
Use relative frequency as an estimate of probability and apply the addition and multiplication rules.

绘制并解读散点图，从图中估计相关系数。
穿过均值点画最佳拟合线，并用它进行内插和外推。
计算并绘制时间序列数据的移动平均，并识别趋势。
构建累积频数表并绘制累积频率曲线。
从累积频率图中读取中位数、四分位数和四分位距。
绘制并比较箱线图。
从分布图或箱线图中识别偏态。
为给定数据集选择并论证最合适的平均数。
使用相对频率估计概率，并应用加法法则和乘法法则。

Remember: Handling Data questions are rewarding because they are highly structured and follow familiar patterns. With consistent practice, these marks are very attainable.

请记住：数据处理题目是高回报率的，因为它们结构清晰且模式固定。通过持续练习，这些分数非常容易获得。

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Handling Data 6 | 数据处理 6