📚 Representations of Data | 数据表示
In Edexcel A-Level Mathematics, representations of data are statistical tools used to summarise, display and compare distributions. Choosing the correct diagram or table can reveal the shape, centre and spread of a data set, and it is essential for answering exam questions on descriptive statistics.
在 Edexcel A-Level 数学中，数据表示是用于总结、展示和比较分布的统计工具。选择正确的图表或表格可以揭示数据集的形状、中心和离散程度，这也是解答描述性统计考题的关键。
1. Stem and Leaf Diagrams | 茎叶图
A stem and leaf diagram keeps every original data value visible while sorting the data into order. The stem represents the leading digit or digits, and the leaf is the final digit. A key must always be given so that the reader knows the place value.
茎叶图在将数据排序的同时保留每个原始数据值。茎代表前一位或多位数字，叶是最后一位数字。必须始终给出图例，以便读者知道每个茎叶所代表的实际数值。
For example, the data 43, 45, 51, 52, 52, 60, 63, 63, 65 can be shown with stem 4 | 3 5, stem 5 | 1 2 2, and stem 6 | 0 3 3 5. A back-to-back stem and leaf diagram can compare two data sets sharing one stem column.
例如，数据 43、45、51、52、52、60、63、63、65 可以表示为茎 4 | 3 5，茎 5 | 1 2 2，茎 6 | 0 3 3 5。背靠背茎叶图可以共用一列茎来比较两组数据。
From an ordered stem and leaf diagram you can easily find the median, quartiles and range, and you can also see the shape of the distribution.
从有序的茎叶图中可以轻松找出中位数、四分位数和极差，同时也能观察分布的形状。
2. Box Plots and Outliers | 箱线图与离群值
A box plot displays the minimum, lower quartile Q₁, median Q₂, upper quartile Q₃ and maximum. It is especially useful for comparing distributions and identifying skewness and outliers.
箱线图展示最小值、下四分位数 Q₁、中位数 Q₂、上四分位数 Q₃ 和最大值。它特别适合比较分布并识别偏度和离群值。
Outliers are usually defined as values less than Q₁ − 1.5 × IQR or greater than Q₃ + 1.5 × IQR, where the interquartile range IQR = Q₃ − Q₁.
离群值通常定义为小于 Q₁ − 1.5 × IQR 或大于 Q₃ + 1.5 × IQR 的值，其中四分位距 IQR = Q₃ − Q₁。
outlier limits: Q₁ − 1.5 × IQR and Q₃ + 1.5 × IQR
离群值界限：Q₁ − 1.5 × IQR 与 Q₃ + 1.5 × IQR
When drawing a box plot, any outliers are marked with crosses and the whiskers extend to the most extreme data values that are not outliers.
绘制箱线图时，离群值用叉号标出，箱须延伸到非离群值中的最极端数据。
3. Histograms and Frequency Density | 直方图与频率密度
A histogram is used for continuous data grouped into classes. Unlike a bar chart, the area of each bar is proportional to the frequency, not the height. Therefore the vertical axis must show frequency density.
直方图用于按区间分组后的连续数据。与条形图不同，直方图每个条形的面积与频率成正比，而不是高度。因此纵轴必须表示频率密度。
frequency density = frequency ÷ class width
频率密度 = 频率 ÷ 组距
If groups have unequal widths, using frequency on the vertical axis would make wider bars look artificially larger. Frequency density corrects this so that the area of each bar equals the frequency.
如果各组宽度不相等，纵轴使用频率会让较宽的条形看起来人为地更大。频率密度可以修正这一点，使每个条形的面积等于频率。
To find the frequency from a histogram, multiply the frequency density by the class width. This is often tested in Edexcel exam questions.
从直方图中求频率时，将频率密度乘以组距。这是 Edexcel 考题中经常考查的内容。
4. Cumulative Frequency Diagrams | 累积频率图
A cumulative frequency diagram plots the running total of frequencies against the upper class boundary. It is used to estimate the median, quartiles and percentiles from grouped data.
累积频率图将频率的累计总数相对于组的上限值绘制成图。它用于从分组数据中估计中位数、四分位数和百分位数。
The median is estimated at the ½nth value, the lower quartile at the ¼nth value and the upper quartile at the ¾nth value, where n is the total frequency.
中位数在 ½n 个值处估计，下四分位数在 ¼n 个值处估计，上四分位数在 ¾n 个值处估计，其中 n 是总频率。
median position = n ÷ 2, Q₁ position = n ÷ 4, Q₃ position = 3n ÷ 4
中位数位置 = n ÷ 2，Q₁ 位置 = n ÷ 4，Q₃ 位置 = 3n ÷ 4
By reading across from the cumulative frequency axis to the curve and down to the data axis, you can estimate the corresponding values. The curve should always be smooth and pass through plotted points at upper boundaries.
从累积频率轴水平延伸到曲线，再向下到数据轴，就可以估计相应的数值。曲线应始终平滑，并经过上组界的已描点。
5. Comparing Distributions | 比较分布
When comparing two or more data sets, a good answer should refer to a measure of central tendency, a measure of spread, the shape of the distribution and any outliers.
在比较两个或多个数据集时，好的答案应提及集中趋势的度量、离散程度的度量、分布的形状以及任何离群值。
For example, you could compare medians and interquartile ranges for skewed data, or means and standard deviations for roughly symmetric data. Always quote numerical values in the context of the question.
例如，对于偏斜数据可以比较中位数和四分位距，对于大致对称的数据可以比较均值和标准差。始终在题目情境中引用具体数值。
Box plots are particularly effective for making comparisons because they show all five key summary statistics side by side.
箱线图在进行比较时特别有效，因为它并排显示了全部五个关键汇总统计量。
A comparison such as ‘Group A has a higher median and a larger IQR than Group B’ is stronger than simply saying ‘Group A is better’.
像“A 组的中位数比 B 组高，四分位距也比 B 组大”这样的比较，比简单说“A 组更好”更有力。
6. Skewness | 偏度
Skewness describes the asymmetry of a distribution. A distribution is positively skewed if the tail is longer on the right, and negatively skewed if the tail is longer on the left.
偏度描述分布的不对称性。如果尾部在右侧较长，分布为正偏；如果尾部在左侧较长，分布为负偏。
For a positively skewed distribution, mean > median > mode. For a negatively skewed distribution, mean < median < mode. In a symmetric distribution, mean = median = mode.
对于正偏分布，均值 > 中位数 > 众数。对于负偏分布，均值 < 中位数 < 众数。在对称分布中，均值 = 中位数 = 众数。
positive skew: mean > median > mode
正偏：均值 > 中位数 > 众数
Skewness can also be seen in a box plot: if the whisker from the median to the maximum is much longer than the whisker from the median to the minimum, the data are likely positively skewed.
偏度也可以从箱线图中看出：如果从中位数到最大值的箱须明显长于从中位数到最小值的箱须，则数据可能为正偏。
7. Choosing the Right Representation | 选择合适的数据表示
The choice of diagram depends on the type of data and the purpose of the analysis. Categorical data are often shown in pie charts or bar charts, while continuous data are shown in histograms or cumulative frequency graphs.
图表的选择取决于数据的类型和分析目的。分类数据通常用饼图或条形图表示，而连续数据用直方图或累积频率图表示。
- Stem and leaf — small data sets, preserving original values.
- Box plot — comparing distributions and identifying outliers.
- Histogram — continuous grouped data with unequal class widths.
- Cumulative frequency — finding medians, quartiles and percentiles from grouped data.
- Scatter diagram — showing relationship between two variables.
- 茎叶图 — 小数据集，保留原始数值。
- 箱线图 — 比较分布和识别离群值。
- 直方图 — 具有不等组距的连续分组数据。
- 累积频率图 — 从分组数据中求中位数、四分位数和百分位数。
- 散点图 — 显示两个变量之间的关系。
8. Grouped Frequency Tables | 分组频率表
Grouped frequency tables condense large data sets into intervals. When working with grouped data, exact values are unknown, so the mean is estimated using midpoints of classes.
分组频率表将大型数据集压缩为区间。处理分组数据时，精确值未知，因此使用组中点来估计均值。
estimated mean = Σ(f × x) ÷ Σf
估计均值 = Σ(f × x) ÷ Σf
where f is the frequency and x is the class midpoint. For grouped data, standard deviation can also be estimated using the same midpoints.
其中 f 是频率，x 是组中点。对于分组数据，标准差也可以使用相同的中点进行估计。
Always use the upper and lower class boundaries correctly when constructing cumulative frequency tables, especially for continuous data such as 0–10, 10–20 and so on.
在构建累积频率表时，要正确使用组的上下界，特别是对于 0–10、10–20 等连续数据。
9. Scatter Diagrams and Correlation | 散点图与相关性
A scatter diagram plots bivariate data as points on a coordinate grid. It is used to identify whether there is a linear relationship between two variables.
散点图将二元数据绘制为坐标网格上的点。它用于判断两个变量之间是否存在线性关系。
Correlation can be positive, negative or zero. Positive correlation means as one variable increases, the other tends to increase. Negative correlation means as one variable increases, the other tends to decrease.
相关性可以是正相关、负相关或零相关。正相关意味着当一个变量增加时，另一个变量也倾向于增加。负相关意味着当一个变量增加时，另一个变量倾向于减少。
The strength of correlation ranges from weak to strong. A line of best fit can be drawn through the points and used to estimate unknown values, but extrapolation beyond the data range is unreliable.
相关性的强度从弱到强。可以通过这些点画出最佳拟合线，并用于估计未知值，但超出数据范围的外推是不可靠的。
10. Common Exam Pitfalls | 常见考试失分点
Students often lose marks by using frequency instead of frequency density on a histogram vertical axis, especially when class widths are unequal.
学生常常因为在直方图纵轴上使用频率而不是频率密度而失分，尤其是当组距不相等时。
Another common error is forgetting to include a key in a stem and leaf diagram. Without a key, the diagram cannot be interpreted correctly.
另一个常见错误是忘记在茎叶图中给出图例。没有图例，图表就无法被正确解读。
When drawing cumulative frequency graphs, points should be plotted at upper class boundaries, not at midpoints. Also, when estimating quartiles, always show construction lines on the graph.
绘制累积频率图时，点应标在组的上界，而不是中点。此外，估计四分位数时，始终在图中画出辅助线。
Finally, when comparing box plots, do not just state which median is larger; mention spread, skewness and outliers for full marks.
最后，在比较箱线图时，不要只说明哪个中位数更大；要提及离散程度、偏度和离群值才能获得满分。
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