KS3统计 频率表 饼图 平均数 散点图
KS3 的统计部分看起来最好拿分，实际是最容易在细节上丢分的一块。数据本身不难算，难的是读懂题目在问什么：它要你描述一个分布，还是比较两组数据，还是从一个散点图里判断两个量有没有关系。方向读错，算得再准也拿不到分。
英国教育部（DfE）官网公布的国家课程文件（DFE-00179-2013）把 KS3 统计内容归成三条：描述、解释并比较单个变量的分布；构造并读取表格与图表；用散点图描述两个变量之间的简单关系。下面按这三条把常考的图表、三种平均数与散点图逐个拆开。
Statistics at key stage 3 looks like easy marks, and it is where careful readers pull ahead. The numbers themselves are simple, so the difficulty sits in working out what the question actually wants. The 2013 national curriculum for England sets out three strands: describing and comparing distributions of one variable, constructing and reading tables and charts, and using scatter graphs to describe simple relationships between two variables.
数据的类型：分类、离散与连续
读一张图表之前，先判断手里是什么类型的数据。分类数据（categorical data）记录的是类别，比如最喜欢的科目、上学的交通方式，类别之间没有大小顺序，也不能相加。离散数据（discrete data）是数出来的整数，比如一个班的人数、家里有几双球鞋。连续数据（continuous data）是量出来的，比如身高 163.4 厘米、跑步用时 12.8 秒，理论上两个值之间还能继续细分。分组数据（grouped data）把连续数据或取值范围很宽的离散数据切成区间，例如 150 到 160 厘米这一组。很多学生以为 0 到 10 这一组的中点是 5，其实要按 0 与 10 的中间位置算。常见的错误是把连续数据四舍五入之后当成离散数据来画竖线图，图形的形状会失真。
Before reading any chart, decide what kind of data you have. Categorical data records labels such as favourite subject, so the categories have no order and cannot be added; discrete data is counted in whole units, like the number of students in a class; continuous data is measured, and a height of 163.4 cm can in principle be split further. Grouped data chops a wide range into intervals such as 150 to 160 cm, and examiners expect the boundaries to be written rather than left ambiguous. Rounding continuous values and then plotting them as if they were discrete changes the shape of the graph, which is a recurring error in exam scripts.
频率表与组中值
频率表把原始数据整理成两列：类别或区间，以及落在里面的次数。手写时先用划记（tally）数一遍，再换成数字，最后核对频数总和是否等于样本量，这一步能拦住大部分抄错。分组频率表还需要一列组中值（class midpoint），算法是区间上下界的平均数，所以 10 到 19 这一组的组中值是 14.5，不是 15。组中值代表这一组所有数据的替身，用它算出来的平均数只能是估计值。常见的错误是把 20 到 29 的组中值写成 24，上限 29 被直接拿来用了。阅卷时会专门看组中值这一列有没有写对，因为后面的平均数全靠它。
A frequency table has two columns: the class or category, and how many values fall into it. Tally marks help you count without losing your place, and checking that the frequencies add up to the sample size catches most copying errors. For grouped data you also need the class midpoint, which is the mean of the boundaries, so the group 10 to 19 has a midpoint of 14.5; that value then stands in for every observation in the group, which is why any mean calculated from it is only an estimate. Writing 24 for the group 20 to 29 shows the upper boundary was used instead of the midpoint.
条形图与分类数据的读法
条形图（bar chart）的横轴是类别，条形之间要留空隙，这一点和直方图不同。复合条形图（composite bar chart）把同一类别再按子组拆开，读的时候要分清题目比的是总高度还是其中一段。题目常问哪一类最多、多多少，答话要带单位和具体差值，只写类别名拿不到分。坐标轴不从零开始是常见陷阱：纵轴从 80 开始，两个相差 3 的条形会被画成三倍高。很多学生以为这种图在造假，其实它只是把差异放大了，读图时先看刻度起点。
In a bar chart the categories sit on the horizontal axis with gaps between the bars, which is the visible difference from a histogram. A composite bar chart splits each category into sub-groups, so decide whether the question is about the whole bar or one section of it; answers naming the largest category need a value and a unit, not just a label. Watch the scale as well, because an axis starting at 80 rather than 0 can make a difference of 3 look three times larger. That is a presentation choice rather than a false chart, and the starting value tells you so.
饼图的角度换算
饼图把总量当成 360 度，每一类的角度等于这一类的频数除以总数再乘 360。三十名学生里有 12 人选足球，角度就是 12/30 乘 360 得 144 度，剩下 18 人合计 216 度。做完要回头加一遍，角度总和不是 360 就说明中间某一步错了。反向题更常考：给出角度求人数，用角度除以 360 再乘总数。常见的错误是拿百分比直接当角度用，30% 写成 30 度，差了三倍多。饼图不适合类别很多的场合，超过六类时角度差异肉眼几乎看不出来，题目却仍可能要求你说出哪一类最少。
A pie chart treats the whole data set as 360 degrees, so each sector is the group frequency divided by the total and multiplied by 360. With 30 students and 12 choosing football, the angle is 144 degrees and the remaining 18 students share 216. Adding the angles back to 360 is a quick check that no step went wrong; the reverse question gives an angle and asks for a frequency, which means dividing by 360 and multiplying by the total. Percentages are not angles, and writing 30% as 30 degrees is a common slip that distorts the sector badly.
象形图与竖线图
象形图（pictogram）用一个符号代表固定的数量，关键是看清单个符号代表几个，以及半个符号是什么意思。符号代表 4 人时，三个半符号是 14 人，不是 3.5 人。竖线图（vertical line chart）用于未分组的数值数据，每条竖线的高度就是频数，线画在具体数值上，不是画在类别上。分组数据画竖线图时把线落在组中值处，轮廓会接近直方图。常见的错误是象形图不按比例画，符号被画得大小不一，读者会误判比较结果。
A pictogram uses one symbol for a fixed number of items, so read the key first and check what half a symbol means. If one symbol stands for 4 people, three and a half symbols are 14 people rather than 3.5. A vertical line chart, by contrast, puts a line at each numerical value with the height giving the frequency, and grouped data puts those lines at the midpoints so the outline begins to look like a histogram. Symbols drawn at different sizes break the scale and make the comparison misleading.
平均数、中位数、众数
平均数（mean）把总和平均分给每一个数据，中位数（median）是排序后正中间那个值，众数（mode）是出现次数最多的值。七个数据排序后取第四个就是中位数，偶数个数据取中间两个的平均。一组测验分数 12、15、15、18、21、22、60，平均数是 23.3，中位数只有 18，众数是 15。三种平均回答的问题不同：平均数用上每一个数据，中位数只看位置，众数可以用在分类数据上。很多学生以为题目说 average 就一定是平均数，英国教材里的 average 通常指三个中的任意一个，要看上下文再决定。
The mean shares the total equally, the median is the middle value once the data is ordered, and the mode is the value that appears most often. With seven values the median is the fourth, and with an even number you average the middle two; for the scores 12, 15, 15, 18, 21, 22 and 60 the mean is 23.3 while the median is 18 and the mode is 15. That gap already hints that one extreme value is pulling the mean upward. The word average in British textbooks can mean any of the three, so read the surrounding wording before choosing.
极差、离群值与平均数被拖动的方向
极差（range）只等于最大值减最小值，中间的数据一个都没用上，所以它对离群值（outlier）极其敏感。上面那组分数里，60 分让极差变成 48；去掉它之后极差是 10，平均数从 23.3 降到 17.2。离群值是不是错误数据，要看记录过程：量错、抄错、样本里混进另一个群体都可能造成离群。题目常见问法是哪一个平均值更能代表这批数据，有离群值时通常选中位数，理由是它只看位置，不会被极端值拖动。常见的错误是只报一个平均数就下结论，比较两组数据时平均数与离散程度都要说到。
The range is simply the largest value minus the smallest, so the middle of the data is ignored and one outlier can dominate it. In the scores above the 60 pushes the range to 48, and removing it drops the range to 10 while the mean falls from 23.3 to 17.2. Whether an outlier is a mistake depends on how the data was recorded, since a misread instrument, a typing error or a mixed population all produce extreme values. Questions asking which average represents the data best usually want the median when an outlier is present, because position is not dragged by extremes.
分组连续数据的平均数估算
分组数据看不到原始值，只能假设每一组的数据都集中在组中值上，用频数乘组中值再除以总频数。组界 0 到 9 频数 4、10 到 19 频数 7、20 到 29 频数 9，组中值分别是 4.5、14.5、24.5，三项乘积是 18、101.5、220.5，总和 340，除以 20 得 17。答案要写成估计值（estimated mean），因为原始数据已经被区间抹掉了。中位数所在组用累积频数找：总数 20，一半是 10，累积频数依次是 4、11、20，第一个超过 10 的组是 10 到 19。众数在分组数据里改称众数组（modal class），取频数最大的那一组。
With grouped data the original values are gone, so each class is treated as if every observation sat at its midpoint. Multiplying frequency by midpoint, adding and dividing by the total gives an estimated mean: 4 times 4.5 plus 7 times 14.5 plus 9 times 24.5 is 340, and dividing by 20 gives 17. The median class is found from cumulative frequencies, taking the first class whose total passes half the sample. For grouped data the mode becomes the modal class, which is simply the class with the highest frequency.
累积频数与中位数位置
累积频数（cumulative frequency）是一列逐行累加的数字，最后一行必须等于总数。它的用途是快速定位中位数所在的组，也能读出有多少个数据小于某个值。十五个数据的中位数位置是第八个，三十个数据取第十五个与第十六个的平均。累积频数画成曲线时，纵轴到总数一半处横着拉到曲线上，再竖着读回横轴，得到的是中位数的估计。常见的错误是把累积频数当成普通频数再平均一次，结果整列数字都错。
A cumulative frequency column adds the frequencies as it goes, and its last entry must equal the total. That column locates the class holding the median and answers questions about how many values fall below a given point. With 15 values the median is the 8th, while 30 values means averaging the 15th and 16th; drawn as a curve, the median is read across at half the total and down to the horizontal axis. Treating the cumulative column as ordinary frequencies and averaging it again is a mistake that spoils every later step.
散点图与两个变量的关系
散点图（scatter graph）把一个个体画成一个点，横轴是第一个变量，纵轴是第二个变量。点从左下往右上排是正相关（positive correlation），从左上往右下排是负相关，散成一团说明看不出关系。相关强度用强、弱、无来描述，不写具体相关系数，KS3 不要求计算相关系数。最佳拟合线（line of best fit）要穿过点的中间，两侧点数大致相当，并且经过两个变量的平均值点。相关不等于因果：冰淇淋销量和溺水人数同时上升，真正的原因是气温。
Each point on a scatter graph represents one individual, with one variable on each axis. Points rising from bottom left to top right show positive correlation, points falling show negative correlation, and a shapeless cloud means no visible relationship. Strength is described as strong, weak or none, since key stage 3 does not ask for a correlation coefficient; a line of best fit passes through the middle of the points with roughly equal numbers on each side. Correlation is not causation, as ice cream sales and drowning figures rise together because of warm weather.
比较两个分布该怎么落笔
比较两组数据的题目，答案至少要两句：一句比平均数，一句比离散程度或极差。只说 A 组比 B 组高通常只拿到一半的分，因为没有说高多少，也没有说数据是否更集中。规范的写法是把两个具体数值放进去，例如 A 组中位数 62 分、极差 18 分，B 组中位数 55 分、极差 34 分，B 组更分散。比较前要确认单位一致，一组用分钟一组用秒的答案会直接判错。题目里出现 describe the distribution 时，按集中趋势、离散程度、形状或异常值三件事各说一句就够。
Comparing two data sets needs at least two statements: one about an average and one about spread. Saying that group A is higher than group B usually earns half the marks, because no number and nothing about consistency has been given. Write both values into the sentence, for example a median of 62 with a range of 18 against a median of 55 with a range of 34, which shows group B is more spread out. Check that the units match before comparing, and when a question says describe the distribution, cover centre, spread and any unusual values.
坐标轴、单位与四舍五入的丢分点
读图题的失分很少出在计算上，多半出在读刻度。纵轴每一小格代表 2 还是 5，要先算清楚，两个刻度之间的值只能估读，答案写成 23.4 会被判为过度精确。单位要和题目一致，速度题写成 12 而不写 km/h 会扣分。四舍五入的位数按题目要求，没要求时保留一位小数，中间结果多留一位再在最后取整。画图时用铅笔和直尺，条形宽度一致，饼图的分界线从圆心出发。阅卷时会先看坐标轴标签和单位，缺一项就少一分，这一步最容易漏。
Marks in graph-reading questions are lost to the scale rather than to arithmetic. Work out what one small square stands for first, since a value between two labelled lines can only be estimated, and an answer of 23.4 from such a graph claims precision the reading does not support. Keep units with every number, because a speed written as 12 without km/h loses a mark. When drawing, use a ruler and pencil with equal bar widths and pie sectors starting from the centre, and label both axes before plotting anything.
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