统计

IGCSE 数学 · 第 9 主题

训练

讲义词汇表

本讲义涵盖主题 9,统计学(Statistics)。标记 (Extended) 的部分只在拓展卷上考查;其他一切对两个层次都适用。

9.1 9.2

统计数据的分类

大纲

Subject content	Notes and examples
Classify and tabulate statistical data.	e.g. tally tables, two-way tables.

Subject content	Notes and examples
1 Read, interpret and draw inferences from tables and statistical diagrams.
2 Compare sets of data using tables, graphs and statistical measures.	e.g. compare averages and ranges between two data sets.
3 Appreciate restrictions on drawing conclusions from given data.

Subject content	Notes and examples
1 Read, interpret and draw inferences from tables and statistical diagrams.
2 Compare sets of data using tables, graphs and statistical measures.	e.g. compare averages and measures of spread between two data sets.
3 Appreciate restrictions on drawing conclusions from given data.

来源:剑桥国际大纲

一大群人 — 数据从人们那里收集——从一个总体抽取的一个样本。

要组织统计(statistical)数据(data),用:

一张计数表(tally table)——为每个值做一个记号,然后数。
一张双向表(two-way table)——一次按两个特征把数据分类(例如,男孩/女孩对步行/公交)。

当你读一个图时,只得出数据真正支持的结论。

探索

Data handling cycle

Follow a statistical question from collection to display.

词汇表训练

英文	中文	拼音
statistical	统计	tǒng jì
data	数据	shù jù
tally table	计数表	jì shù biǎo
two-way table	双向表	shuāng xiàng biǎo

9.3

平均数与极差

大纲

Subject content	Notes and examples
Calculate the mean, median, mode and range for individual data and distinguish between the purposes for which these are used.	Data may be in a list or frequency table, but will not be grouped.

Subject content	Notes and examples
1 Calculate the mean, median, mode, quartiles, range and interquartile range for individual data and distinguish between the purposes for which these are used.
2 Calculate an estimate of the mean for grouped discrete or grouped continuous data.
3 Identify the modal class from a grouped frequency distribution.

来源:剑桥国际大纲

三个平均描述数据的一个典型值,而极差显示它有多分散。每一个用于一个不同的目的:

数据 :平均数、中位数、众数、极差 — 一个小数据集的平均数、中位数、众数和极差

平均数(mean)$= \dfrac{\text{sum of all values}}{\text{how many values}}$。
中位数(median)$=$ 数据按顺序排列时中间的值。
众数(mode)$=$ 出现最频繁的值。
极差(range)$=$ 最大值 $-$ 最小值(它显示数据有多分散)。

Worked example. 求 $4, 7, 7, 2, 5$ 的平均数、中位数、众数和极差。

把数据排序:$2, 4, 5, 7, 7$。

$$\text{mean} = \frac{4 + 7 + 7 + 2 + 5}{5} = \frac{25}{5} = 5, \quad \text{median} = 5, \quad \text{mode} = 7, \quad \text{range} = 7 - 2 = 5.$$

探索

Average choice lab

Choose the average that fits the data situation.

词汇表训练

英文	中文	拼音
mean	平均数	píng jūn shù
median	中位数	zhōng wèi shù
mode	众数	zhòng shù
range	极差	jí chà

9.3

平均数与极差

大纲

Subject content	Notes and examples
Calculate the mean, median, mode and range for individual data and distinguish between the purposes for which these are used.	Data may be in a list or frequency table, but will not be grouped.

Subject content	Notes and examples
1 Calculate the mean, median, mode, quartiles, range and interquartile range for individual data and distinguish between the purposes for which these are used.
2 Calculate an estimate of the mean for grouped discrete or grouped continuous data.
3 Identify the modal class from a grouped frequency distribution.

来源:剑桥国际大纲

当数据带它的频数(frequency)(每个值出现多少次)列出时,平均数是

$$\text{mean} = \frac{\sum (\text{value} \times \text{frequency})}{\sum \text{frequency}}.$$

Worked example. 值 $1, 2, 3$ 以频数 $4, 5, 1$ 出现。求平均数。

$$\text{mean} = \frac{1(4) + 2(5) + 3(1)}{4 + 5 + 1} = \frac{17}{10} = 1.7.$$

Grouped data (Extended)

对于分组数据(grouped data),你不能求出精确的平均数,所以用每组的中点作为值来估计它。众数组(modal class)就是频数最高的组。

词汇表训练

英文	中文	拼音
frequency	频数	pín shuò
grouped data	分组数据	fēn zǔ shù jù
modal class	众数组	zhòng shù zǔ

9.3

平均数与离散程度的度量

大纲

Subject content	Notes and examples
1 Calculate the mean, median, mode, quartiles, range and interquartile range for individual data and distinguish between the purposes for which these are used.
2 Calculate an estimate of the mean for grouped discrete or grouped continuous data.
3 Identify the modal class from a grouped frequency distribution.

来源:剑桥国际大纲

当数据按顺序排列时,四分位数(quartiles)把它切成四个相等的部分。下四分位数(LQ)在四分之一处;上四分位数(UQ)在四分之三处。四分位距(interquartile range)衡量中间一半的分散:

$$\text{interquartile range} = \text{UQ} - \text{LQ}.$$

四分位距有用,因为与极差不同,它忽略极端值。

一个箱线图,显示最小值、下四分位数、中位数、上四分位数和最大值,四分位距横跨箱子被标记 — 一个箱线图显示五数概括;箱子长度是四分位距。

探索

Box-and-whisker plot

Build a box plot from the five-number summary: the box is the middle 50% (the interquartile range) and the median's position shows the skew.

词汇表训练

英文	中文	拼音
quartile	四分位数	sì fēn wèi shù
interquartile range	四分位距	sì fēn wèi jù

9.4

统计图表

大纲

Subject content	Notes and examples
Draw and interpret: (a) bar charts (b) pie charts (c) pictograms (d) stem-and-leaf diagrams (e) simple frequency distributions.	Includes composite (stacked) and dual (side-by-side) bar charts. Stem-and-leaf diagrams should have ordered data with a key.

来源:剑桥国际大纲

一个形成钟形的高尔顿板 — 一个高尔顿板显示数据如何堆积成一个分布。

一个象形图,每个圆代表一个物品 — 一个象形图用一个符号代表若干物品

一个最喜欢颜色的条形图 — 一个条形图显示每组的频数

图	它显示什么
条形图(bar chart)	每个类别一根条;高度是频数
饼图(pie chart)	一个圆被分成片,每片是 $360^{\circ}$ 的一个分数
象形图(pictogram)	用一个符号代表若干物品
茎叶图(stem-and-leaf diagram)	保留有序数据的数字,带一个图例
频数分布(frequency distribution)	一张值和它们频数的表

Worked example (pie chart). $120$ 人中,$30$ 人选择茶。求茶片的角度。

$$\frac{30}{120} \times 360^{\circ} = 90^{\circ}.$$

份饮料分成茶、咖啡、果汁和水的一个饼图,茶片被标记为一个度的四分之一 — 每片是它占 $360^\circ$ 的分数;茶是 $\tfrac{30}{120}\times360^\circ=90^\circ$。

探索

Pie-chart angles

Each slice's angle is its share of the whole turned into degrees — frequency ÷ total × 360 — and the slices always add to 360°.

探索

Chart choice lab

Choose the chart that fits the kind of data.

词汇表训练

英文	中文	拼音
bar chart	条形图	tiáo xíng tú
pie chart	饼图	bǐng tú
pictogram	象形图	xiàng xíng tú
stem-and-leaf diagram	茎叶图	jīng yè tú
frequency distribution	频数分布	pín shuò fēn bù

9.5

散点图

大纲

Subject content	Notes and examples
1 Draw and interpret scatter diagrams.	Plotted points should be clearly marked, for example as small crosses (×).
2 Understand what is meant by positive, negative and zero correlation.
3 Draw by eye, interpret and use a straight line of best fit.	A line of best fit: • should be a single ruled line drawn by inspection • should extend across the full data set • does not need to coincide exactly with any of the points but there should be a roughly even distribution of points either side of the line over its entire length.

Subject content	Notes and examples
1 Draw and interpret scatter diagrams.	Plotted points should be clearly marked, for example as small crosses (x).
2 Understand what is meant by positive, negative and zero correlation.
3 Draw by eye, interpret and use a straight line of best fit.	A line of best fit: • should be a single ruled line drawn by inspection • should extend across the full data set • does not need to coincide exactly with any of the points but there should be a roughly even distribution of points either side of the line over its entire length.

来源:剑桥国际大纲

一个散点图(scatter diagram)把成对的值描成点以显示两样东西是否有联系。这个联系是相关性(correlation):

正相关(positive correlation)——一个上升时,另一个上升。
负相关(negative correlation)——一个上升时,另一个下降。
零相关(zero correlation)——没有清晰的联系。

三个散点图:一个点一起上升、一个点下降,和一个散乱的点显示没有趋势 — 正相关一起上升;负相关随着另一个上升而下降;零相关显示没有清晰的联系。

若有相关性,画一条最佳拟合线(line of best fit):穿过点的中间的一条直的画线,每一侧大约相同数量的点。用它来预测值。

一个复习时间对测试分数的散点图,一条直的最佳拟合线穿过点的中间画出 — 一条最佳拟合线是穿过点的中间的一条直的画线;用它来预测。

探索

Scatter and correlation

Change the strength of the relationship and add a line of best fit to spot the trend.

词汇表训练

英文	中文	拼音
scatter diagram	散点图	sàn diǎn tú
correlation	相关性	xiāng guān xìng
positive correlation	正相关	zhèng xiāng guān
negative correlation	负相关	fù xiāng guān
zero correlation	零相关	líng xiāng guān
line of best fit	最佳拟合线	zuì jiā nǐ hé xiàn

9.6

累积频数图

大纲

Subject content	Notes and examples
1 Draw and interpret cumulative frequency tables and diagrams.	Plotted points on a cumulative frequency diagram should be clearly marked, for example as small crosses (x), and be joined with a smooth curve.
2 Estimate and interpret the median, percentiles, quartiles and interquartile range from cumulative frequency diagrams.

来源:剑桥国际大纲

累积频数(cumulative frequency)是频数的一个连续总计。把它相对于每组的上端描出并用一条光滑曲线连接这些点。

从曲线你能读出中位数(在总数的一半处)、四分位数(在四分之一和四分之三处),以及任何百分位数(percentile)(例如,第 $90$ 百分位数在总数的 $90\%$ 处)。

一条 S 形的累积频数曲线,虚线从竖轴向下到值读出下四分位数、中位数和上四分位数 — 在总数的一半处读中位数,在四分之一和四分之三处读四分位数:横过到曲线,然后向下。

探索

Cumulative frequency route

Follow raw data into a cumulative frequency curve and percentile reading.

词汇表训练

英文	中文	拼音
cumulative frequency	累积频数	lěi jī pín shuò
percentile	百分位数	bǎi fēn wèi shù

9.7

直方图

大纲

Subject content	Notes and examples
1 Draw and interpret histograms.	On histograms, the vertical axis is labelled 'Frequency density'.
2 Calculate with frequency density.	Frequency density is defined as $\text{frequency density} = \text{frequency} \div \text{class width}$.

来源:剑桥国际大纲

一个直方图(histogram)看起来像一个条形图,但这些条能有不同的宽度,而每根条的面积(不是它的高度)显示频数。竖轴是频数密度(frequency density):

$$\text{frequency density} = \frac{\text{frequency}}{\text{class width}}.$$

一个直方图,条的宽度不同,竖轴是频数密度,所以每根条的面积给出它的频数 — 用不相等的组宽,条的面积(不是它的高度)是频数,所以坐标轴是频数密度。

Worked example. 一个组有组距(class width)$10$ 和频数 $25$。求频数密度。

$$\text{frequency density} = \frac{25}{10} = 2.5.$$

探索

Frequency-density histogram

With unequal class widths the bar height is frequency density, so the area (not the height) represents the frequency.

词汇表训练

英文	中文	拼音
histogram	直方图	zhí fāng tú
frequency density	频数密度	pín shuò mì dù
class width	组距	zǔ jù

9.7

考试技巧

知道三个平均:平均数(加起来 ÷ 有多少个)、中位数(按顺序时中间的值)、众数(最常见的)。极差是最高 − 最低。
从一张频数表,平均数是 $\frac{\sum fx}{\sum f}$ ——除以总频数,不是行数。
在一个散点图上,把相关性描述为正、负或没有,并穿过平均点画最佳拟合线。
对于一个组宽不相等的直方图,高度是频数密度(频数 ÷ 组距),不是频数。

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