CIE S1 知识点及题型总结第三篇

来来来，同学们看过来，这里继续为大家总结S1数据的位置和测量的第五个知识点。...



来来来，同学们看过来，这里继续为大家总结S1数据的位置和测量的第五个知识点。这个知识点包括了数据的集中图解，在考试中属于必考的题型，一般占分5-7分。

知识点5

Construct and interpret diagrams to represent data—stem and leafdiagram, box-and-whisker plots, histogram, and cumulative frequency graphs

数据的图解表示

1.      Stemand leaf diagram 茎叶图

A way ofgrouping data into intervals while still retaining the original data is to drawa stem-and-leaf diagram, also knownas a stemplot.

Theseare the marks of 20 students in an assignment:

84 17 3845 47 53 76 54 75 32

66 65 5554 51 44 39 19 54 72

Instem-and-leaf diagram all the intervals must be of equal width, so it seems sensible to choose intervals 10-19, 20-29,30-39,…,80-89 for this data, so tale the stem to represent the tens and the leaf to represent the units.

Youenter the numbers one by one. When all the numbers have been entered, you mustarrange the entries in each row in numerical order with the smallest nest tothe stem.

Important: You must always give a Key toexplain what the stem and leaf represent.

Stem-and-leafdiagrams can be used to compare two sets of data by showing them together on aback-to-back stem-and-leaf diagram.

2.      Histogram

Groupeddata can be displayed in a histogram.

Histogramsresemble bar charts, but there are two important differences:

Thereare no gaps between the bars.

The area of the bar is proportional to thefrequency that it represents.

If theheight of the bar is adjusted so that the area is equal to the frequency in that interval, then 

So ,this is known as the frequency density.

3.      Box-and-whiskerplots 盒式图

In abox-and-whisker plot the median and quartiles are shown, as well as the minimumand maximum values of the distribution. It gives a very good visual summary ofa distribution and is particularly useful when comparing set of data. 

4.      Cumulativefrequency graphs 

Cumulativefrequency is the total frequency up to a particular item. Cumulative frequencyis particularly useful when finding the median and quartiles.

Cumulativefrequency table can be illustrated on a cumulative frequency graph

in which thecumulative frequencies are plotted against the upper class boundaries. 

The points are joined either with a curve or with straight lines.

Values can be estimated from thegraph, for example:

(a)  Estimatehow many plants had a height less than 10.5 cm.

  ▶ From 10.5 on the x-axis draw a verticalline up to the curve.

  ▶ Now draw a horizontal line to the y-axisand read off the value

(b)  10% ofplants had a height of at least x cm. estimate x.

  ▶ 10% of 30=3

  ▶ 3plants had height of at least x cm, so 27plants had a height less than x cm.

  ▶ From 27 on the y-axis draw a horizontalline to the curve.

  ▶ Now draw a vertical line down to the x-axisand read off the value.

  ▶ From the graph, 10% of the plants had aheight of at least 16.5.

Example 1

CIE-2013-spring-62

答案及解析：

该题考察的是stem-and-leaf diagram的作图，以及Q1, Q2, Q3 IQR的计算

1. Quartiles(四分位数)：the lower quartile and the upper quartiles are not influencedby extreme values. They are values such that, together with the median, theysplit a distribution into four parts. 四分位数把一组数据平分成四个部分。Interquartilerange=Q3-Q1

该题最容易失分的点：茎叶图的作图最后必须Key, 在写Key的时候数字必须带单位（$140）。

Example 2

CIE-2014-spring-63

答案与解析：

该题考察的是Back-to-back stem-and-leaf diagram 的作图，以及两组数据的比较。

需要注意的点是在比较两组数据的时候，要从两个方面进行，一是两组数据的平均（median）, 二是两组数据的分布（skewness）.

Example 3

CIE-2013-Ocotober-62

答案与解析：

该题考察的是histogram中frequency 和area成正比，计算频数即只需要计算面积。

对于分组的frequency table或者是histogram，计算mean和standard deviation 都假设同一组的所有数据都等于该组的中点。因此计算出来的值是近似值。

Example 4

CIE-2014-May-62

答案与解析：

该题考察的是累计频数分布和频数分布的转换；

Histogram的作图，以及mean 和standard deviation 的计算。

需要注意的是画图前要先计算出Frequency density, 另外，计算mean 和standard deviation时要用中点。

Example 5

CIE-2014-May-61

答案与解析：

该题考察的是histogram 的作图，mean的计算，以及quartile的知识点。

Example 6

答案与解析：

该题考察的是quartile的知识点，以及histogram 的作图，关键是求出frequency density.

Example 7

CIE-2013-May-61

答案与解析：

该题考察的是盒式图的作图，需要注意的是盒式图要先画出坐标轴，并且坐标轴上面要标明数据的单位。

Example 8

CIE-2013-October-61

答案：

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