---
url: /learn/pv8ahawu/index.md
---
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## Chap01 - 抽样与数据

:::note Chapter Objectives

* 认识、辨别关键术语
* 将不同采样方法与不同类型的数据联系起来
* 制作频率分布表

:::

### 统计学、概率论及相关术语的定义

统计学是一门研究数据的收集、分析、解读和呈现的学科。在日常生活中，我们无时无刻不在接触和使用数据。

:::echarts

```js
option = {
    title: {
        text: 'Male and female height and weight distribution',
        subtext: 'Data from: Heinz 2003'
    },
    grid: {
        left: '3%',
        right: '7%',
        bottom: '7%',
        containLabel: true
    },
    tooltip: {
        // trigger: 'axis',
        showDelay: 0,
        formatter: function (params) {
            if (params.value.length > 1) {
                return (
                    params.seriesName +
                    ' :<br/>' +
                    params.value[0] +
                    'cm ' +
                    params.value[1] +
                    'kg '
                );
            } else {
                return (
                    params.seriesName +
                    ' :<br/>' +
                    params.name +
                    ' : ' +
                    params.value +
                    'kg '
                );
            }
        },
        axisPointer: {
            show: true,
            type: 'cross',
            lineStyle: {
                type: 'dashed',
                width: 1
            }
        }
    },
    toolbox: {
        feature: {
            dataZoom: {},
            brush: {
                type: ['rect', 'polygon', 'clear']
            }
        }
    },
    brush: {},
    legend: {
        data: ['Female', 'Male'],
        left: 'center',
        bottom: 10
    },
    xAxis: [
        {
            type: 'value',
            scale: true,
            axisLabel: {
                formatter: '{value} cm'
            },
            splitLine: {
                show: false
            }
        }
    ],
    yAxis: [
        {
            type: 'value',
            scale: true,
            axisLabel: {
                formatter: '{value} kg'
            },
            splitLine: {
                show: false
            }
        }
    ],
    series: [
        {
            name: 'Female',
            type: 'scatter',
            emphasis: {
                focus: 'series'
            },
            // prettier-ignore
            data: [[161.2, 51.6], [167.5, 59.0], [159.5, 49.2], [157.0, 63.0], [155.8, 53.6],
                [170.0, 59.0], [159.1, 47.6], [166.0, 69.8], [176.2, 66.8], [160.2, 75.2],
                [172.5, 55.2], [170.9, 54.2], [172.9, 62.5], [153.4, 42.0], [160.0, 50.0],
                [147.2, 49.8], [168.2, 49.2], [175.0, 73.2], [157.0, 47.8], [167.6, 68.8],
                [159.5, 50.6], [175.0, 82.5], [166.8, 57.2], [176.5, 87.8], [170.2, 72.8],
                [174.0, 54.5], [173.0, 59.8], [179.9, 67.3], [170.5, 67.8], [160.0, 47.0],
                [154.4, 46.2], [162.0, 55.0], [176.5, 83.0], [160.0, 54.4], [152.0, 45.8],
                [162.1, 53.6], [170.0, 73.2], [160.2, 52.1], [161.3, 67.9], [166.4, 56.6],
                [168.9, 62.3], [163.8, 58.5], [167.6, 54.5], [160.0, 50.2], [161.3, 60.3],
                [167.6, 58.3], [165.1, 56.2], [160.0, 50.2], [170.0, 72.9], [157.5, 59.8],
                [167.6, 61.0], [160.7, 69.1], [163.2, 55.9], [152.4, 46.5], [157.5, 54.3],
                [168.3, 54.8], [180.3, 60.7], [165.5, 60.0], [165.0, 62.0], [164.5, 60.3],
                [156.0, 52.7], [160.0, 74.3], [163.0, 62.0], [165.7, 73.1], [161.0, 80.0],
                [162.0, 54.7], [166.0, 53.2], [174.0, 75.7], [172.7, 61.1], [167.6, 55.7],
                [151.1, 48.7], [164.5, 52.3], [163.5, 50.0], [152.0, 59.3], [169.0, 62.5],
                [164.0, 55.7], [161.2, 54.8], [155.0, 45.9], [170.0, 70.6], [176.2, 67.2],
                [170.0, 69.4], [162.5, 58.2], [170.3, 64.8], [164.1, 71.6], [169.5, 52.8],
                [163.2, 59.8], [154.5, 49.0], [159.8, 50.0], [173.2, 69.2], [170.0, 55.9],
                [161.4, 63.4], [169.0, 58.2], [166.2, 58.6], [159.4, 45.7], [162.5, 52.2],
                [159.0, 48.6], [162.8, 57.8], [159.0, 55.6], [179.8, 66.8], [162.9, 59.4],
                [161.0, 53.6], [151.1, 73.2], [168.2, 53.4], [168.9, 69.0], [173.2, 58.4],
                [171.8, 56.2], [178.0, 70.6], [164.3, 59.8], [163.0, 72.0], [168.5, 65.2],
                [166.8, 56.6], [172.7, 105.2], [163.5, 51.8], [169.4, 63.4], [167.8, 59.0],
                [159.5, 47.6], [167.6, 63.0], [161.2, 55.2], [160.0, 45.0], [163.2, 54.0],
                [162.2, 50.2], [161.3, 60.2], [149.5, 44.8], [157.5, 58.8], [163.2, 56.4],
                [172.7, 62.0], [155.0, 49.2], [156.5, 67.2], [164.0, 53.8], [160.9, 54.4],
                [162.8, 58.0], [167.0, 59.8], [160.0, 54.8], [160.0, 43.2], [168.9, 60.5],
                [158.2, 46.4], [156.0, 64.4], [160.0, 48.8], [167.1, 62.2], [158.0, 55.5],
                [167.6, 57.8], [156.0, 54.6], [162.1, 59.2], [173.4, 52.7], [159.8, 53.2],
                [170.5, 64.5], [159.2, 51.8], [157.5, 56.0], [161.3, 63.6], [162.6, 63.2],
                [160.0, 59.5], [168.9, 56.8], [165.1, 64.1], [162.6, 50.0], [165.1, 72.3],
                [166.4, 55.0], [160.0, 55.9], [152.4, 60.4], [170.2, 69.1], [162.6, 84.5],
                [170.2, 55.9], [158.8, 55.5], [172.7, 69.5], [167.6, 76.4], [162.6, 61.4],
                [167.6, 65.9], [156.2, 58.6], [175.2, 66.8], [172.1, 56.6], [162.6, 58.6],
                [160.0, 55.9], [165.1, 59.1], [182.9, 81.8], [166.4, 70.7], [165.1, 56.8],
                [177.8, 60.0], [165.1, 58.2], [175.3, 72.7], [154.9, 54.1], [158.8, 49.1],
                [172.7, 75.9], [168.9, 55.0], [161.3, 57.3], [167.6, 55.0], [165.1, 65.5],
                [175.3, 65.5], [157.5, 48.6], [163.8, 58.6], [167.6, 63.6], [165.1, 55.2],
                [165.1, 62.7], [168.9, 56.6], [162.6, 53.9], [164.5, 63.2], [176.5, 73.6],
                [168.9, 62.0], [175.3, 63.6], [159.4, 53.2], [160.0, 53.4], [170.2, 55.0],
                [162.6, 70.5], [167.6, 54.5], [162.6, 54.5], [160.7, 55.9], [160.0, 59.0],
                [157.5, 63.6], [162.6, 54.5], [152.4, 47.3], [170.2, 67.7], [165.1, 80.9],
                [172.7, 70.5], [165.1, 60.9], [170.2, 63.6], [170.2, 54.5], [170.2, 59.1],
                [161.3, 70.5], [167.6, 52.7], [167.6, 62.7], [165.1, 86.3], [162.6, 66.4],
                [152.4, 67.3], [168.9, 63.0], [170.2, 73.6], [175.2, 62.3], [175.2, 57.7],
                [160.0, 55.4], [165.1, 104.1], [174.0, 55.5], [170.2, 77.3], [160.0, 80.5],
                [167.6, 64.5], [167.6, 72.3], [167.6, 61.4], [154.9, 58.2], [162.6, 81.8],
                [175.3, 63.6], [171.4, 53.4], [157.5, 54.5], [165.1, 53.6], [160.0, 60.0],
                [174.0, 73.6], [162.6, 61.4], [174.0, 55.5], [162.6, 63.6], [161.3, 60.9],
                [156.2, 60.0], [149.9, 46.8], [169.5, 57.3], [160.0, 64.1], [175.3, 63.6],
                [169.5, 67.3], [160.0, 75.5], [172.7, 68.2], [162.6, 61.4], [157.5, 76.8],
                [176.5, 71.8], [164.4, 55.5], [160.7, 48.6], [174.0, 66.4], [163.8, 67.3]
            ],
            markArea: {
                silent: true,
                itemStyle: {
                    color: 'transparent',
                    borderWidth: 1,
                    borderType: 'dashed'
                },
                data: [
                    [
                        {
                            name: 'Female Data Range',
                            xAxis: 'min',
                            yAxis: 'min'
                        },
                        {
                            xAxis: 'max',
                            yAxis: 'max'
                        }
                    ]
                ]
            },
            markPoint: {
                data: [
                    { type: 'max', name: 'Max' },
                    { type: 'min', name: 'Min' }
                ]
            },
            markLine: {
                lineStyle: {
                    type: 'solid'
                },
                data: [{ type: 'average', name: 'AVG' }, { xAxis: 160 }]
            }
        },
        {
            name: 'Male',
            type: 'scatter',
            emphasis: {
                focus: 'series'
            },
            // prettier-ignore
            data: [[174.0, 65.6], [175.3, 71.8], [193.5, 80.7], [186.5, 72.6], [187.2, 78.8],
                [181.5, 74.8], [184.0, 86.4], [184.5, 78.4], [175.0, 62.0], [184.0, 81.6],
                [180.0, 76.6], [177.8, 83.6], [192.0, 90.0], [176.0, 74.6], [174.0, 71.0],
                [184.0, 79.6], [192.7, 93.8], [171.5, 70.0], [173.0, 72.4], [176.0, 85.9],
                [176.0, 78.8], [180.5, 77.8], [172.7, 66.2], [176.0, 86.4], [173.5, 81.8],
                [178.0, 89.6], [180.3, 82.8], [180.3, 76.4], [164.5, 63.2], [173.0, 60.9],
                [183.5, 74.8], [175.5, 70.0], [188.0, 72.4], [189.2, 84.1], [172.8, 69.1],
                [170.0, 59.5], [182.0, 67.2], [170.0, 61.3], [177.8, 68.6], [184.2, 80.1],
                [186.7, 87.8], [171.4, 84.7], [172.7, 73.4], [175.3, 72.1], [180.3, 82.6],
                [182.9, 88.7], [188.0, 84.1], [177.2, 94.1], [172.1, 74.9], [167.0, 59.1],
                [169.5, 75.6], [174.0, 86.2], [172.7, 75.3], [182.2, 87.1], [164.1, 55.2],
                [163.0, 57.0], [171.5, 61.4], [184.2, 76.8], [174.0, 86.8], [174.0, 72.2],
                [177.0, 71.6], [186.0, 84.8], [167.0, 68.2], [171.8, 66.1], [182.0, 72.0],
                [167.0, 64.6], [177.8, 74.8], [164.5, 70.0], [192.0, 101.6], [175.5, 63.2],
                [171.2, 79.1], [181.6, 78.9], [167.4, 67.7], [181.1, 66.0], [177.0, 68.2],
                [174.5, 63.9], [177.5, 72.0], [170.5, 56.8], [182.4, 74.5], [197.1, 90.9],
                [180.1, 93.0], [175.5, 80.9], [180.6, 72.7], [184.4, 68.0], [175.5, 70.9],
                [180.6, 72.5], [177.0, 72.5], [177.1, 83.4], [181.6, 75.5], [176.5, 73.0],
                [175.0, 70.2], [174.0, 73.4], [165.1, 70.5], [177.0, 68.9], [192.0, 102.3],
                [176.5, 68.4], [169.4, 65.9], [182.1, 75.7], [179.8, 84.5], [175.3, 87.7],
                [184.9, 86.4], [177.3, 73.2], [167.4, 53.9], [178.1, 72.0], [168.9, 55.5],
                [157.2, 58.4], [180.3, 83.2], [170.2, 72.7], [177.8, 64.1], [172.7, 72.3],
                [165.1, 65.0], [186.7, 86.4], [165.1, 65.0], [174.0, 88.6], [175.3, 84.1],
                [185.4, 66.8], [177.8, 75.5], [180.3, 93.2], [180.3, 82.7], [177.8, 58.0],
                [177.8, 79.5], [177.8, 78.6], [177.8, 71.8], [177.8, 116.4], [163.8, 72.2],
                [188.0, 83.6], [198.1, 85.5], [175.3, 90.9], [166.4, 85.9], [190.5, 89.1],
                [166.4, 75.0], [177.8, 77.7], [179.7, 86.4], [172.7, 90.9], [190.5, 73.6],
                [185.4, 76.4], [168.9, 69.1], [167.6, 84.5], [175.3, 64.5], [170.2, 69.1],
                [190.5, 108.6], [177.8, 86.4], [190.5, 80.9], [177.8, 87.7], [184.2, 94.5],
                [176.5, 80.2], [177.8, 72.0], [180.3, 71.4], [171.4, 72.7], [172.7, 84.1],
                [172.7, 76.8], [177.8, 63.6], [177.8, 80.9], [182.9, 80.9], [170.2, 85.5],
                [167.6, 68.6], [175.3, 67.7], [165.1, 66.4], [185.4, 102.3], [181.6, 70.5],
                [172.7, 95.9], [190.5, 84.1], [179.1, 87.3], [175.3, 71.8], [170.2, 65.9],
                [193.0, 95.9], [171.4, 91.4], [177.8, 81.8], [177.8, 96.8], [167.6, 69.1],
                [167.6, 82.7], [180.3, 75.5], [182.9, 79.5], [176.5, 73.6], [186.7, 91.8],
                [188.0, 84.1], [188.0, 85.9], [177.8, 81.8], [174.0, 82.5], [177.8, 80.5],
                [171.4, 70.0], [185.4, 81.8], [185.4, 84.1], [188.0, 90.5], [188.0, 91.4],
                [182.9, 89.1], [176.5, 85.0], [175.3, 69.1], [175.3, 73.6], [188.0, 80.5],
                [188.0, 82.7], [175.3, 86.4], [170.5, 67.7], [179.1, 92.7], [177.8, 93.6],
                [175.3, 70.9], [182.9, 75.0], [170.8, 93.2], [188.0, 93.2], [180.3, 77.7],
                [177.8, 61.4], [185.4, 94.1], [168.9, 75.0], [185.4, 83.6], [180.3, 85.5],
                [174.0, 73.9], [167.6, 66.8], [182.9, 87.3], [160.0, 72.3], [180.3, 88.6],
                [167.6, 75.5], [186.7, 101.4], [175.3, 91.1], [175.3, 67.3], [175.9, 77.7],
                [175.3, 81.8], [179.1, 75.5], [181.6, 84.5], [177.8, 76.6], [182.9, 85.0],
                [177.8, 102.5], [184.2, 77.3], [179.1, 71.8], [176.5, 87.9], [188.0, 94.3],
                [174.0, 70.9], [167.6, 64.5], [170.2, 77.3], [167.6, 72.3], [188.0, 87.3],
                [174.0, 80.0], [176.5, 82.3], [180.3, 73.6], [167.6, 74.1], [188.0, 85.9],
                [180.3, 73.2], [167.6, 76.3], [183.0, 65.9], [183.0, 90.9], [179.1, 89.1],
                [170.2, 62.3], [177.8, 82.7], [179.1, 79.1], [190.5, 98.2], [177.8, 84.1],
                [180.3, 83.2], [180.3, 83.2]
            ],
            markArea: {
                silent: true,
                itemStyle: {
                    color: 'transparent',
                    borderWidth: 1,
                    borderType: 'dashed'
                },
                data: [
                    [
                        {
                            name: 'Male Data Range',
                            xAxis: 'min',
                            yAxis: 'min'
                        },
                        {
                            xAxis: 'max',
                            yAxis: 'max'
                        }
                    ]
                ]
            },
            markPoint: {
                data: [
                    { type: 'max', name: 'Max' },
                    { type: 'min', name: 'Min' }
                ]
            },
            markLine: {
                lineStyle: {
                    type: 'solid'
                },
                data: [{ type: 'average', name: 'Average' }, { xAxis: 170 }]
            }
        }
    ]
};
```

:::

#### 概率

> 一枚硬币抛 4 次，结果不一定是 2 次正面和 2 次反面。但如果是 4000 次，结果就很可能是 2000 次正面和
> 2000 次反面。

概率是一种用于研究随机现象的数学工具。它用来描述某事件发生的概率或可能性。

「概率论」最初是随着对扑克等概率游戏的研究而产生的。各种预测实际上都是以概率的形式来表达的。无论是预测地震或降雨的可能性，还是预测你在这门课程中能否取得
A 级成绩，我们都使用概率来进行分析。医生们也利用概率来评估接种疫苗后，是否有可能反而引发该疫苗原本旨在预防的疾病。

#### 关键术语

在统计学中，我们通常希望研究整个总体。所谓「总体」，指的是我们所研究的所有人、物或对象的集合。

为了研究这个总体，我们需要选取一个样本。「抽样」的目的就是从总体中挑选出一部分作为样本，通过研究这个样本来了解总体的情况。数据其实就是从总体中抽样所得的结果。

> 由于对整个人群进行调查需要耗费大量时间和金钱，因此抽样是一种非常实用的方法。如果你想计算学校全体学生的平均成绩，那么选取一部分在校学生作为样本是比较合理的。

从样本数据中，我们可以计算出各种「统计量」。统计量是一种用来表示样本特征的数值。例如，如果我们把某个数学班视为所有数学班的样本，那么该班级学生在学期末的平均成绩就是一个统计量的例子。

统计量实际上是对总体参数的估计。「参数」是指整个总体的某种数值特征，而统计量则可以用来估计这些参数。由于我们将所有数学班视为总体，那么所有数学班中每名学生的平均成绩则属于参数的范畴。

> 在「英雄联盟」中，某些版本英雄与总体英雄的参数严重不符（梅尔），我们称之为「超标」。

在统计学领域，人们最关心的问题之一就是：某个统计量在估计某个参数时的「准确性」如何。这种准确性实际上取决于样本在多大程度上能够代表总体。只有当样本具备总体的各种特征时，它才能被视为具有代表性的样本。在推断统计学中，我们既关注样本统计量，也关注总体参数。在后面的章节中，我们将利用样本统计量来检验所估计的总体参数是否准确。

「变量」通常用大写字母来表示，比如 X 和 Y。变量指的是可以针对总体中的每一个成员来确定的某种特征或数值。变量可以是数值型的，也可以是分类型的。

数值型变量具有相同的计量单位，比如以磅为单位的重量、以小时为单位的时间等。分类型变量则将人或事物归入不同的类别。

> 如果让 X 表示某位数学学生在学期末所获得的分数，那么 X 就是数值型变量。如果让 Y 表示一个人的政治党派归属，那么
> Y 的值可以是共和党、民主党或无党派人士。Y 就是分类型变量。我们可以对 X 的值进行数学运算（比如计算平均分），但对
> Y 的值进行数学运算则没有意义（因为无法计算出“平均党派归属”这样的概念）。

切比雪夫不等式

$$\mu({x\in X:|f(x)|\geq t})\leq\frac{1}{t^2}\int\_Xf^2d\mu.$$