數(shù)理統(tǒng)計(jì)CH3描述統(tǒng)計(jì)32課件

1、第三章 描述統(tǒng)計(jì) Descriptive Statistics9/25/20221王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)本章內(nèi)容3.1 離散樣本的頻數(shù)統(tǒng)計(jì)3.2 連續(xù)樣本的頻數(shù)統(tǒng)計(jì)3.3 三維統(tǒng)計(jì)圖3.4 基于觀測的統(tǒng)計(jì)計(jì)算3.5 基于頻數(shù)的統(tǒng)計(jì)計(jì)算3 描述統(tǒng)計(jì)9/25/20222王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)3.4 基于觀測的統(tǒng)計(jì)計(jì)算Statistic Calculating based Observation3 描述統(tǒng)計(jì)9/25/20223王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì) 基于觀測的統(tǒng)計(jì)計(jì)算，指直接利用抽樣獲得的n個(gè)樣本觀察值(observation) x1,x2,xn計(jì)算均值、方差、標(biāo)準(zhǔn)差、變異系

2、數(shù)、協(xié)方差、相關(guān)系數(shù)、偏度和峰度等樣本統(tǒng)計(jì)量觀察值(statistic observation)引言3.4 基于觀測的統(tǒng)計(jì)計(jì)算9/25/20224王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì) 無論離散變量還是連續(xù)變量，樣本觀察值(observation)、統(tǒng)計(jì)出的頻數(shù)和頻率都是有限離散的。因此，無論離散變量還是連續(xù)變量，統(tǒng)計(jì)量觀察值(statistic observation)的計(jì)算本質(zhì)上都是離散運(yùn)算，其算法及公式均相同。后面討論的問題均屬于離散運(yùn)算，如求和、求均值、求方差、求協(xié)方差等。引言3.4 基于觀測的統(tǒng)計(jì)計(jì)算9/25/20225王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)樣本均值反映樣本觀察值的集中趨勢（cent

3、ral tendency）樣本均值mean比較(1)計(jì)算樣本均值簡化：3.4 基于觀測的統(tǒng)計(jì)計(jì)算9/25/20226王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)(1)計(jì)算樣本均值序號xyx2y2xy19698921696049408299100980110000990031041081081611664112324969792169409931259493883686498742合計(jì)4894964788549326485943.4 基于觀測的統(tǒng)計(jì)計(jì)算9/25/20227王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)比較樣本方差反映樣本觀察值的離散程度（variation level）樣本標(biāo)準(zhǔn)差樣本方差(2)計(jì)算樣本方差及標(biāo)

4、準(zhǔn)差3.4 基于觀測的統(tǒng)計(jì)計(jì)算9/25/20228王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)樣本方差variance樣本標(biāo)準(zhǔn)差 standard deviation(2)計(jì)算樣本方差及標(biāo)準(zhǔn)差Corrected Sum of Square 先計(jì)算3.4 基于觀測的統(tǒng)計(jì)計(jì)算9/25/20229王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)(2)計(jì)算樣本方差及標(biāo)準(zhǔn)差Corrected Sum of Square 校正的平方和離差平方和3.4 基于觀測的統(tǒng)計(jì)計(jì)算9/25/202210王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)(2)計(jì)算樣本方差及標(biāo)準(zhǔn)差序號xyx2y2xy196989216960494082991009801100009900

5、31041081081611664112324969792169409931259493883686498742合計(jì)4894964788549326485943.4 基于觀測的統(tǒng)計(jì)計(jì)算9/25/202211王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)(2)計(jì)算樣本方差及標(biāo)準(zhǔn)差序號xyx2y2xy19698921696049408299100980110000990031041081081611664112324969792169409931259493883686498742合計(jì)4894964788549326485943.4 基于觀測的統(tǒng)計(jì)計(jì)算9/25/202212王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)(2)計(jì)算

6、樣本方差及標(biāo)準(zhǔn)差序號xyx2y2xy19698921696049408299100980110000990031041081081611664112324969792169409931259493883686498742合計(jì)4894964788549326485943.4 基于觀測的統(tǒng)計(jì)計(jì)算9/25/202213王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)變異系數(shù)variation coefficient(3)計(jì)算樣本變異系數(shù)序號xyx2y2xy1969892169604940829910098011000099003104108108161166411232496979216940993125949388

7、3686498742合計(jì)4894964788549326485943.4 基于觀測的統(tǒng)計(jì)計(jì)算9/25/202214王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)樣本協(xié)方差(covariance)協(xié)方差計(jì)算公式(4)計(jì)算樣本協(xié)方差Corrected Crossproduct 校正的叉積和定義來自SAS3.4 基于觀測的統(tǒng)計(jì)計(jì)算9/25/202215王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)(4)計(jì)算樣本協(xié)方差Corrected Crossproduct 校正的乘積和(校正的叉積和)3.4 基于觀測的統(tǒng)計(jì)計(jì)算9/25/202216王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)先計(jì)算兩樣本的觀測和與叉積和(4)計(jì)算樣本協(xié)方差序號xyx2y2x

8、y19698921696049408299100980110000990031041081081611664112324969792169409931259493883686498742合計(jì)4894964788549326485943.4 基于觀測的統(tǒng)計(jì)計(jì)算9/25/202217王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)(4)計(jì)算樣本協(xié)方差序號xyx2y2xy19698921696049408299100980110000990031041081081611664112324969792169409931259493883686498742合計(jì)489496478854932648594再計(jì)算校正的叉積和3

9、.4 基于觀測的統(tǒng)計(jì)計(jì)算9/25/202218王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)序號xyx2y2xy19698921696049408299100980110000990031041081081611664112324969792169409931259493883686498742合計(jì)489496478854932648594(4)計(jì)算樣本協(xié)方差3.4 基于觀測的統(tǒng)計(jì)計(jì)算9/25/202219王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)校正的叉積和x的離差平方和y的離差平方和由離差平方和計(jì)算相關(guān)系數(shù)(5)計(jì)算樣本相關(guān)系數(shù)Pearson Correlation Coefficient樣本相關(guān)系數(shù)Correlat

10、ion Coefficient3.4 基于觀測的統(tǒng)計(jì)計(jì)算9/25/202220王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)最后計(jì)算次計(jì)算先計(jì)算校正的叉積和x的離差平方和y的離差平方和(5)計(jì)算樣本相關(guān)系數(shù)Pearson Correlation Coefficient3.4 基于觀測的統(tǒng)計(jì)計(jì)算9/25/202221王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)協(xié)方差由協(xié)方差計(jì)算相關(guān)系數(shù)(5)計(jì)算樣本相關(guān)系數(shù)Pearson Correlation Coefficientx或y的樣本方差3.4 基于觀測的統(tǒng)計(jì)計(jì)算9/25/202222王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)(5)計(jì)算樣本相關(guān)系數(shù)Pearson Correlation Co

11、efficient樣本相關(guān)系數(shù)Correlation Coefficient樣本方差樣本協(xié)方差3.4 基于觀測的統(tǒng)計(jì)計(jì)算9/25/202223王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)(5)計(jì)算樣本相關(guān)系數(shù)序號xyx2y2xy19698921696049408299100980110000990031041081081611664112324969792169409931259493883686498742合計(jì)489496478854932648594先計(jì)算樣本和、樣本平方和與樣本叉積和3.4 基于觀測的統(tǒng)計(jì)計(jì)算9/25/202224王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)計(jì)算x與y的校正叉積和(5)計(jì)算樣本相關(guān)系

12、數(shù)計(jì)算校正的叉積和3.4 基于觀測的統(tǒng)計(jì)計(jì)算9/25/202225王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)計(jì)算x的離差平方和(5)計(jì)算樣本相關(guān)系數(shù)計(jì)算x的離差平方和3.4 基于觀測的統(tǒng)計(jì)計(jì)算9/25/202226王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)計(jì)算y的離差平方和(5)計(jì)算樣本相關(guān)系數(shù)計(jì)算y的離差平方和3.4 基于觀測的統(tǒng)計(jì)計(jì)算9/25/202227王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)(5)計(jì)算樣本相關(guān)系數(shù)序號xyx2y2xy19698921696049408299100980110000990031041081081611664112324969792169409931259493883686498742合計(jì)4

13、89496478854932648594最后計(jì)算相關(guān)系數(shù)3.4 基于觀測的統(tǒng)計(jì)計(jì)算9/25/202228王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)(6)計(jì)算樣本偏度Skewness總體的偏度或偏斜系數(shù)定義如下：偏度或偏度系數(shù)是三階中心矩與標(biāo)準(zhǔn)差三次方之比中心偏差三次方的期望與標(biāo)準(zhǔn)差三次方之比3.4 基于觀測的統(tǒng)計(jì)計(jì)算9/25/202229王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)正態(tài)分布的偏度或偏斜系數(shù)為：(6)計(jì)算樣本偏度Skewness3.4 基于觀測的統(tǒng)計(jì)計(jì)算9/25/202230王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)(6)計(jì)算樣本偏度SkewnessSPSS的樣本偏度計(jì)算：3.4 基于觀測的統(tǒng)計(jì)計(jì)算9/25/2022

14、31王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)SAS的樣本偏度計(jì)算：(6)計(jì)算樣本偏度Skewness3.4 基于觀測的統(tǒng)計(jì)計(jì)算9/25/202232王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)序號xyx2y2xy19698921696049408299100980110000990031041081081611664112324969792169409931259493883686498742合計(jì)489496478854932648594x的離差三次方和(6)計(jì)算樣本偏度Skewness3.4 基于觀測的統(tǒng)計(jì)計(jì)算9/25/202233王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)序號xyx2y2xy19698921696049408

15、299100980110000990031041081081611664112324969792169409931259493883686498742合計(jì)489496478854932648594(6)計(jì)算樣本偏度Skewnessx的離差平方和3.4 基于觀測的統(tǒng)計(jì)計(jì)算9/25/202234王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)序號xyx2y2xy19698921696049408299100980110000990031041081081611664112324969792169409931259493883686498742合計(jì)489496478854932648594(6)計(jì)算樣本偏度Skewn

16、ess3.4 基于觀測的統(tǒng)計(jì)計(jì)算9/25/202235王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)序號xyx2y2xy19698921696049408299100980110000990031041081081611664112324969792169409931259493883686498742合計(jì)489496478854932648594均值右側(cè)數(shù)據(jù)較分散(6)計(jì)算樣本偏度Skewness3.4 基于觀測的統(tǒng)計(jì)計(jì)算9/25/202236王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)(7)計(jì)算樣本峰度Kurtosis總體的峰度或峰突系數(shù)的定義如下： 峰度或峰突系數(shù)是：4階中心矩與標(biāo)準(zhǔn)差4次方之比再減3；中心偏差4次方

17、的期望與標(biāo)準(zhǔn)差4次方之比再減3。3.4 基于觀測的統(tǒng)計(jì)計(jì)算9/25/202237王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)(7)計(jì)算樣本峰度Kurtosis正態(tài)分布的峰度或峰突系數(shù)為3.4 基于觀測的統(tǒng)計(jì)計(jì)算9/25/202238王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)(7)計(jì)算樣本峰度KurtosisSPSS的樣本峰度計(jì)算：3.4 基于觀測的統(tǒng)計(jì)計(jì)算9/25/202239王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)SAS的樣本峰度計(jì)算：(7)計(jì)算樣本峰度Kurtosis3.4 基于觀測的統(tǒng)計(jì)計(jì)算9/25/202240王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)序號xyx2y2xy1969892169604940829910098011000

18、0990031041081081611664112324969792169409931259493883686498742合計(jì)489496478854932648594x的離差四次方和(7)計(jì)算樣本峰度Kurtosis3.4 基于觀測的統(tǒng)計(jì)計(jì)算9/25/202241王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)序號xyx2y2xy19698921696049408299100980110000990031041081081611664112324969792169409931259493883686498742合計(jì)489496478854932648594x的離差平方和(7)計(jì)算樣本峰度Kurtosis3.4

19、 基于觀測的統(tǒng)計(jì)計(jì)算9/25/202242王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)序號xyx2y2xy19698921696049408299100980110000990031041081081611664112324969792169409931259493883686498742合計(jì)489496478854932648594(7)計(jì)算樣本峰度Kurtosis3.4 基于觀測的統(tǒng)計(jì)計(jì)算9/25/202243王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)序號xyx2y2xy1969892169604940829910098011000099003104108108161166411232496979216940993

20、1259493883686498742合計(jì)489496478854932648594略低于正態(tài)分布(7)計(jì)算樣本峰度Kurtosis3.4 基于觀測的統(tǒng)計(jì)計(jì)算9/25/202244王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)3.5 基于頻數(shù)的統(tǒng)計(jì)計(jì)算Statistic Calculating based Frequency Distribution3 描述統(tǒng)計(jì)9/25/202245王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)引言3.5 基于頻數(shù)的統(tǒng)計(jì)計(jì)算 基于頻數(shù)的統(tǒng)計(jì)計(jì)算，先利用抽樣獲得的n個(gè)樣本觀察值(observation) x1,x2,xn做出頻數(shù)分布表，再利用該表的組中值(或觀測)和頻數(shù)數(shù)據(jù)計(jì)算均值、方差、標(biāo)準(zhǔn)

21、差、變異系數(shù)、偏度和峰度等統(tǒng)計(jì)量觀察值(statistic observation)。9/25/202246王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)(1)計(jì)算樣本均值樣本值頻數(shù)頻率xn(x)fn(x)040.0201180.0902310.1553410.2054380.1905290.1456210.1057100.050850.025920.0101010.0053.5 基于頻數(shù)的統(tǒng)計(jì)計(jì)算9/25/202247王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)樣本值頻數(shù)頻率xn(x)fn(x)040.0201180.0902310.1553410.2054380.1905290.1456210.1057100.0508

22、50.025920.0101010.005(1)計(jì)算樣本均值3.5 基于頻數(shù)的統(tǒng)計(jì)計(jì)算9/25/202248王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)組中值頻數(shù)頻率xn(x)fn(x)45.510.0149.050.0552.5110.1156.0230.2359.5350.3563.0130.1366.570.0770.020.0273.530.03(1)計(jì)算樣本均值3.5 基于頻數(shù)的統(tǒng)計(jì)計(jì)算9/25/202249王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)組中值頻數(shù)頻率xn(x)fn(x)45.510.0149.050.0552.5110.1156.0230.2359.5350.3563.0130.1366.570

23、.0770.020.0273.530.03(1)計(jì)算樣本均值3.5 基于頻數(shù)的統(tǒng)計(jì)計(jì)算9/25/202250王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)(2)計(jì)算樣本方差和標(biāo)準(zhǔn)差樣本值頻數(shù)頻率xn(x)fn(x)040.0201180.0902310.1553410.2054380.1905290.1456210.1057100.050850.025920.0101010.0053.5 基于頻數(shù)的統(tǒng)計(jì)計(jì)算9/25/202251王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)樣本值頻數(shù)頻率xn(x)fn(x)040.0201180.0902310.1553410.2054380.1905290.1456210.1057100.

24、050850.025920.0101010.005(2)計(jì)算樣本方差和標(biāo)準(zhǔn)差3.5 基于頻數(shù)的統(tǒng)計(jì)計(jì)算9/25/202252王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)組中值頻數(shù)頻率xn(x)fn(x)45.510.0149.050.0552.5110.1156.0230.2359.5350.3563.0130.1366.570.0770.020.0273.530.03(2)計(jì)算樣本方差和標(biāo)準(zhǔn)差3.5 基于頻數(shù)的統(tǒng)計(jì)計(jì)算9/25/202253王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)組中值頻數(shù)頻率xn(x)fn(x)45.510.0149.050.0552.5110.1156.0230.2359.5350.3563.0

25、130.1366.570.0770.020.0273.530.03(2)計(jì)算樣本方差和標(biāo)準(zhǔn)差3.5 基于頻數(shù)的統(tǒng)計(jì)計(jì)算9/25/202254王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)(3)計(jì)算變異系數(shù)樣本值頻數(shù)頻率xn(x)fn(x)040.0201180.0902310.1553410.2054380.1905290.1456210.1057100.050850.025920.0101010.0053.5 基于頻數(shù)的統(tǒng)計(jì)計(jì)算9/25/202255王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)組中值頻數(shù)頻率xn(x)fn(x)45.510.0149.050.0552.5110.1156.0230.2359.5350.35

26、63.0130.1366.570.0770.020.0273.530.03(3)計(jì)算變異系數(shù)3.5 基于頻數(shù)的統(tǒng)計(jì)計(jì)算9/25/202256王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)(4)計(jì)算樣本偏度SAS的樣本偏度(skewness)計(jì)算3.5 基于頻數(shù)的統(tǒng)計(jì)計(jì)算9/25/202257王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)樣本值頻數(shù)頻率xn(x)fn(x)040.0201180.0902310.1553410.2054380.1905290.1456210.1057100.050850.025920.0101010.005(4)計(jì)算樣本偏度右偏3.5 基于頻數(shù)的統(tǒng)計(jì)計(jì)算9/25/202258王玉順：數(shù)理統(tǒng)計(jì)03

27、_描述統(tǒng)計(jì)組中值頻數(shù)頻率xn(x)fn(x)45.510.0149.050.0552.5110.1156.0230.2359.5350.3563.0130.1366.570.0770.020.0273.530.03(4)計(jì)算樣本偏度右偏3.5 基于頻數(shù)的統(tǒng)計(jì)計(jì)算9/25/202259王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)(5)計(jì)算樣本峰度SAS的樣本峰度(kurtosis)計(jì)算3.5 基于頻數(shù)的統(tǒng)計(jì)計(jì)算9/25/202260王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)樣本值頻數(shù)頻率xn(x)fn(x)040.0201180.0902310.1553410.2054380.1905290.1456210.105710

28、0.050850.025920.0101010.005(5)計(jì)算樣本峰度略高于正態(tài)分布3.5 基于頻數(shù)的統(tǒng)計(jì)計(jì)算9/25/202261王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)組中值頻數(shù)頻率xn(x)fn(x)45.510.0149.050.0552.5110.1156.0230.2359.5350.3563.0130.1366.570.0770.020.0273.530.03(5)計(jì)算樣本峰度略低于正態(tài)分布3.5 基于頻數(shù)的統(tǒng)計(jì)計(jì)算9/25/202262王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)(6)關(guān)于協(xié)方差和相關(guān)系數(shù) 協(xié)方差和相關(guān)系數(shù)表征兩變量的線性相關(guān)關(guān)系，由于兩變量是在相同條件下成對觀測的，而且一般尺度也不

29、同，故它們各自統(tǒng)計(jì)的頻數(shù)分布已破壞了兩變量的相關(guān)關(guān)系。因此，不存在“基于頻數(shù)的協(xié)方差計(jì)算和相關(guān)系數(shù)計(jì)算”。計(jì)算協(xié)方差和相關(guān)系數(shù)，采用基于樣本觀測的算法。3.5 基于頻數(shù)的統(tǒng)計(jì)計(jì)算9/25/202263王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)3.6本章小結(jié)Summarization3 描述統(tǒng)計(jì)9/25/202264王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)3.6 本章小結(jié)離散樣本的頻數(shù)頻率分布表樣本值頻數(shù)累積頻數(shù)頻率累積頻率0440.0200.020118220.0900.110231530.1550.265341940.2050.4704381320.1900.6605291610.1450.8056211820.1050.9107101920.0500.960851970.0250.985921990.0100.9951012000.0051.000(1)離散樣本的頻數(shù)頻率統(tǒng)計(jì)9/25/202265王玉順：數(shù)理統(tǒng)計(jì)03_描述統(tǒng)計(jì)3.6 本章小結(jié)(1)離散樣本的頻數(shù)頻率統(tǒng)計(jì)離散樣本頻數(shù)分布柱