教程m1统计学基础basic statistics04

04:统计学基础生活中充斥着变异variationGE

Global

Research

Green

BeltDMAICCourseCorrectDecision(1-

a)Type

IIErrorbType

IErroraCorrectDecision(1-

b)90807060Y

=

50X1X2Fuel

EconomyOctane

Level

Air

Temp2520151054.34.24.14.03.93.83.7Subgroup

0Sample

Mean110.60.50.40.30.20.10.0Sample

StDev1Objectives基本的统计学概念用统计学的方式思考数据类型描述统计学正态分布The

Normal

Distribution可能性ProbabilityZ值的计算Z-Calculations让你的数据可视化散点图Scatter

Plot箱线图Box

Plot柱状图Histogram矩阵图Matrix

Plot目录IndicatesMinitab

File04:

基本的统计学概念Page用统计学的方式思考4数据类型5统计学概念6-7描述统计学–集中趋势Central

Tendency8-10描述统计学–变异性Variability11-13描述统计学–练习Exercise14标准正态分布The

Normal

and

Standardized

Normal

Distribution15Z值的计算Z-Calculation16Z值的计算Z-Calculation–练习17让数据可视化–Histogram,

Box

Plot,

Scatter

Plot,

Matrix

Plot18-24参考文献25附录–Z-Tables26-28用统计学的方式思考如果我们的过程中存在太大的变异，那么变异就会阻止我们满足顾客需求生活中充满了变异variation–我们需要做的是:理解并且确定变异variation理解变异是如何影响质量的学会识别引起变异的根本原因能够减少和控制变异数据类型正态分布或者其他连续型分布类型

(e.g.,寿命试验中的威布尔分布

Weibull)连续性Variable可以用连续刻度的标尺测量二项式分布

binomialdistribution泊松分布

poissondistribution良品/不良品离散型Attribute

(aka

Discrete)根据属性计数数据类型不良个数

countdefects统计的概念总体Population

or

Universe所有产品，或所有被测量者Size

=

N样本Sample是总体的一部分Size

=

n推论统计–这个方法允许我们通过分析一部分样本来推断总体的情况统计的概念易拉罐里面的饮料体积都一样吗？目标值Target

Value=500

ml样本大小Sample

Size

=

n

=

24485490495500505510515大多数数据显示:能够看到大多数数据集中在一个平均值的位置呈现出一种变异性(或称为：散布dispersion/spread)描述统计学–集中趋势Central

Tendency均值(平均值)Mean(Average

Value)总体的均值Population

Mean总体大小Population

size=N测量总体中所有的成员m

=总体均值=所有测量值之和除以总体大小样本均值Sample

Mean

=

492.6

+502.9

+504.1++503.8

=501.924在我们这个例子中样本大小Sample

Size=n=2424个样本每个都进行测量X=样本均值Sample

Mean=所有测量值之和除以样本大小_样本的均值是总体均值的估计值中位数Median

(Population

and

Sample)将数据按照从小到大的顺序排列之后，中间那个数的数值如果样本大小是奇数，那么中位数Median=中间那个数值如果样本大小是偶数，那么中位数=中间那两个数的平均值502.75描述统计学–集中趋势Central

Tendency描述统计学–变异性Variability极差Range

(Population

and

sample)X最大值–X最小值=

509.6

–

491.4

=

18.2四分位数间距Interquartile

RangeQ3

–

Q1

=

505.1

–

497.8

=

7.3Q1(497.8)一分位数First

Quartile

-

Q125%的数比它小，75%的数比它大.Q2(Median

–

502.75)(中位数)二分位数

Second

Quartile-Q250%的数比它小，50%的数比它大.Q3(505.1)三分位数Third

Quartile

-

Q375%的数比它小，25%的数比它大.在Minitab中，用

“显示统计量

Descriptive

Statistics”计算四分位数样本大小n=

6X

=500.55500.4502.8499.8499.1498.1503.12

X

-X

0.0235.0630.5632.1036.5036.003差的平方和Sum

of

Squares

20.258样本方差Sample

Variance–S2(对总体方差的估计)5n-1S2

=

Sum

of

Squares

=

20.258

=4.05样本标准偏差Standard

Deviation

–

SS2S

= =

2.01描述统计学–变异性Variability总体方差Population

VarianceNσ2=

Sum

of

Squares

2

μ-X

每个数对平均值的差的平方和总体大小总体的标准偏差Population

Standard

Deviationσ2σ

=描述统计学–变异性Variability描述统计学–练习Exerciseinnx

=

Xi

-1

n12i

=1

n

-1

(

X

-

X

)s

=xx

-

x(x

-

x

)212345678910SXs

2s

现在有一组数：98,

98,

90,

104,

106,110,101,

104,

102,96请计算:均值Mean中位数Median极差Range四分位间距InterquartileRange方差Variance标准偏差StandardDeviation标准正态分布标准正态分布Mean

=

0s

=

1曲线所包含的面积=1Normal

Distribution0.10.080.060.040.020-50

-30

-10

10

30

50

70

90

110

130

150

170xProbability

Density

Functionsigma

=5sigma

=10sigma

=20sigma

=30非常有趣的是，许多过程分布的频率都可以被描述成正态分布.一个正态分布的基本参数是平均值和标准偏，正态分布的特点是以平均值为对称轴左右对称.所有正态分布都可以转换成标准正态分布Standardized

Normal

Distribution.所有正态分布都满足如图所示曲线的特点.68.3%95.4%99.7%0Z

=目标值和平均值之间包含的标准偏差的个数-3s-2s-1sm+1s+2s+3sZ=-3-2-1+1+2+3对于这个正态分布来说Z值的计算如果想知道在Z=2.5的右侧曲线下面的面积是多少，可以查标准正态分布的Z值表“Z-

tables”(见附录).也可以在Excel中用函数计算,Area=1-NORMSDIST(Z)For

Z

=

2.5,

Area

=

0.00621非标准正态分布中，Z=2.5右侧曲线下面包含的面积占总体的比例(即可能性)，和标准正态分布是一样的σZ

=

29.25

-

20.5

=2.53.5Z

=

X

-

μProbability=

0.00621202530151029.25m

=

20.52.5ss

=

3.5标准正态分布+2s

+3s-3s-2s

-1s

m

+1s2.5sorZ

=

2.5Z值的计算-练习5101520253023.3m

=

18.61s

=

2.68针对上图所示的正态分布23.3所在位置的Z值是多少?在分布中>23.3的可能性有多大?从总体中随机挑选一个样品，>23.3的可能性有多大?让数据可视化–箱线图Box

Plot什么时候会用箱线图?有一组或多组数.需要形象的观察数据的分布状态.想看不同组之间数据的差异.三分位数75th

Percentile(Q3)中位数Median/50th

Percentile(Q2)一分位数25th

Percentile(Q1)在下线之内的最小值LowerLimit

=

Q1

-

1.5

(Q3

-

Q1)异常点Outlier任何超出上下限的点Lower

orUpper

Limit.在上限之内的最大值UpperLimit

=

Q3

+

1.5(Q3

-

Q1)*箱线图练习某个化学反应过程里使用5%的NaOH溶液.在过去的一年里，向不同的供应商购买了总共100批次的NaOH溶液.目前已经发现了某些问题，初步怀疑购买的溶液浓度可能存在一定的变异.那么我们来看一看数据是什么样的.MinitabFileOpen

WorksheetNaOH_Analyses.MTWGraphBox

PlotSelect

Y

=

C5 X=

C6OK12343.54.55.56.57.5VendorNaOH-W%3.54.56.50510155.5Vendor

4Frequency让数据可视化–柱状图Histogram什么时候应用柱状图?有一组或几组数据.希望产看分布频度的趋势，尤其是散布的大小.Minitab

如何制作柱状图的？计算每个数据值的个数确定总的数据极差大小Range,

R,确定总的分区个数#of

classes,K.确定分区宽度=R/K计算每个分区内的数据个数.将计算出来的数据数制作成柱状图Note分区数是可以调整的(在“options”选项里面)柱状图练习为每个供应商的NaOH浓度制作柱状图1,2,3,4.MinitabFileOpen

WorksheetNaOH_Analyses.MTWGraphHistogramSelectC1,C2,C3,C4OK3.54.56.50510155.5Vendor

4Frequency让数据可视化–散点图Scatter

Plot什么时候应用?两组一一对应的连续性数据.打算查看两组数之间有无相关性.散点图练习MinitabFileOpen

WorksheetAGE_YRGE.MTWGraphPlotSelect

Data

(Graph

Variables)OK为“Age”和“Years

at

GE”两列数据制作散点图然数据可视化–矩阵图Matrix

Plot什么时候应用?有多组一一对应的(连续性)数据.希望在一张大图“Big

Picture”

里面观察整体的两两对应的相关性.矩阵图练习有这样一个项目，提高野生黑熊的数量,你的团队要到森林里面去对黑熊做标记,首先要将它们麻醉，然后称重，另外还要测量尺寸.现在想根据黑熊的体重来估计它们的身体尺寸.第一步，我们想在一张大图里面看一看这些数据之间有无相关性.MinitabFileOpen

WorksheetBears.MTWGraphMatrix

PlotSelect:

Age,

Head.L,

Head.W

&

WeightOK134.7550.2516.12511.3758.55.550.25134.7539214811.37516.1255.58.5148392AgeHead.LHead.WWeight参考文献Breyfogle,

Forrest

W.

III.

Implementing

Six

Sigma

–

Smarter

SolutionsUsingStatistical

Methods.

John

Wiley

and

Sons;

2nd

edition,

2003.Levine,

David

M.,

Ramsey,

Patricia

P.

&

Smidt,

Robert

K.

AppliedStatistics

For

Engineers

and

Scientists.

Prentice

Hall;

2001.Ryan,

Barbara &

Joiner,

Brian

L.

Minitab™

Handbook.

Brooks

Cole;

4thedition,

2000.StatSoft,

Inc.

Electronic

Statistics

Textbook.

StatSoft,

Tulsa,

OK;

2000.Web:

Click

eStatSoft

BookGE

PowerSystems.

GE

DMAIC

Book

of

Knowledge

Navigator.

2001.Web:

Click

SixSigmaCafé

Link04:统计学基础生活中充满变异附录Z00.010.020.030.040.050.060.070.080.0900.50000.49600.49200.48800.48400.48010.47610.47210.46810.46410.10.46020.45620.45220.44830.44430.44040.43640.43250.42860.42470.20.42070.41680.41290.40900.40520.40130.39740.39360.38970.38590.30.38210.37830.37450.37070.36690.36320.35940.35570.35200.34830.40.34460.34090.33720.33360.33000.32640.32280.31920.31560.31210.50.30850.30500.30150.29810.29460.29120.28770.28430.28100.27760.60.27430.27090.26760.26430.26110.25780.25460.25140.24830.24510.70.24200.23890.23580.23270.22960.22660.22360.22060.21770.21480.80.21190.20900.20610.20330.20050.19770.19490.19220.18940.18670.90.18410.18140.17880.17620.17360.17110.16850.16600.16350.161110.15870.15620.15390.15150.14920.14690.14460.14230.14010.13791.10.13570.13350.13140.12920.12710.12510.12300.12100.11900.11701.20.11510.11310.11120.10930.10750.10560.10380.10200.10030.09851.30.09680.09510.09340.09180.09010.08850.08690.08530.08380.08231.40.08080.07930.07780.07640.07490.07350.07210.07080.06940.06811.50.06680.06550.06430.06300.06180.06060.05940.05820.05710.05591.60.05480.05370.05260.05160.05050.04950.04850.04750.04650.04551.70.04460.04360.04270.04180.04090.04010.03920.03840.03750.03671.80.03590.03510.03440.03360.03290.03220.03140.03070.03010.02941.90.02870.02810.02740.02680.02620.02560.02500.02440.02390.023320.02280.02220.02170.02120.02070.02020.01970.01920.01880.01832.10.01790.01740.01700.01660.01620.01580.01540.01500.01460.01432.20.01390.01360.01320.01290.01250.01220.01190.01160.01130.01102.30.01070.01040.01020.00990.00960.00940.00910.00890.00870.00842.40.00820.00800.00780.00750.00730.00710.00690.00680.00660.00642.50.00620.00600.00590.00570.00550.00540.00520.00510.00490.00482.60.00470.00450.00440.00430.00410.00400.00390.00380.00370.00362.70.00350.00340.00330.00320.00310.00300.00290.00280.00270.00262.80.00260.00250.00240.00230.00230.00220.00210.00210.00200.00192.90.00190.00180.00180.00170.00160.00160.00150.00150.00140.001430.00130.00130.00130.00120.00120.00110.00110.00110.00100.00103.19.68E-049.36E-049.04E-048.74E-048.45E-048.16E-047.89E-047.62E-047.36E-047.11E-043.26.87E-046.64E-046.41E-046.19E-045.98E-045.77E-045.57E-045.38E-045.19E-045.01E-043.34.83E-044.67E-044.50E-044.34E-044.19E-044.04E-043.90E-043.76E-043.62E-043.50E-043.43.37E-043.25E-043.13E-043.02E-042.91E-042.80E-042.70E-042.60E-042.51E-042.42E-043.52.33E-042.24E-042.16E-042.08E-042.00E-041.93E-041.85E-041.79E-041.72E-041.65E-043.61.59E-041.53E-041.47E-041.42E-041.36E-041.31E-041.26E-041.21E-041.17E-041.12E-043.71.08E-041.04E-049.96E-059.58E-059.20E-058.84E-058.50E-058.16E-057.84E-057.53E-053.87.24E-056.95E-056.67E-056.41E-056.15E-055.91E-055.67E-055.44E-055.22E-055.01E-053.94.81E-054.62E-054.43E-054.25E-054.08E-053.91E-053.75E-053.60E-053.45E-053.31E-05Single-Tail

Z

Table (values

from

0.00

to

3.99)Calculates

theprobability

of

a

USLdefect

above

thegiven

value

of

zzCalculates

theprobability

of

a

USLdefect

above

thegiven

value

of

zZ00.010.020.030.040.050.060.070.080.0943.17E-053.04E-052.91E-052.79E-052.67E-052.56E-052.45E-052.35E-052.25E-052.16E-054.12.07E-051.98E-051.90E-051.81E-051.74E-051.66E-051.59E-051.52E-051.46E-051.40E-054.21.34E-051.28E-051.22E-051.17E-051.12E-051.07E-051.02E-059.78E-069.35E-068.94E-064.38.55E-068.17E-067.81E-067.46E-067.13E-066.81E-066.51E-066.22E-065.94E-065.67E-064.45.42E-065.17E-064.94E-064.72E-064.50E-064.30E-064.10E-063.91E-063.74E-063.56E-064.53.40E-063.24E-063.09E-062.95E-062.82E-062.68E-062.56E-062.44E-062.33E-062.22E-064.62.11E-062.02E-061.92E-061.83E-061.74E-061.66E-061.58E-061.51E-061.44E-061.37E-064.71.30E-061.24E-061.18E-061.12E-061.07E-061.02E-069.69E-079.22E-078.78E-078.35E-074.87.94E-077.56E-077.19E-076.84E-076.50E-076.18E-075.88E-075.59E-075.31E-075.05E-074.94.80E-074.56E-074.33E-074.12E-073.91E-073.72E-073.53E-073.35E-073.18E-073.02E-0752.87E-072.73E-072.59E-072.46E-072.33E-072.21E-072.10E-071.99E-071.89E-071.79E-075.11.70E-071.61E-071.53E-071.45E-071.38E-071.30E-071.24E-071.17E-071.11E-071.05E-075.29.98E-089.46E-088.96E-088.49E-088.04E-087.62E-087.22E-086.84E-086.47E-086.13E-085.35.80E-085.49E-085.20E-084.92E-084.66E-084.41E-084.17E-083.95E-083.73E-083.53E-085.43.34E-083.16E-082.99E-082.82E-082.67E-082.52E-082.39E-082.26E-082.13E-082.01E-085.51.90E-081.80E-081.70E-081.61E-081.52E-081.43E-081.35E-081.28E-081.21E-081.14E-085.61.07E-081.01E-089.57E-099.04E-098.53E-098.04E-097.59E-097.16E-096.75E-096.37E-095.76.01E-095.67E-095.34E-095.04E-094.75E-094.48E-094.22E-093.98E-093.75E-093.53E-095.83.33E-093.13E-092.95E-092.78E-092.62E-092.47E-092.32E-092.19E-092.06E-091.94E-095.91.82E-091.72E-091.62E-091.52E-091.43E-091.35E-091.27E-091.19E-091.12E-091.05E-0969.90E-109.31E-108.75E-108.23E-107.73E-107.27E-106.83E-106.42E-106.03E-105.67E-106.15.32E-105.00E-104.70E-104.41E-104.14E-103.89E-103.65E-103.43E-103.22E-103.02E-106.22.83E-102.66E-102.50E-102.34E-102.20E-102.06E-101.93E-101.81E-101.70E-101.59E-106.31.49E-101.40E-101.31E-101.23E-101.15E-101.08E-101.01E-109.49E-118.89E-118.33E-116.47.80E-117.31E-116.85E-116.41E-116.00E-115.62E-115.26E-114.92E-114.61E-114.31E-116.54.04E-113.78E-113.53E-113.30E-113.09E-112.89E-112.70E-112.53E-112.36E-112.21E-116.62.07E-111.93E-111.81E-111.