2-1 data summary and display population mean for a finite population with n measurements, the mean...

2-1 Data Summary and Display


Population Mean

For a finite population with N measurements, the mean is

The sample mean is a reasonable estimate of the population mean.


Sample Variance and Sample Standard Deviation


The sample variance is

The sample standard deviation is


Computational formula for s2


Population Variance

When the population is finite and consists of N values, we may define the population variance as

The sample variance is a reasonable estimate of the population variance.

2-2 Stem-and-Leaf Diagram

Steps for Constructing a Stem-and-Leaf Diagram


Median = (40th + 41st )/2=(160+163)/2=161.5

Q1 = (n+1)/4=20.25 btn 20th & 21st

Q1= (143+145)/2 = 144Q2 = medianQ3 = 3(n+1)/4 = 60.75 Q3 = (181+181)/2 = 181

IQR = interquartile range = Q3-Q1

Percentiles, quartiles, and the medianrange

2-3 Histograms

A histogram is a more compact summary of data than a stem-and-leaf diagram. To construct a histogram for continuous data, we must divide the range of the data into intervals, which are usually called class intervals, cells, or bins. If possible, the bins should be of equal width to enhance the visual information in the histogram.

2-3 Histograms

2-3 Histograms

An important variation of the histogram is the Pareto chart. This chart is widely used in quality and process improvement studies where the data usually represent different types of defects, failure modes, or other categories of interest to the analyst. The categories are ordered so that the category with the largest number of frequencies is on the left, followed by the category with the second largest number of frequencies, and so forth.

2-3 Histograms

2-4 Box Plots

• The box plot is a graphical display that simultaneously describes several important features of a data set, such as center, spread, departure from symmetry, and identification of observations that lie unusually far from the bulk of the data.

• Whisker• Outlier• Extreme outlier

2-4 Box Plots

2-4 Box Plots

1st quartile = 143.5 3rd quartile = 181

2nd quartile = median = 161.5

IQR = Q3 – Q1 = 181 – 143.5 = 37.51.5 IQR = 56.25Q3 + 1.5 IQR = 237.25

IQR = Q3 – Q1 = 181 – 143.5 = 37.51.5 IQR = 56.25Q1 - 1.5 IQR = 143.5 – 56.25 = 87.25

2-4 Box Plots

OPTIONS NODATE NOOVP NONUMBER;DATA STRENGTH; INPUT STRENGTH @@;CARDS; 105 221 183 186 121 181 180 143 97 154 153 174 120 168 167 141 245 228 174 199 181 158 176 110 163 131 154 115 160 208 158 133 207 180 190 193 194 133 156 123 134 178 76 167 184 135 229 146 218 157 101 171 165 172 158 169 199 151 142 163 145 171 148 158 160 175 149 87 160 237 150 135 196 201 200 176 150 170 118 149 PROC UNIVARIATE DATA=STRENGTH PLOT NORMAL FREQ; VAR STRENGTH; histogram strength/vscale=count;TITLE 'DESCRIPTIVE STATISTICS AND GRAPHS';

/* PROC CHART DATA=STRENGTH; VBAR STRENGTH; VBAR STRENGTH/TYPE=PCT; HBAR STRENGTH/TYPE=CPCT DISCRETE; TITLE 'HISTOGRAM'; */

RUN; QUIT;

SAS code and output

DESCRIPTIVE STATISTICS AND GRAPHS

UNIVARIATE 프로시저 변수 : STRENGTH

적률 N 80 가중합 80 평균 162.6625 관측치 합 13013 표준 편차 33.7732363 분산 1140.63149 왜도 -0.0250253 첨도 0.24027117 제곱합 2206837 수정 제곱합 90109.8875 변동계수 20.7627672 평균의 표준 오차 3.7759626

기본 통계 측도 위치측도 변이측도 평균 162.6625 표준 편차 33.77324 중위수 161.5000 분산 1141 최빈값 158.0000 범위 169.00000 사분위 범위 37.00000

위치모수 검정 : Mu0=0 검정 --통계량 --- -------p 값 ------- 스튜던트의 t t 43.07842 Pr > |t| <.0001 부호 M 40 Pr >= |M| <.0001 부호 순위 S 1620 Pr >= |S| <.0001

정규성 검정

검정 ----통계량 ---- -------p 값 -------

Shapiro-Wilk W 0.991699 Pr < W 0.8911 Kolmogorov-Smirnov D 0.057091 Pr > D >0.1500 Cramer-von Mises W-Sq 0.046374 Pr > W-Sq >0.2500 Anderson-Darling A-Sq 0.270335 Pr > A-Sq >0.2500

분위수 ( 정의 5)

분위수 추정값

100% 최댓값 245.0 99% 245.0 95% 224.5 90% 204.0 75% Q3 181.0 50% 중위수 161.5 25% Q1 144.0 10% 119.0 5% 103.0 1% 76.0 0% 최솟값 76.0

DESCRIPTIVE STATISTICS AND GRAPHS UNIVARIATE 프로시저 변수 : STRENGTH

극 관측치 -----최소 ---- -----최대 ---- 값 관측치 값 관측치 76 43 221 2 87 68 228 18 97 9 229 47 101 51 237 70 105 1 245 17

빈도 수 백분율 백분율 백분율 값 빈도 셀 누적 값 빈도 셀 누적 값 빈도 셀 누적 76 1 1.3 1.3 149 2 2.5 31.3 180 2 2.5 73.8 87 1 1.3 2.5 150 2 2.5 33.8 181 2 2.5 76.3 97 1 1.3 3.8 151 1 1.3 35.0 183 1 1.3 77.5 101 1 1.3 5.0 153 1 1.3 36.3 184 1 1.3 78.8 105 1 1.3 6.3 154 2 2.5 38.8 186 1 1.3 80.0 110 1 1.3 7.5 156 1 1.3 40.0 190 1 1.3 81.3 115 1 1.3 8.8 157 1 1.3 41.3 193 1 1.3 82.5 118 1 1.3 10.0 158 4 5.0 46.3 194 1 1.3 83.8 120 1 1.3 11.3 160 3 3.8 50.0 196 1 1.3 85.0 121 1 1.3 12.5 163 2 2.5 52.5 199 2 2.5 87.5 123 1 1.3 13.8 165 1 1.3 53.8 200 1 1.3 88.8 131 1 1.3 15.0 167 2 2.5 56.3 201 1 1.3 90.0 133 2 2.5 17.5 168 1 1.3 57.5 207 1 1.3 91.3 134 1 1.3 18.8 169 1 1.3 58.8 208 1 1.3 92.5 135 2 2.5 21.3 170 1 1.3 60.0 218 1 1.3 93.8 141 1 1.3 22.5 171 2 2.5 62.5 221 1 1.3 95.0 142 1 1.3 23.8 172 1 1.3 63.8 228 1 1.3 96.3 143 1 1.3 25.0 174 2 2.5 66.3 229 1 1.3 97.5 145 1 1.3 26.3 175 1 1.3 67.5 237 1 1.3 98.8 146 1 1.3 27.5 176 2 2.5 70.0 245 1 1.3 100.0 148 1 1.3 28.8 178 1 1.3 71.3

SAS code and output


UNIVARIATE 프로시저 변수 : STRENGTH

줄기 잎 # 상자그림 24 5 1 0 23 7 1 0 22 189 3 | 21 8 1 | 20 0178 4 | 19 034699 6 | 18 0011346 7 +-----+ 17 0112445668 10 | | 16 0003357789 10 *--+--* 15 001344678888 12 | | 14 12356899 8 +-----+ 13 133455 6 | 12 013 3 | 11 058 3 | 10 15 2 | 9 7 1 | 8 7 1 0 7 6 1 0 ----+----+----+----+ 값 : (줄기 . 잎 )*10**+1

SAS code and output

정규 확률도 245+ +*+ | *++ | ***+ | *+ | *** | *** | *** | **** | **** | ***** | ****+ | ****+ | **+ | *** | +** | +++* | +++ * 75++* +----+----+----+----+----+----+----+----+----+----+ -2 -1 0 +1 +2

SAS code and output

2-5 Time Series Plots

• A time series or time sequence is a data set in which the observations are recorded in the order in which they occur. • A time series plot is a graph in which the vertical axis denotes the observed value of the variable (say x) and the horizontal axis denotes the time (which could be minutes, days, years, etc.). • When measurements are plotted as a time series, weoften see

•trends, •cycles, or •other broad features of the data

2-5 Time Series Plots

OPTIONS NODATE NOOVP NONUMBER LS=80;DATA STRENGTH; INPUT STRENGTH @@; N=_N_;CARDS; 105 221 183 186 121 181 180 143 97 154 153 174 120 168 167 141 245 228 174 199 181 158 176 110 163 131 154 115 160 208 158 133 207 180 190 193 194 133 156 123 134 178 76 167 184 135 229 146 218 157 101 171 165 172 158 169 199 151 142 163 145 171 148 158 160 175 149 87 160 237 150 135 196 201 200 176 150 170 118 149 SYMBOL INTERPOL=JOIN VALUE=DOT HEIGHT=1 LINE=1; PROC GPLOT DATA=STRENGTH; PLOT STRENGTH*N; TITLE 'TIME SERIES GRAPH FOR STRENGTH'; RUN; QUIT;

SAS code and output


87.5 112.5 137.5 162.5 187.5 212.5 237.5

0

5

10

15

20

25

30

35

백분

율

STRENGTH

SAS code and output

2-6 Multivariate Data

• The dot diagram, stem-and-leaf diagram, histogram, and box plot are descriptive displays for univariate data; that is, they convey descriptive information about a single variable. •Many engineering problems involve collecting and analyzing multivariate data, or data on several different variables.•In engineering studies involving multivariate data, often the objective is to determine the relationships among the variables or to build an empirical model.


Sample Correlation CoefficientThe strength of a linear relationship between two variables


Strong when 0.8≤ r ≤ 1, weak 0 ≤ r ≤ 0.5, and moderate otherwise

OPTIONS NODATE NOOVP NONUMBER LS=80;DATA SHAMPOO;INPUT FOAM SCENT COLOR RESIDUE REGION QUALITY;CARDS;6.3 5.3 4.8 3.1 1 914.4 4.9 3.5 3.9 1 873.9 5.3 4.8 4.7 1 825.1 4.2 3.1 3.6 1 835.6 5.1 5.5 5.1 1 834.6 4.7 5.1 4.1 1 844.8 4.8 4.8 3.3 1 906.5 4.5 4.3 5.2 1 848.7 4.3 3.9 2.9 1 978.3 3.9 4.7 3.9 1 935.1 4.3 4.5 3.6 1 823.3 5.4 4.3 3.6 1 845.9 5.7 7.2 4.1 2 877.7 6.6 6.7 5.6 2 807.1 4.4 5.8 4.1 2 845.5 5.6 5.6 4.4 2 846.3 5.4 4.8 4.6 2 824.3 5.5 5.5 4.1 2 794.6 4.1 4.3 3.1 2 813.4 5 3.4 3.4 2 836.4 5.4 6.6 4.8 2 815.5 5.3 5.3 3.8 2 844.7 4.1 5 3.7 2 834.1 4 4.1 4 2 80PROC CORR DATA=SHAMPOO; VAR FOAM SCENT COLOR RESIDUE REGION QUALITY; TITLE 'CORRELATIONS OF VARIABLES';PROC SGSCATTER DATA=SHAMPOO; MATRIX FOAM SCENT COLOR RESIDUE REGION QUALITY; TITLE 'MATRIX OF SCATTER PLOTS FOR THE SHAMPOO DATA';SYMBOL INTERPOL=NONE;PROC GPLOT DATA=SHAMPOO; PLOT QUALITY*FOAM=REGION; TITLE 'SCATTER PLOT OF SHAMPOO QUALITY VS. FORM';RUN; QUIT:

SAS code and output

CORRELATIONS OF VARIABLES

CORR 프로시저

6 개의 변수 : FOAM SCENT COLOR RESIDUE REGION QUALITY

단순 통계량 변수 N 평균 표준편차 합 최솟값 최댓값 FOAM 24 5.50417 1.44808 132.10000 3.30000 8.70000 SCENT 24 4.90833 0.67303 117.80000 3.90000 6.60000 COLOR 24 4.90000 1.02278 117.60000 3.10000 7.20000 RESIDUE 24 4.02917 0.69750 96.70000 2.90000 5.60000 REGION 24 1.50000 0.51075 36.00000 1.00000 2.00000 QUALITY 24 84.50000 4.36388 2028 79.00000 97.00000

피어슨 상관 계수 , N = 24 H0: Rho=0 가정하에서 Prob > |r|

FOAM SCENT COLOR RESIDUE REGION QUALITY

FOAM 1.00000 0.00219 0.32849 0.19315 -0.03233 0.51224 0.9919 0.1171 0.3658 0.8808 0.0105

SCENT 0.00219 1.00000 0.59877 0.49960 0.27826 -0.25166 0.9919 0.0020 0.0129 0.1880 0.2355

COLOR 0.32849 0.59877 1.00000 0.52414 0.45776 -0.19385 0.1171 0.0020 0.0086 0.0245 0.3641

RESIDUE 0.19315 0.49960 0.52414 1.00000 0.16476 -0.48923 0.3658 0.0129 0.0086 0.4417 0.0153

REGION -0.03233 0.27826 0.45776 0.16476 1.00000 -0.50718 0.8808 0.1880 0.0245 0.4417 0.0114

QUALITY 0.51224 -0.25166 -0.19385 -0.48923 -0.50718 1.00000 0.0105 0.2355 0.3641 0.0153 0.0114

SAS code and output

SAS code and output

2-1 data summary and display population mean for a finite population with n measurements, the mean...

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