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.

2-1 Data Summary and Display Sample Variance and Sample Standard Deviation

2-1 Data Summary and Display The sample variance is The sample standard deviation is

2-1 Data Summary and Display Computational formula for s 2

2-1 Data Summary and Display 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

2-2 Stem-and-Leaf Diagram

Median = (40 th + 41 st )/2=( )/2=161.5 Q 1 = (n+1)/4=20.25  btn 20 th & 21 st Q1= ( )/2 = 144 Q 2 = median Q 3 = 3(n+1)/4 = Q3 = ( )/2 = 181 IQR = interquartile range = Q3-Q1 Percentiles, quartiles, and the median range

2-2 Stem-and-Leaf Diagram

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

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

1 st quartile = rd quartile = nd quartile = median = IQR = Q 3 – Q 1 = 181 – = IQR = Q IQR = IQR = Q 3 – Q 1 = 181 – = IQR = Q IQR = – = 87.25

2-4 Box Plots

OPTIONS NODATE NOOVP NONUMBER; DATA STRENGTH; INPUT STRENGTH CARDS; 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 관측치 합 표준 편차 분산 왜도 첨도 제곱합 수정 제곱합 변동계수 평균의 표준 오차 기본 통계 측도 위치측도 변이측도 평균 표준 편차 중위수 분산 1141 최빈값 범위 사분위 범위 위치모수 검정 : Mu0=0 검정 -- 통계량 p 값 스튜던트의 t t 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 Pr < W Kolmogorov-Smirnov D Pr > D > Cramer-von Mises W-Sq Pr > W-Sq > Anderson-Darling A-Sq Pr > A-Sq > 분위수 ( 정의 5) 분위수 추정값 100% 최댓값 % % % % Q % 중위수 % Q % % % % 최솟값 76.0

DESCRIPTIVE STATISTICS AND GRAPHS UNIVARIATE 프로시저 변수 : STRENGTH 극 관측치 최소 최대 ---- 값 관측치 값 관측치 빈도 수 백분율 백분율 백분율 값 빈도 셀 누적 값 빈도 셀 누적 값 빈도 셀 누적 SAS code and output

DESCRIPTIVE STATISTICS AND GRAPHS UNIVARIATE 프로시저 변수 : STRENGTH 줄기 잎 # 상자그림 | | | | | | *--+--* | | | | | | | 값 : ( 줄기. 잎 )*10**+1 SAS code and output 정규 확률도 *+ | *++ | ***+ | *+ | *** | **** | ***** | ****+ | **+ | *** | +** | +++* 75++*

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, we often 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; 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

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.

2-6 Multivariate Data

Sample Correlation Coefficient The strength of a linear relationship between two variables

2-6 Multivariate Data Strong when 0.8≤ r ≤ 1, weak 0 ≤ r ≤ 0.5, and moderate otherwise

2-6 Multivariate Data

OPTIONS NODATE NOOVP NONUMBER LS=80; DATA SHAMPOO; INPUT FOAM SCENT COLOR RESIDUE REGION QUALITY; CARDS; PROC 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 SCENT COLOR RESIDUE REGION QUALITY 피어슨 상관 계수, N = 24 H0: Rho=0 가정하에서 Prob > |r| FOAM SCENT COLOR RESIDUE REGION QUALITY FOAM SCENT COLOR RESIDUE REGION QUALITY SAS code and output