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2/20/2014 Statistics The collection, tabulation, analysis, interpretation, and presentation of numerical data Deductive Statistics (descriptive statistics) à Describe a population or a complete group of data à Each entity in the population must be studied Inductive Statistics à Deals with a limited amount of data or a representative sample of the population à Used for samples to predict the population Basic Concepts of Variation Variation defined = Change over time Everything varies – no two things are exactly the same Variation can be measured (quantified) Statistical Thinking A PHILOSOPHY of learning and action based on the following fundamental principles: Work needs to be viewed as a process that can be studied and improved All work occurs in a system of interconnected processes Variation exists in all processes Understanding and reducing variation are keys to success* DATA Data Sources = computer or manual reports, log books, special studies, vendor data, memos and notes, peoples’ memory* Variation has a pattern Groups of like measurements tend to cluster around a middle value dd e a ue VARIABLE = Continuous Scale, Data that is measured The shape of a distribution curve can be determined for any process ATTRIBUTE = Go/No‐Go, Data that is counted, Characteristic with limited choices/aspects Variations from assignable causes tend to distort the normal distribution curve Common Terms Tolerance – range of acceptance Specifications – given values by a customer Population – all of a group, collection of all possible elements, values, or items associated with a situation Sample – part of a group, subset of elements or measurements taken from a population, Must be randomized to represent the population Quality Tool #6: CONTROL CHARTS Has average and upper & lower control limits, tolerances (range of acceptance) Used to determine: à Process centering, averages and ranges à Variability of process à Capability of a process à Control of a process à Non-normal patterns or trends 1 2/20/2014 Control Chart showing centerline, UCL, LCL Run Chart Data over time Donna C.S. Summers Quality, 3e Copyright ©2003 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Control Chart of Inner/Outer Diameter Concentricity Donna C.S. Summers Quality, 3e Copyright ©2003 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Quality Tool #7: HISTOGRAM Pictorial representation of a set of data using a bar graph which divides the measurements into cells Indicate status, not cause of problem Process: à Determine/Interpret the shape of the data set (show frequency distributions and range) à Determine dispersion & central tendency à Compare to specifications Donna C.S. Summers Quality, 3e Copyright ©2003 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Percentage of Measurements Falling Within Each Standard Deviation Donna C.S. Summers Quality, 3e Copyright ©2003 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. 2 2/20/2014 HISTOGRAM TYPES CENTRAL TENDENCY “Middle” value of a distribution (determined from the mean, median, or mode) GENERAL LEFT SKEW RIGHT PRECIPICE Typical value representing a population “Mean” is not enough to go off of COMB PLATEAU TWIN PEAK Different Distributions with Same Averages and Ranges ISOLATED PEAK Construction - Clutch Plate Thickness Frequency Distribution for Clutch Plate Thickness # of Categorizations affect what we see and don’t see!! Here, using too few of cells hides a potential problem Check this out Donna C.S. Summers Quality, 3e Copyright ©2003 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. HISTOGRAM Construction à Find the largest and smallest values, subtract to calculate range à Select the # cells (divisions) à Determine the width of each cell Divide range by # cells Round off to convenient (odd) number à Compute cell boundaries Use smallest value of set as midpoint of first cell Subtract and add half of cell width to midpoint for first cell boundary Add cell width to each upper boundary until value is greater than the largest value of set à Use tick marks to assign each measurement to it’s cell à Count the tick marks to complete frequency chart à Construct the graph Vertical axis is frequency, Horizontal axis shows cell boundary Draw bars à Overlay specification limits à Interpret capability and shape Donna C.S. Summers Quality, 3e Copyright ©2003 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Construction - Number of Cells à NOTE: Different rules for this – affects what we see in histograms (no single value can fit into 2 cells) # Data Points Under 50 50 – 100 100 – 250 Over 250 # Classes K 5-7 6-10 7-12 10-20 One rule = (Sqroot of n) 3 2/20/2014 1.005 0.995 1.002 1.006 1.000 0.984 0.987 0.992 0.994 1.002 0.994 0.994 0.988 0.990 1.006 0.998 1.002 1.015 0.991 1.007 1.006 0.997 0.987 1.002 0.993 1.002 1.008 1.006 0.988 1.007 0.995 0.987 0.991 1.004 1.008 0.989 1.004 1.001 0.993 0.990 0.982 0.992 0.996 1.003 1.001 0.995 1.002 0.997 0.992 0.999 1.002 0.992 0.984 1.010 1.004 0.990 1.003 1.000 0.996 1.010 1.021 0.992 0.985 0.984 0.984 0.995 0.992 1.019 1.005 0.997 0.987 0.990 1.002 1.016 1.008 0.989 1.014 0.986 1.012 L S 1.021 -0.982 0.039 Range No. Cells (sum of values divided by the # of values) 0.039 10 Cell width = 0.0039 Round to .004 0.982 -0.0005 0.9815 STATISTICAL MEASURES Central Tendency à Mean ‐ average of a set of values Largest Smallest Range Smallest Half of last place Midpoint 0.9815 -0.0020 0.9795 1 2 3 4 5 6 7 8 9 10 CELL START X X X Σ X n 11 + 12 + 13 + 13 + 14 + 15 6 78 = 6 = 13 = MID POINT TALLY ////\ /// ////\ //// ////\ ////\ ////\ ////\ ////\ //// ////\ ////\ ////\ ////\ ////\ / /// // 0.9815 0.9855 0.9895 0.9935 0.9975 1.0015 1.0055 1.0095 1.0135 1.0175 LSL .986 16 FREQUENCY ////\ // ////\ / ////\ //// / 8 9 17 16 9 19 11 6 3 2 USL 1.012 12 1/2 Cell width Lower boundary = 0.9835 0.9875 0.9915 0.9955 0.9995 1.0035 1.0075 1.0115 1.0155 1.0195 20 1 decimal place beyond! Why not use half of .004??? 8 4 0.9815 +0.0020 1/2 Cell width 0.9835 Upper boundary X CELL END 0.9795 0.9835 0.9875 0.9915 0.9955 0.9995 1.0035 1.0075 1.0115 1.0155 1.0175 1.002 0.992 0.985 0.985 1.0135 1.000 1.007 1.001 1.005 1.0015 0.995 0.997 1.013 1.012 CELL # .9815 1.002 1.000 0.997 0.990 HISTOGRAM Construction Calculate First Cell Boundaries .9895 HISTOGRAM Construction Mean Formula Statistical Formula à Median ‐ The middle number in a set of values Æ 10, 11, 12, 13, 14 à Mode ‐ The most often occurring value in a set of values Æ 1,2,3,3,4,4,4,5,5,6,7,8.9 Dispersion à Range ‐ The largest value in a sample minus R = X l arg e − X small the smallest à Variance ‐ The sum of the differences of each Σ( X − X n )2 Σ( X − X n ) 2 2 or value and the average squared divided by the S = n n −1 degrees of freedom (number of values or number of values minus 1) 2 2 à Standard Deviation ‐ The square root of the S = Σ ( X − X n ) or Σ ( X − X n ) n −1 n variance Median Donna C.S. Summers Quality, 3e ΣX n 11 + 12 + 13 + 13 + 14 + 15 X= 6 78 X= 6 X = 13 X= Simplified Formula Mode(s) Copyright ©2003 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Donna C.S. Summers Quality, 3e Copyright ©2003 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. 4 2/20/2014 Normal Curve Skewness Median Mode Mean Median Copyright ©2003 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. R = X l arg e − X small Seven Numbers: 36 35 Find: 39 Range 40 Average 35 Standard Deviation 38 41 AVERAGE STANDARD DEVIATION S= S= RANGE R = 41 − 35 R=6 ΣX n 36 + 35 + 39 + 40 + 35 + 38 + 41 X= 7 264 X= 7 X = 37.7 X= Donna C.S. Summers Quality, 3e Mean Mode Copyright ©2003 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Manual Calculation of Standard Deviation Three numbers: 7 10 13 AVERAGE STANDARD DEVIATION Σ( X − Xn) 2 n −1 (37.7 − 36)2 + (37.7 − 35)2 + (37.7 − 39)2 + (37.7 − 40)2 + (37.7 − 35)2 + (37.7 − 38)2 + (37.7 − 41)2 n −1 S = 2.43 TERMS à Data Variable ‐ Quality characteristics that can be measured Attribute ‐ Quality characteristics that are observed to be either present or absent, conforming or nonconforming Relative ‐ Quality characteristics which are assigned a value which cannot be actually measured à Accuracy How far from the actual or real value the measurement is The location of X or X bar à Precision The ability to repeat a series of measurements and get the same value each time Repeatability The variability of measurements à Measurement Error The difference between a value measured and the true value 5