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STATISTICS
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HISTOGRAM Definition Pictorial representation of a set of date using a bar graph which divides the measurements into cells Purpose Determine the shape of the data set Interpret the shape of the data set Determine dispersion Determine central tendency Compare to specifications Construction Find the largest and smallest values Subtract to calculate range Select the number of cells
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HISTOGRAM - QUINCUNX
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HISTOGRAM Determine the width of each cell Divide range by # cells
Round off to convenient (odd) number Compute cell boundary 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 Use tic marks to assign each measurement to it’s cell Count the tic marks to complete frequency chart Construct the graph Vertical axis is frequency Horizontal axis shows cell boundary Draw bars Overlay specification limits Interpret capability and shape
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Number of Cells # Data Points # Classes K Under 50 5-7 50 – 100 6-10
50 – 100 – Over
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Figure 4.15 Histogram Cell Description
Donna C.S. Summers Quality, 3e Copyright ©2003 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.
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Figure 4.16 Cell Boundaries and Midpoints
Donna C.S. Summers Quality, 3e Copyright ©2003 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.
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Figure 4.17 Clutch Plate Thickness Histogram
Donna C.S. Summers Quality, 3e Copyright ©2003 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.
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HISTOGRAM Calculate First Cell Boundaries
HISTOGRAM Calculate First Cell Boundaries Largest Smallest Range Range No. Cells Cell width = Round to S Smallest Half of last place Midpoint 0.9815 /2 Cell width Lower boundary L 0.9815 /2 Cell width Upper boundary
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HISTOGRAM CELL # CELL START CELL END MID POINT TALLY FREQUENCY
0.9815 0.9855 0.9895 0.9935 0.9975 1.0015 1.0055 1.0095 1.0135 1.0175 8 9 17 16 19 11 6 3 2 ////\ /// ////\ //// ////\ ////\ ////\ // ////\ ////\ ////\ / ////\ ////\ ////\ //// ////\ ////\ / ////\ / /// // 20 LSL .986 USL 1.012 16 12 8 4 .9855 .9895 .9935 .9975 1.0015 1.0055 1.0095 1.0135 1.0175 .9815
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FREQUENCY TABLE 9.00-9.19 9.1 l 1 9.20-9.39 9.3 lllll llll 9
lllll lllll lllll l 16 lllll lllll lllll lllll lllll ll 27 lllll lllll lllll lllll lllll lllll l 31 lllll lllll lllll lllll ll 22 lllll lllll ll 12 ll 2 llll 5 CLASS NO. BOUNDARIES MID POINT FREQUENCY TOTAL
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9.60-9.79 9.7 lllll lllll lllll lllll lllll ll 27
lllll lllll lllll lllll lllll lllll l 31 lllll lllll lllll lllll ll 22 lllll lllll ll 12 ll 2 llll 5 CLASS NO. BOUNDARIES MID POINT FREQUENCY TOTAL SPECIFICATIONS 9.0 10.5 7.5 10 20 30
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HISTOGRAM SPECIFICATIONS 9+-1.5 9.0 10.5 7.5 10 20 30
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Histogram Types & Interpretations
Sorted or Fixed Process Limit Normally Distributed GENERAL RIGHT PRECIPICE Process Shifting (tool wears part gets bigger) Poor Measurement Discrimination LEFT SKEW COMB
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Mixed Lots – Bimodal Distribution
Averages have Shifted further Averages are Close to the same PLATEAU TWIN PEAK If this is Nominal, Check for “Salting” Very Different Processes ISOLATED PEAK
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Figure 4.25 Discrepancies in Histograms
Donna C.S. Summers Quality, 3e Copyright ©2003 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.
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Plastic used for dashboards are examined for cracking
Lot A Lot B Two lots of plastic Mixed together Lots Separated Donna C.S. Summers Quality, 3e
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Figure 4.9 Tally Sheet for Thickness of Clutch Plate
Donna C.S. Summers Quality, 3e Copyright ©2003 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.
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Figure 4.10 Frequency Distribution for Clutch Plate Thickness
Check this out Donna C.S. Summers Quality, 3e Copyright ©2003 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.
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Figure 4.11 Monogram and Embroidery Arm
Donna C.S. Summers Quality, 3e Copyright ©2003 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.
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Histogram Cell Width 11.145 11.165 11.185 11.205 11.225 11.245 11.265 11.285 11.305 Donna C.S. Summers Quality, 3e Copyright ©2003 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.
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Figure 4.18 Histogram for Monogramming and Embroidery Arm Data
Donna C.S. Summers Quality, 3e Copyright ©2003 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.
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NORMAL DISTRIBUTION
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CENTRAL TENDENCY
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Figure 4.33 Different Distributions with Same Averages and Ranges
Donna C.S. Summers Quality, 3e Copyright ©2003 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.
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? TIME TIME
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Figure 4.34 Frequency Diagrams of the Amount of Pipe Laid per Day in Feet
Donna C.S. Summers Quality, 3e Copyright ©2003 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.
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Figure 4.39 Distribution of Sample Averages
Donna C.S. Summers Quality, 3e Copyright ©2003 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.
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Figure 4.20 Symmetrical Histogram with Smooth Curve Overlay
Donna C.S. Summers Quality, 3e Copyright ©2003 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.
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ANALYSIS OF CURVE SHAPE LOCATION SPREAD
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Figure 4.41 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 All rights reserved.
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Six Sigma Normal Curve by Gerald Lee Quality Digest November 30, 2007
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Area Under Tail in Curve -Z
1 2 3 -1 -3 -2 2.3% Z 15.9% 1 2 3 -1 -3 -2 2.3% Z 15.9%
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Area Under Tail in Curve -Z
-1 1 Z 1.5 6.7% 2 -2 -3 3
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Figure 4.42 Normal Curve for Left-Reading Z Table
Donna C.S. Summers Quality, 3e Copyright ©2003 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.
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Figure 4.43 Example 4.34: Area Under the Curve, Xi = 0.0624
Donna C.S. Summers Quality, 3e Copyright ©2003 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.
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Figure 4.44 Example 4.34: Area Under the Curve, Xi = 0.0629
Donna C.S. Summers Quality, 3e Copyright ©2003 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.
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Figure 4. 45 Example 4. 34: Area Under the Curve Between 0. 0623 and 0
Donna C.S. Summers Quality, 3e Copyright ©2003 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.
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STATISTICAL MEASURES Central Tendency Dispersion
Mean - average of a set of values (sum of values divided by the # of values) Median - The middle number in a set of values Mode - The most often occurring value in a set of values Dispersion Range - The largest value in a sample minus the smallest Variance - The sum of the differences of each value and the average squared divided by the degrees of freedom (number of values or number of values minus 1) Standard Deviation - The square root of the variance 10, 11, 12, 13, 14 1,2,3,3,4,4,4,5,5,6,7,8.9
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Mean Formula Statistical Formula Simplified Formula
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Figure 4.29 Calculating Medians
Donna C.S. Summers Quality, 3e Copyright ©2003 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.
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Figure 4.30 Calculating Modes
Donna C.S. Summers Quality, 3e Copyright ©2003 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.
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Figure 4.40 The Normal Curve
Donna C.S. Summers Quality, 3e Copyright ©2003 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.
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Median Mode Mean Median Mean Mode Figure 4.21 Skewness
Donna C.S. Summers Quality, 3e Copyright ©2003 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.
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Figure 4.32 Comparison of Mean, Mode, and Median for the Clutch Plate
Donna C.S. Summers Quality, 3e Copyright ©2003 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.
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Example Calculations RANGE 36 35 39 40 Find: Range Average 38
41 Find: Range Average Standard Deviation AVERAGE STANDARD DEVIATION
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Manual Calculation of Standard Deviation
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TERMS Statistics Population Sample
The collection, tabulation, analysis, interpretation, and presentation of numerical data Population A collection of all possible elements, values, or items associated with a situation Sample A subset of elements or measurements taken from a population Must be randomized to represent the population Deductive Statistics (descriptive statistics) Describe a population or a complete group of data Each entity in the population must be studied Inductive Statistics Deals with a limited amount of data or a representative sample of the population Used for samples to predict the population
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TERMS Data Accuracy Precision Measurement Error Variable
Those quality characteristics that can be measured Attribute Those quality characteristics that are observed to be either present or absent, conforming or nonconforming Relative Those quality characteristics which are assigned a value which cannot be actually measured Accuracy How far from the actual or real value the measurement is The location of X or X bar Precision The ability to repeat a series of measurements and get the same value each time Repeatability The variability of measurements Measurement Error The difference between a value measured and the true value
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Accuracy VS Precision High Accuracy Low High Precision Low
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