STATISTICS

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

HISTOGRAM - QUINCUNX

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

Number of Cells # Data Points # Classes K Under 50 5-7 50 – 100 6-10 50 – 100 6-10 100 – 250 7-12 Over 250 10-20

Figure 4.15 Histogram Cell Description Donna C.S. Summers Quality, 3e Copyright ©2003 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Figure 4.16 Cell Boundaries and Midpoints Donna C.S. Summers Quality, 3e Copyright ©2003 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Figure 4.17 Clutch Plate Thickness Histogram Donna C.S. Summers Quality, 3e Copyright ©2003 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

HISTOGRAM Calculate First Cell Boundaries 1.002 0.995 1.000 1.002 1.005 1.000 0.997 1.007 0.992 0.995 0.997 1.013 1.001 0.985 1.002 0.990 1.012 1.005 0.985 1.006 1.000 1.002 1.006 1.007 0.993 0.984 0.994 0.998 1.006 1.002 0.987 0.994 1.002 0.997 1.008 0.992 0.988 1.015 0.987 1.006 0.994 0.990 0.991 1.002 0.988 1.007 1.008 0.990 1.001 0.999 0.995 0.989 0.982 0.995 1.002 0.987 1.004 0.992 1.002 0.992 0.991 1.001 0.996 0.997 0.984 1.004 0.993 1.003 0.992 1.010 1.004 1.010 0.984 0.997 1.008 0.990 1.021 0.995 0.987 0.989 1.003 0.992 0.992 0.990 1.014 1.000 0.985 1.019 1.002 0.986 0.996 0.984 1.005 1.016 1.012 HISTOGRAM Calculate First Cell Boundaries 1.021 Largest -0.982 Smallest 0.039 Range Range 0.039 No. Cells 10 Cell width = 0.0039 Round to .004 S 0.982 Smallest -0.0005 Half of last place 0.9815 Midpoint 0.9815 -0.0020 1/2 Cell width 0.9795 Lower boundary L 0.9815 +0.0020 1/2 Cell width 0.9835 Upper boundary

HISTOGRAM CELL # CELL START CELL END MID POINT TALLY FREQUENCY 1 0.9795 0.9835 2 0.9835 0.9875 3 0.9875 0.9915 4 0.9915 0.9955 5 0.9955 0.9995 6 0.9995 1.0035 7 1.0035 1.0075 8 1.0075 1.0115 9 1.0115 1.0155 10 1.0155 1.0195 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

FREQUENCY TABLE 9.00-9.19 9.1 l 1 9.20-9.39 9.3 lllll llll 9 9.40-9.59 9.5 lllll lllll lllll l 16 9.60-9.79 9.7 lllll lllll lllll lllll lllll ll 27 9.80-9.99 9.9 lllll lllll lllll lllll lllll lllll l 31 10.10-10.19 10.1 lllll lllll lllll lllll ll 22 10.20-10.39 10.3 lllll lllll ll 12 10.40-10.59 10.5 ll 2 10.60-10.79 10.7 llll 5 10.80-10.99 10.9 0 CLASS NO. BOUNDARIES MID POINT FREQUENCY TOTAL

9.60-9.79 9.7 lllll lllll lllll lllll lllll ll 27 9.80-9.99 9.9 lllll lllll lllll lllll lllll lllll l 31 10.10-10.19 10.1 lllll lllll lllll lllll ll 22 10.20-10.39 10.3 lllll lllll ll 12 10.40-10.59 10.5 ll 2 10.60-10.79 10.7 llll 5 10.80-10.99 10.9 0 CLASS NO. BOUNDARIES MID POINT FREQUENCY TOTAL SPECIFICATIONS 9+-1.5 9.0 10.5 7.5 10 20 30

HISTOGRAM SPECIFICATIONS 9+-1.5 9.0 10.5 7.5 10 20 30

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

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

Figure 4.25 Discrepancies in Histograms Donna C.S. Summers Quality, 3e Copyright ©2003 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

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

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 07458 All rights reserved.

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 07458 All rights reserved.

Figure 4.11 Monogram and Embroidery Arm Donna C.S. Summers Quality, 3e Copyright ©2003 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

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 07458 All rights reserved.

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 07458 All rights reserved.

NORMAL DISTRIBUTION

CENTRAL TENDENCY

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 07458 All rights reserved.

? TIME TIME

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 07458 All rights reserved.

Figure 4.39 Distribution of Sample Averages Donna C.S. Summers Quality, 3e Copyright ©2003 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

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 07458 All rights reserved.

ANALYSIS OF CURVE SHAPE LOCATION SPREAD

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 07458 All rights reserved.

Six Sigma Normal Curve by Gerald Lee Quality Digest November 30, 2007

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%

Area Under Tail in Curve -Z -1 1 Z 1.5 6.7% 2 -2 -3 3

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 07458 All rights reserved.

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 07458 All rights reserved.

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 07458 All rights reserved.

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 07458 All rights reserved.

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

Mean Formula Statistical Formula Simplified Formula

Figure 4.29 Calculating Medians Donna C.S. Summers Quality, 3e Copyright ©2003 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Figure 4.30 Calculating Modes Donna C.S. Summers Quality, 3e Copyright ©2003 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Figure 4.40 The Normal Curve Donna C.S. Summers Quality, 3e Copyright ©2003 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

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 07458 All rights reserved.

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 07458 All rights reserved.

Example Calculations RANGE 36 35 39 40 Find: Range Average 38 41 Find: Range Average Standard Deviation AVERAGE STANDARD DEVIATION

Manual Calculation of Standard Deviation

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

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

Accuracy VS Precision High Accuracy Low High Precision Low