11/8/2015 1 statistics. 11/8/2015 2 histogram u definition –pictorial representation of a set of...

04/21/2304/21/23 11

STATISTICS

04/21/2304/21/23 22

HISTOGRAM DefinitionDefinition

– Pictorial representation of a set of date using a bar graph which Pictorial representation of a set of date using a bar graph which divides the measurements into cellsdivides the measurements into cells

PurposePurpose– Determine the shape of the data setDetermine the shape of the data set– Interpret the shape of the data setInterpret the shape of the data set– Determine dispersionDetermine dispersion– Determine central tendencyDetermine central tendency– Compare to specificationsCompare to specifications

ConstructionConstruction– Find the largest and smallest valuesFind the largest and smallest values– Subtract to calculate rangeSubtract to calculate range– Select the number of cellsSelect the number of cells

04/21/2304/21/23 33

HISTOGRAM - QUINCUNX

04/21/2304/21/23 44

– Determine the width of each cellDetermine the width of each cell Divide range by # cellsDivide range by # cells Round off to convenient (odd) numberRound off to convenient (odd) number

– Compute cell boundaryCompute cell boundary Use smallest value of set as midpoint of first cellUse smallest value of set as midpoint of first cell Subtract and add half of cell width to midpoint for first cell Subtract and add half of cell width to midpoint for first cell

boundaryboundary Add cell width to each upper boundary until value is greater than Add cell width to each upper boundary until value is greater than

the largest value of setthe largest value of set– Use tic marks to assign each measurement to it’s cellUse tic marks to assign each measurement to it’s cell– Count the tic marks to complete frequency chartCount the tic marks to complete frequency chart– Construct the graphConstruct the graph

Vertical axis is frequencyVertical axis is frequency Horizontal axis shows cell boundaryHorizontal axis shows cell boundary Draw barsDraw bars

– Overlay specification limitsOverlay specification limits– Interpret capability and shapeInterpret capability and shape

HISTOGRAM

04/21/2304/21/23 55

Number of Cells

# Data Points# Data Points # Classes K# Classes KUnder 50Under 50 5-75-750 – 10050 – 100 6-106-10100 – 250100 – 250 7-127-12Over 250Over 250 10-2010-20

04/21/2304/21/23 66

Figure 4.15 Histogram Cell Description

Donna C.S. SummersQuality, 3e

Copyright ©2003 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458

All rights reserved.

04/21/2304/21/23 77

Figure 4.16 Cell Boundaries and Midpoints




04/21/2304/21/23 88

Figure 4.17 Clutch Plate Thickness Histogram




04/21/2304/21/23 99

HISTOGRAM1.002 0.995 1.000 1.002 1.0051.000 0.997 1.007 0.992 0.9950.997 1.013 1.001 0.985 1.0020.990 1.012 1.005 0.985 1.006

1.000 1.002 1.006 1.007 0.9930.984 0.994 0.998 1.006 1.0020.987 0.994 1.002 0.997 1.0080.992 0.988 1.015 0.987 1.0060.994 0.990 0.991 1.002 0.988

1.007 1.008 0.990 1.001 0.9990.995 0.989 0.982 0.995 1.0020.987 1.004 0.992 1.002 0.9920.991 1.001 0.996 0.997 0.9841.004 0.993 1.003 0.992 1.010

1.004 1.010 0.984 0.997 1.0080.990 1.021 0.995 0.987 0.9891.003 0.992 0.992 0.990 1.0141.000 0.985 1.019 1.002 0.9860.996 0.984 1.005 1.016 1.012

S

L

1.021 Largest-0.982 Smallest 0.039 Range

0.982 Smallest-0.0005 Half of last place 0.9815 Midpoint

0.9815+0.0020 1/2 Cell width 0.9835 Upper boundary

Range 0.039No. Cells 10

Cell width = 0.0039Round to .004

Calculate First Cell Boundaries

0.9815-0.0020 1/2 Cell width 0.9795 Lower boundary

04/21/2304/21/23 1010.981

5

.985

5

.989

5

.993

5

.997

5

1.00

15

1.00

55

1.00

95

1.01

35

1.01

75

4

8

12

16

20LSL.986

USL1.012

HISTOGRAMCELL # CELL START CELL END MID POINT TALLY FREQUENCY

1 0.9795 0.98352 0.9835 0.98753 0.9875 0.99154 0.9915 0.99555 0.9955 0.99956 0.9995 1.00357 1.0035 1.00758 1.0075 1.01159 1.0115 1.015510 1.0155 1.0195

0.98150.98550.98950.99350.99751.00151.00551.00951.01351.0175

89171691911632

////\ ///////\ ////////\ ////\ ////\ //////\ ////\ ////\ /////\ ////////\ ////\ ////\ ////////\ ////\ /////\ //////

04/21/2304/21/23 1111

FREQUENCY TABLE

11 9.00-9.199.00-9.19 9.19.1 ll 1 122 9.20-9.399.20-9.39 9.39.3 lllll lllllllll llll 9 933 9.40-9.599.40-9.59 9.59.5 lllll lllll lllll llllll lllll lllll l 161644 9.60-9.799.60-9.79 9.79.7 lllll lllll lllll lllll lllll lllllll lllll lllll lllll lllll ll 272755 9.80-9.999.80-9.99 9.99.9 lllll lllll lllll lllll lllll lllll llllll lllll lllll lllll lllll lllll l 313166 10.10-10.1910.10-10.19 10.110.1 lllll lllll lllll lllll lllllll lllll lllll lllll ll 222277 10.20-10.3910.20-10.39 10.310.3 lllll lllll lllllll lllll ll 121288 10.40-10.5910.40-10.59 10.510.5 llll 2 299 10.60-10.7910.60-10.79 10.710.7 llllllll 5 51010 10.80-10.9910.80-10.99 10.910.9 0 0

CLASSCLASSNO.NO.

CLASSCLASS BOUNDARIESBOUNDARIES

MIDMIDPOINTPOINT FREQUENCYFREQUENCY TOTALTOTAL

04/21/2304/21/23 1212

11 9.00-9.199.00-9.19 9.19.1 ll 1 122 9.20-9.399.20-9.39 9.39.3 lllll lllllllll llll 9 933 9.40-9.599.40-9.59 9.59.5 lllll lllll lllll llllll lllll lllll l 161644 9.60-9.799.60-9.79 9.79.7 lllll lllll lllll lllll lllll lllllll lllll lllll lllll lllll ll 272755 9.80-9.999.80-9.99 9.99.9 lllll lllll lllll lllll lllll lllll llllll lllll lllll lllll lllll lllll l 313166 10.10-10.1910.10-10.19 10.110.1 lllll lllll lllll lllll lllllll lllll lllll lllll ll 222277 10.20-10.3910.20-10.39 10.310.3 lllll lllll lllllll lllll ll 121288 10.40-10.5910.40-10.59 10.510.5 llll 2 299 10.60-10.7910.60-10.79 10.710.7 llllllll 5 51010 10.80-10.9910.80-10.99 10.910.9 0 0

CLASSCLASSNO.NO.

CLASSCLASS BOUNDARIESBOUNDARIES

MIDMIDPOINTPOINT FREQUENCYFREQUENCY TOTALTOTAL

SPECIFICATIONS 9+-1.5SPECIFICATIONS 9+-1.5

9.09.0 10.510.57.57.5

001010

2020

3030

04/21/2304/21/23 1313

HISTOGRAM

SPECIFICATIONSSPECIFICATIONS9+-1.59+-1.5

9.09.0 10.510.57.57.5

00

1010

2020

3030

04/21/2304/21/23 1414

Histogram Types & Interpretations

GENERAL

LEFT SKEW

RIGHT PRECIPICE

COMB

Normally Normally DistributedDistributed

Sorted orSorted orFixed Process LimitFixed Process Limit

Process ShiftingProcess Shifting(tool wears part gets bigger)(tool wears part gets bigger)

Poor Measurement Poor Measurement DiscriminationDiscrimination

04/21/2304/21/23 1515

PLATEAU TWIN PEAK

ISOLATED PEAK

Mixed Lots – Bimodal DistributionMixed Lots – Bimodal Distribution

Averages are Averages are Close to the sameClose to the same

Averages have Averages have Shifted furtherShifted further

Very Different Very Different ProcessesProcesses

If this is Nominal, If this is Nominal, Check for “Salting”Check for “Salting”

04/21/2304/21/23 1616

Figure 4.25 Discrepancies in Histograms




04/21/2304/21/23 1717

Two lots of plasticTwo lots of plasticMixed togetherMixed together

Lot ALot A

Lot BLot B

Plastic used for dashboards are Plastic used for dashboards are examined for crackingexamined for cracking


Lots SeparatedLots Separated

04/21/2304/21/23 1818

Figure 4.9 Tally Sheet for Thickness of Clutch Plate




04/21/2304/21/23 1919

Figure 4.10 Frequency Distribution for Clutch Plate Thickness




Check this outCheck this out

04/21/2304/21/23 2020

Figure 4.11 Monogram and Embroidery Arm




04/21/2304/21/23 2121Donna C.S. SummersQuality, 3e



Histogram Cell Width

305.1102.0285.11

285.1102.0265.11

265.1102.0245.11

245.1102.0225.11

225.1102.0205.11

205.1102.0185.11

185.1102.0165.11

165.1102.0145.11

145.11)int(01.0155.11

155.11005.16.11

01.2

02.0

02.07

16.1129.11

ervalhalf

11.1

4511

.145

11.1

6511

.165

11.1

8511

.185

11.2

0511

.205

11.2

2511

.225

11.2

4511

.245

11.2

6511

.265

11.2

8511

.285

11.3

0511

.305

04/21/2304/21/23 2222

Figure 4.18 Histogram for Monogramming and Embroidery Arm Data




04/21/2304/21/23 2323

NORMAL DISTRIBUTION

04/21/2304/21/23 2424

CENTRAL TENDENCY

04/21/2304/21/23 2525

Figure 4.33 Different Distributions with Same Averages and Ranges




?TIME

TIME

04/21/2304/21/23 2727

Figure 4.34 Frequency Diagrams of the Amount of Pipe Laid per Day in Feet




04/21/2304/21/23 2828

Figure 4.39 Distribution of Sample Averages




04/21/2304/21/23 2929

Figure 4.20 Symmetrical Histogram with Smooth Curve Overlay




04/21/2304/21/23 3030

ANALYSIS OF CURVE

SHAPESHAPE LOCATIONLOCATION SPREADSPREAD

04/21/2304/21/23 3131

Figure 4.41 Percentage of Measurements Falling Within Each Standard Deviation




04/21/2304/21/23 3232

Six Sigma Normal Curve

by Gerald Leeby Gerald LeeQuality DigestQuality DigestNovember 30, 2007November 30, 2007

04/21/2304/21/23 3333

Area Under Tail in Curve-Z

11

22

33

-1-1

-3-3

-2-2

X

2.3%2.3%

ZZ

ZZ

15.9%15.9%

11

22

33

-1-1

-3-3

-2-2

X

2.3%2.3%

ZZ

ZZ

15.9%15.9%

04/21/2304/21/23 3434

Area Under Tail in Curve-Z

11

22

33

-1-1

-3-3

-2-2

X Z 1.5Z 1.5

6.7%6.7%

04/21/2304/21/23 3535

Figure 4.42 Normal Curve for Left-Reading Z Table




04/21/2304/21/23 3636

Figure 4.43 Example 4.34: Area Under the Curve, Xi = 0.0624




04/21/2304/21/23 3737

Figure 4.44 Example 4.34: Area Under the Curve, Xi = 0.0629




04/21/2304/21/23 3838

Figure 4.45 Example 4.34: Area Under the Curve Between 0.0623 and 0.0626




04/21/2304/21/23 3939

STATISTICAL MEASURES

Central TendencyCentral Tendency– Mean - average of a set of values Mean - average of a set of values

(sum of values divided by the # of values)(sum of values divided by the # of values)

– MedianMedian - The middle number in a set of values - The middle number in a set of values

– ModeMode - The most often occurring value in a set - The most often occurring value in a set of valuesof values

DispersionDispersion– Range - The largest value in a sample minus Range - The largest value in a sample minus

the smallest the smallest

– Variance - The sum of the differences of each Variance - The sum of the differences of each value and the average squared divided by the value and the average squared divided by the degrees of freedom (number of values or degrees of freedom (number of values or number of values minus 1)number of values minus 1)

– Standard Deviation - The square root of the Standard Deviation - The square root of the variancevariance

XX

n

X

X

X

11 12 13 13 14 15

678

6

13

10, 11, 10, 11, 1212, 13, 14, 13, 14

1,2,3,3,1,2,3,3,4,4,44,4,4,5,5,6,7,8.9,5,5,6,7,8.9

R X Xl e small arg

SX X

nor

X X

nn n2

2 2

1

( ) ( )

SX X

nor

X X

nn n

( ) ( )2 2

1

04/21/2304/21/23 4040

Mean Formula

XX

n

X

X

X

11 12 13 13 14 15

678

6

13

Statistical FormulaStatistical Formula

Simplified FormulaSimplified Formula

04/21/2304/21/23 4141

Figure 4.29 Calculating Medians




04/21/2304/21/23 4242

Figure 4.30 Calculating Modes




04/21/2304/21/23 4343

Figure 4.40 The Normal Curve




04/21/2304/21/23 4444

Figure 4.21 Skewness




MeanMean

MedianMedian

ModeMode

ModeMode

MedianMedian

MeanMean

04/21/2304/21/23 4545

Figure 4.32 Comparison of Mean, Mode, and Median for the Clutch Plate




04/21/2304/21/23 4646

XX

n

X

X

X

36 35 39 40 35 38 41

7264

7

37 7.

SX Xn

n

Sn

S

( )

. . . . . . .

.

2

2 2 2 2 2 2 2

1

37 7 36 37 7 35 37 7 39 37 7 40 37 7 35 37 7 38 37 7 41

12 43

6

3541

arg

R

R

XXR smallel

3636353539394040353538384141

Find:Find:RangeRangeAverageAverageStandard DeviationStandard Deviation

RANGERANGE

AVERAGEAVERAGE

STANDARD DEVIATIONSTANDARD DEVIATION

Example CalculationsExample Calculations

04/21/2304/21/23 4747

Manual Calculation of Standard Deviation

04/21/2304/21/23 4848

TERMS StatisticsStatistics– The collection, tabulation, analysis, interpretation, and presentation of The collection, tabulation, analysis, interpretation, and presentation of

numerical datanumerical data

PopulationPopulation– A collection of all possible elements, values, or items associated with a A collection of all possible elements, values, or items associated with a

situationsituation

SampleSample– A subset of elements or measurements taken from a populationA subset of elements or measurements taken from a population

– Must be randomized to represent the populationMust be randomized to represent the population

Deductive Statistics (descriptive statistics)Deductive Statistics (descriptive statistics)– Describe a population or a complete group of dataDescribe a population or a complete group of data

– Each entity in the population must be studiedEach entity in the population must be studied

Inductive StatisticsInductive Statistics– Deals with a limited amount of data or a representative sample of the Deals with a limited amount of data or a representative sample of the

populationpopulation

– Used for samples to predict the populationUsed for samples to predict the population

04/21/2304/21/23 4949

TERMS– DataData VariableVariable

– Those quality characteristics that can be measuredThose quality characteristics that can be measured AttributeAttribute

– Those quality characteristics that are observed to be either present or Those quality characteristics that are observed to be either present or absent, conforming or nonconformingabsent, conforming or nonconforming

RelativeRelative

– Those quality characteristics which are assigned a value which cannot be Those quality characteristics which are assigned a value which cannot be actually measuredactually measured

– AccuracyAccuracy How far from the actual or real value the measurement isHow far from the actual or real value the measurement is The location of X or X barThe location of X or X bar

– PrecisionPrecision The ability to repeat a series of measurements and get the same value each timeThe ability to repeat a series of measurements and get the same value each time RepeatabilityRepeatability The variability of measurementsThe variability of measurements

– Measurement ErrorMeasurement Error The difference between a value measured and the true valueThe difference between a value measured and the true value

04/21/2304/21/23 5050

High Precision Low

Accuracy

Low

High

Accuracy VS Precision

11/8/2015 1 statistics. 11/8/2015 2 histogram u definition –pictorial representation of a set of...

Documents