What is a Control Chart?A Control Chart is a statistical tool used to distinguish between process variation resulting from common causes and variation resulting from special causes.

Statistical Methods

What Are the Control Chart Types?•X-Bar and R Chart•Individual X and Moving Range ChartOther Control Chart types:•c Chart •p Chart

Statistical Methods

Are youchartingattributedata?

Data arevariables

Data

Is sampleSize equal

to 1?

Use XmRchart forvariables

data

Use P chart and C chart

For sample sizebetween 2 and15, use X-Barand R Chart

NO

YES

YES

NO

Control Chart Decision Tree

Steps in drawing an XmR Chart1.Decide what are the data to be collected2.Collect Data3.Arrange data in their chronological sequence4.Calculate Average ( X bar )5.Compute Moving Range ( MR )6.Calculate Average Moving Range ( MR bar )7.Substitute in the formulae8.UCL = Xbar+2.66(MRbar)9.LCL = Xbar-2.66(MRbar)

XmR Chart

XmR ChartExample

Date 1/8 2/8 3/8 4/8 5/8 6/8 7/8 8/8 9/8 10/8Minutes 19 22 16 18 19 23 18 15 19 18

Calculating MR bar Compute differences between successive values arranged in their chronological sequence and take the modulus of those values ( MR )Calculate the average of the differences (MR bar)Example

XmR Chart

Date 1/8 2/8 3/8 4/8 5/8 6/8 7/8 8/8 9/8 10/8Minutes 19 22 16 18 19 23 18 15 19 18Ranges 3 6 2 1 4 5 3 4 1

Average = 187 / 10 = 18.7 ; MRbar = 29 / 9 = 3.2

Steps in drawing an XmR ChartSubstituting in the formulaeUCL = Xbar+2.66(MRbar) = 18.7+2.66(3.2) = 27.27LCL = Xbar-2.66(MRbar) = 18.7-2.66(3.2) = 10.13Plotting the Chart we get…

XmR Chart

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Constructing an X – bar and R Chart• Step 1 - Determine the data to be collected.• Step 2 - Collect and enter data by subgroup• Step 3 - Calculate and enter subgroup averages• Step 4 - Calculate and enter subgroup ranges• Step 5 - Calculate grand mean• Step 6 - Calculate average of subgroup ranges• Step 7 - Calculate UCL and LCL for subgroup

averages• Step 8 - Select scales and plot• Step 9 - Document the chart

Average X = X1 + X2 + X3 +…..+Xn

nWhere X is the averageand X1… are the individual values in the subgroupn is the total values in the subgroup

X = X1 + X2 + X3 + ……+Xn nRange R = Largest value in each Subgroup – Smallest value in each Subgroup

Average R = R1 + R2 + R3 +…..+Rn nWhere R is the average Rangeand R1… are the individual Ranges of the subgroupsn is the total number of Subgroups

UCL = X + A2 RLCL = X – A2 RUse value of A2 based on number of values in subgroup = n

n A2

2 1.8803 1.0234 0.7295 0.5776 0.4837 0.4198 0.3739 0.33710 0.30811 0.28512 0.26613 0.24914 0.235

Subgroup 1 2 3 4 5 6 7 8 9X1 15.3 14.4 15.3 15.0 15.3 14.9 15.6 14.0 14.0

X2 14.9 15.5 15.1 14.8 16.4 15.3 16.4 15.8 15.2

X3 15.0 14.8 15.3 16.0 17.2 14.9 15.3 16.4 13.6

X4 15.2 15.6 18.5 15.6 15.5 16.5 15.3 16.4 15.0

X5 16.4 14.9 14.9 15.4 15.5 15.1 15.0 15.3 15.0

Averages 15.36 15.04 15.82 15.36 15.98 15.34 15.52 15.58 14.56

Ranges 1.5 1.2 3.6 1.2 1.9 1.6 1.4 2.4 1.6

Grand Average = 15.40Range Average = 1.82

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Control Chart used where the Bulk Sample is the Same

c ChartUCL = c + 3 c

LCL = c - 3

c = Average Number of Blemishes

cA Two Yard sample of fabric was being observed for blemishes every half an hour in a factory and the following observations were recorded . Draw a Control Chart for the same .

Sample 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17Defects 2 4 1 1 4 5 2 1 2 4 4 3 5 2 1 1 2

Sample 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32Defects 3 2 4 3 2 4 3 2 3 5 1 4 3 4 2

Sample 33 34 35 36 37 38 39 40Defects 3 6 4 0 1 2 3 1

Control Chart used where the Bulk Sample is the Same

c Chartc = Average Number of Blemishes = Total number of

blemishes / Total number of Samples

= 109 / 40 = 2.725

UCL = c + 3 c LCL = c - 3 c

= 2.725 + 3 2.725

= 2.725 + 4.952

= 7.68

= 2.725 - 3 2.725

= 2.725 - 4.952

= - 2.232 LCL = 0

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