1. Exercise for X bar - R Chart
10.4 13.0 10.4 7.4 6.3 6.1 11.9 9.6 8.2 8.8
10.6 8.6 11.9 11.0 10.0 6.6 8.2 9.5 12.4 9.4
9.6 10.3 8.9 9.1 10.6 11.7 9.4 11.4 10.9 9.3
10.9 9.4 9.1 8.7 11.2 9.7 10.2 9.2 9.8 8.4
9.3 10.5 10.6 8.5 9.5 10.4 10.4 9.8 9.9 5.8
Prepare X bar - R chart using following Xs.
Analyze the data and present the same

2. X bar - R Chart
• Collect the data
• Calculate avarage (X bar) and range (R) of each
sub-group
• X bar = (X1+X2+……+Xn)/n
• R = Xhighest - Xlowest
Calculate X double bar and R bar
X double bar = (X1 bar + X2 bar + ………..+Xn bar) / n
R bar = (R1 + R2 +……….+ Rn) / n

3. Constants for Control Limits : X bar - R Chart
Sub Group
Size
A2 D4 D3
2 1.880 3.267 0
3 1.023 2.527 0
4 0.729 2.282 0
5 0.577 2.115 0
6 0.483 2.004 0
7 0.419 1.924 0.076

4. Formula for Control Limits : X bar - R Chart
• For Average control chart
Upper Control Limit, UCLx = X(Double bar) + A2 x R bar
Lower Control Limit, LCLx = X(Double bar) - A2 x R bar
• For range control chart
Upper Control Limit, UCLr = D4 x R bar
Lower Control Limit, LCLr = D3 x R bar

5. X Bar - R Charts
Draw lines for the following on the control chart
X double bar
R bar
control limits

6. 6
PROCESS CAPABILTY RATIOS
SPC
Cp & Cpk are two key measures of process capability
Cp = USL – LSL ie Total Tolerance
6 Sigma Process Spread
Cpk = Min X bar - LSL , USL – X bar Cpk accounts for process centering and spread
3 sigma 3 sigma
Cpk will always be equal to or less than Cp
Standard Deviation, Sigma = R bar/ d2
d2 is a constant varying by sample size,
n 2 3 4 5 6 7 8 9 10
d2 1.13 1.69 2.06 2.33 2.53 2.70 2.85 2.97 3.08