Quality Management

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Control Chart

X-Bar Chart and R-Chart

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(Ref: https://sixsigmastudyguide.com/x-bar-r-control-charts/)

• Choosing X-Bar and R Chart.
• X-Bar Chart and R Chart.
• X-Bar Chart and R Chart formulas.
• X-Bar Chart Calculation.
• R Chart Calculation.
• Comparing X-Bar and R Chart.

X-MR/i-MR Chart

X-bar R Chart

X-bar S Chart

U

Chart

C

Chart

P

Chart

NP

Chart

Subgroup

n = 1

eg: 20 Lots (20 Subgroups) and each Lot (Subgroup) has 1 Sample.

Subgroup

n = 2 to 9

eg: 20 Lots (20 Subgroups) and each Lot (Subgroups) has 2 to 9 Samples.

Subgroup

n > 10

eg: 20 Lots (20 Subgroups) and each Lot (Subgroup) has more than 10 Samples.

Sample Size

Variable

eg: 20 Lots (20 subgroups) and each Lot (Subgroup) uses different # of Samples.

Sample Size

Fixed

eg: 20 Lots (20 subgroups) and each Lot (Subgroup) uses same # of Samples.

Sample Size

Variable

eg: 20 Lots (20 subgroups) and each Lot (Subgroup) uses different # of Samples.

Sample Size

Fixed

eg: 20 Lots (20 subgroups) and each Lot (Subgroup) uses same # of Samples.

Subgroup size

eg: 10 Lots and each lot take 5 samples,

each Lot’s number contain 5 samples and that 5 sample is referred to as the Subgroup’s size.

Defects

eg: in a Subgorup of 5 Samples, found 8 kinds of Defects and we don’t know which Sample is Defective but within the 5 Samples found 8 kinds of Defects.

Defective

in Sample Size of 20, 5 are identified to be Defective meaning we exactly know among the 20, exactly which one is Defective.

Continuous Data

eg: measurements of each Sample can be 15.43mm, 18.99mm, 9.11mm.

Discrete Data

eg: defects/defectives or non-conformances counting like

1. in a Sample of 20 items found 5 Defective items.

2. in a Sample of 5 found 8 Defects.

Control Charts Selection

• Choosing X-Bar and R Chart.
• X-Bar Chart and R Chart.
• X-Bar Chart and R Chart formulas.
• X-Bar Chart Calculation.
• R Chart Calculation.
• Comparing X-Bar and R Chart.

X-Bar Chart and R Chart

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• X-Bar chart and R chart, are Control Charts combination that helps manufacturers to understand the stability of their processes and to pinpoint variations.
• X-Bar and R chart pair to visualize continuous data collected at regular intervals in sample subgroups. The size of the subgroups is also very important, it needs to be between 2 and 10. If your sample size is 1 or more than 10, you need to select different control charts.
• The X-Bar Chart helps to monitor the average or the mean of the process and how this changed over time.
• The R-Chart shows the sample range, which represents the difference between the highest and lowest value in each sample.
• When working with this Chart pair to visualize your data, start by examining the R-chart first because the control limits for the X-bar are derived from average range values (shown on the R-chart). Only if the values of the R-chart are in control, you can interpret the X-Bar. If the values are out of control, this is a sign that the X-Bar control limits are inaccurate.

• Choosing X-Bar and R Chart.
• X-Bar Chart and R Chart.
• X-Bar Chart and R Chart formulas.
• X-Bar Chart Calculation.
• R Chart Calculation.
• Comparing X-Bar and R Chart.

In order to Simplify the calculations for X-Bar Control Chart’s UCL and LCL

and R Control Chart’s UCL and LCL,

the tables to reference the numbers for A2, D3, D4 can be used

X bar chart

Reference for R2

R chart

Reference for D3, D4

Control Chart Constants are approximate values to measure the control limits for the X bar S chart and other control charts based on subgroup size.

• Choosing X-Bar and R Chart.
• X-Bar Chart and R Chart.
• X-Bar Chart and R Chart formulas.
• X-Bar Chart Calculation.
• R Chart Calculation.
• Comparing X-Bar and R Chart.

This example uses 20 Lots where each with 5 Samples

X-bar is the mean of of 5 samples for each Lot. Eg: Lot 1

(40+27+33+34+35)/5 = 33.8

X D-bar is the average of X-bar

where Sum of X-bar / k

ie:

(33.8 + 41.4 + ….. +21.0 + 30.0 / 20

= 31.28

Range is the difference between Max and Min for Samples in a Lot

eg: Lot 1 5 Samples

Max is 40

Min is 27

Max - Main = 40 - 27 = 13

Max

Min

R-bar is

the Average of R where

= Sum of R / k

= (13+30+...+17+24)/20

= 20.35

For 5 Samples, if number of Samples are 5 then A2 is 0.577

For 5 Samples, if number of Samples are 5 then A2 is 0.577

• Choosing X-Bar and R Chart.
• X-Bar Chart and R Chart.
• X-Bar Chart and R Chart formulas.
• X-Bar Chart Calculation.
• R Chart Calculation.
• Comparing X-Bar and R Chart.

This example uses 20 Lots where each with 5 Samples

Range is the difference between Max and Min for Samples in a Lot

eg: Lot 1 5 Samples

Max is 40

Min is 27

Max - Main = 40 - 27 = 13

Max

Min

R-bar is

the Average of R where

= Sum of R / k

= (13+30+...+17+24)/20

= 20.35

For 5 Samples, if number of Samples are 5 then D4 is 2.114

For 5 Samples, if number of Samples are 5 then D3 is 0

• Choosing X-Bar and R Chart.
• X-Bar Chart and R Chart.
• X-Bar Chart and R Chart formulas.
• X-Bar Chart Calculation.
• R Chart Calculation.
• Comparing X-Bar and R Chart.

The X-bar helps to monitor the average or the mean of the process and how this changed over time.

The R-chart shows the sample range, which represents the difference between the highest and lowest value in each sample.