Control Chart : Xbar Chart

CONTROL CHART

A control chart, also known as a Shewhart chart or process-behavior chart, is a statistical tool used in quality control to monitor process performance over time. The purpose of a control chart is to determine whether a process is in control (i.e., operating within statistical control limits) or out of control (i.e., exhibiting significant variation due to assignable causes).

A control chart is a powerful tool in quality control, as it allows for real-time monitoring of process performance and early detection of potential problems. By using control charts, organizations can identify and correct issues with their processes before they result in significant problems or defects. This leads to improved process stability and reduced variability, ultimately resulting in higher quality products and services.

There are several types of control charts, including Xbar-R charts, Xbar-S charts, and individuals control charts. The choice of which control chart to use depends on the type of data being collected and the goals of the process improvement effort.

It is important to understand that control charts are not a one-time solution, but rather a continuous monitoring tool. Regular use of control charts can help organizations identify and address process issues in a timely manner, leading to continuous improvement over time.

When using control charts, it is also important to have a well-defined data collection process, and to ensure that the data collected is representative of the process being monitored. In addition, the control limits should be periodically re-calculated to reflect any changes in the process.

In conclusion, the use of control charts is a valuable tool for organizations looking to improve the quality of their products and services. By monitoring process performance over time and detecting and correcting issues quickly, organizations can achieve continuous improvement and deliver high-quality products and services to their customers.

How to create Control Chart: Data Science

Xbar-R Charts

An Xbar-R chart is a type of control chart that is used to monitor the mean and the variability of a process over time. The Xbar chart is used to monitor the process mean, and the R chart is used to monitor the process variability. The Xbar chart plots the average of each subgroup of data, and the R chart plots the range of each subgroup.

Sample figure Xbar-R Chart: Six Sigma Study Guide

STEPS TO CONTRUCT X-BAR CHART 

Here is how to construct an Xbar And R chart:

1. Collect data: Collect data from the process over a period of time to be plotted on the Xbar-R chart. The data should be collected in subgroups of a constant size, typically between 2 to 10 data points per subgroup.
2. Calculate subgroup averages (X-bar): Calculate the average of each subgroup to obtain the X-bar values.
3. Calculate subgroup ranges (R): Calculate the range of each subgroup to obtain the R values. The range is the difference between the highest and lowest values in the subgroup.
4. Calculate control limits: Use statistical formulas to calculate the control limits for the Xbar and R charts. The control limits define the upper and lower boundaries that define what is considered a normal variation of the process.
5. Plot data points: Plot the Xbar and R values on the appropriate chart, using the control limits as reference.
6. Evaluate patterns: Evaluate the patterns of the plotted data to determine whether the process is in control or out of control. If the process is in control, the plotted data should fall within the control limits, and there should not be any distinct patterns or trends. If the process is out of control, the plotted data may fall outside of the control limits, or there may be distinct patterns or trends that suggest the presence of assignable causes.
7. Make process improvements: If the process is out of control, use the information from the Xbar-R chart to make process improvements to bring the process back into control.

HOW TO CALCULATE CONTROL LIMIT (i,e UCL and LCL for X-bar R Chart)

The formula to calculate the Upper Control Limit (UCL) and Lower Control Limit (LCL) for an Xbar-R chart are:
 
where Xbar is the average of the sample means, R bar is the average of the sample ranges, and A2 is a constant based on the sample size (n) and the confidence level desired. The value of A2 can be looked up in tables for control chart constants, or calculated using statistical software.

EXAMPLE -01

In this example, the data has been collected in subgroups of 10 data points each. Calculate the X-bar column ( the average of each subgroup ), and the R column (shows the range of each subgroup). This data can be plotted on an Xbar-R chart, along with the calculated control limits, to monitor the mean and variability of the process over time.

Subgroup	S1	S2	S3	S4	S5	S6	S7	S8	S9	S10
1	42	45	38	41	44	39	46	37	43	40.7
2	39	40	36	38	37	42	41	35	40	38.5
3	45	43	40	44	41	46	42	39	44	42.4
4	40	37	39	38	41	36	42	39	40	39
5	46	43	42	45	44	47	41	44	43	44.2
6	37	39	36	38	37	41	39	35	40	38
7	44	41	42	43	42	46	39	44	41	42
8	36	35	34	38	36	39	37	35	38	36.8
9	41	40	39	40	38	43	40	39	41	40.4

Step # 01: Calculate mean and Range for each sample (Subgroup).

Step # 02: Calculate Average X bar (Average Of Mean Observation) and Average Range for each sample (Subgroup).

Step # 03: Find UCL and LCL using Statistical Formula:

The formula to calculate the Upper Control Limit (UCL) and Lower Control Limit (LCL) for an Xbar-R chart are:

UCL (Xbar) = Xbar + A2 * Rbar

LCL (Xbar) = Xbar - A2 * Rbar

where Xbar is the average of the sample means, Rbar is the average of the sample ranges, and A2 is a constant based on the sample size (n) and the confidence level desired. The value of A2 can be looked up in tables for control chart constants, or calculated using statistical software.

The formula for the control chart constant A2 used to calculate the Upper Control Limit (UCL) and Lower Control Limit (LCL) of an Xbar-R chart is:

A2 = d2 * sqrt(n)

where d2 is a control chart constant for Xbar-R charts and depends on the sample size (n) and the desired confidence level, and sqrt(n) is the square root of the sample size (n). The value of d2 can be found in control chart constant tables or calculated using statistical software.

But For Simplicity we take statistical Approximation

We use the Following Formulas to compute UCL(Upper Control Limit) and UCL (Lowe Control Limit)
\(UCL_{\overline{\overline{x}}} = \overline{\overline{x}}+ 3\frac{\widehat{\sigma}}{n}\) and

\(LCL_{\overline{\overline{x}}} = \overline{\overline{x}}- 3\frac{\widehat{\sigma}}{n}\)  Where \(\widehat{\sigma}\) is the estimated standard deviation.

Step# 04: Finally Construct the Chart Using Excel Chart Tool and Also Done Manually , Check Pattern And Comments On Finding. 

Repeat the Above four Step for the Following Problems.

Problem # 01: Following data represent 20 Subgroup of five measurements on the critical dimension of a part produced by Machining Process

Example Question For X Bar Chart taken from By Douglas C. Montgomery

Set up Xbar Charts on this process. Verify that the process is in Statistical Control.

GOOD BOOK FOR QC TOOLS

Introduction to Statistical Quality Control By Douglas C. Montgomery

SOFTWARE HELP (MINITAB)