Calculating Ucl And Lcl For X Bar Chart
X bar chart is an important statistical tool used to monitor the mean of a process. It helps to identify any changes or variations in the process mean over time. The chart consists of a central line representing the mean and upper control limit (UCL) and lower control limit (LCL) lines representing the acceptable range of variation around the mean. In this article, we will discuss how to calculate UCL and LCL for X bar chart.
What is UCL and LCL?
The upper control limit (UCL) is the highest limit of acceptable variation around the mean of a process. Any data point that falls above this limit indicates that the process is out of control and requires investigation. The lower control limit (LCL) is the lowest limit of acceptable variation around the mean. Any data point that falls below this limit also indicates that the process is out of control and requires investigation.
Calculating UCL and LCL
UCL and LCL are calculated using the following formulas:
UCL = X̄ + A2 * R
LCL = X̄ - A2 * R
where:
- X̄ is the mean of the process
- A2 is a constant value that depends on the sample size and the chosen confidence level
- R is the range of the sample
The value of A2 can be obtained from a table of constants. For example, if the sample size is 5 and the chosen confidence level is 95%, then the value of A2 is 0.577.
Example Calculation
Let's assume that we have collected 5 samples from a process and calculated their mean and range as follows:
- Sample 1: X̄ = 10, R = 2
- Sample 2: X̄ = 12, R = 3
- Sample 3: X̄ = 9, R = 4
- Sample 4: X̄ = 11, R = 2
- Sample 5: X̄ = 10, R = 1
First, we need to calculate the overall mean and range:
- X̄ = (10 + 12 + 9 + 11 + 10) / 5 = 10.4
- R = max(2, 3, 4, 2, 1) = 4
Next, we need to calculate the value of A2 for a sample size of 5 and a confidence level of 95%. From the table of constants, we find that A2 = 0.577.
Finally, we can calculate the UCL and LCL:
- UCL = 10.4 + 0.577 * 4 = 12.31
- LCL = 10.4 - 0.577 * 4 = 8.49
Therefore, the X bar chart for this process would have a central line at 10.4, an upper control limit at 12.31, and a lower control limit at 8.49.
Conclusion
Calculating UCL and LCL is an important step in creating an X bar chart. It helps to establish the acceptable range of variation around the mean of a process and identify any changes or variations in the process mean over time. By following the steps outlined in this article, you can easily calculate UCL and LCL for your own X bar charts.