A new process is started, and the sum of the sample standard deviations for 25 subgroups of size 4 is 750. If the specifications are 700 +- 80, what is the process capability index?
What I know:
D2=2.059
USL = 780; LSL = 620; Tolerance = 160
To calculate the process capability index (Cpk), we need to first calculate the process standard deviation (σ) and then use the formula:
Cpk = min((USL - µ) / (3σ), (µ - LSL) / (3σ))
Where:
- USL is the upper specification limit
- LSL is the lower specification limit
- σ is the process standard deviation
- µ is the process mean (average)
To find the process standard deviation (σ), we are given the sum of the sample standard deviations for 25 subgroups of size 4, which is 750. However, we need to convert this sum to the standard deviation of the process. The sum of the sample standard deviations is the sum of the standard deviations of the individual subgroups.
Since we have each subgroup's sample size (n) and sample standard deviation (s), we can calculate the process standard deviation using the following steps:
1. Calculate the within-subgroup standard deviation (s_within) using the formula:
s_within = sqrt(sum(s^2) / (n * (k - 1)))
Where:
- sum(s^2) is the sum of the squared sample standard deviations (750 in this case)
- n is the sample size (4 in this case)
- k is the number of subgroups (25 in this case)
2. Calculate the within-subgroup range (R_within) using the formula:
R_within = D2 * s_within
Where D2 is a constant (in this case, you mentioned it as 2.059)
3. Calculate the process standard deviation (σ) using the formula:
σ = R_within / d2
Where d2 is a constant related to the subgroup size (in this case, 2.704)
4. Calculate the process mean (µ) as the average of all subgroup means.
Once we have σ and µ, we can substitute these values into the formula mentioned earlier to calculate Cpk.
Let me know if you'd like to proceed with the calculations, or if you need further clarification on any step!