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
Cp=USL-LSL/6 sigma note
= 160/6sigma note
sigma note=s(bar)/c4 ; c from the table for sybgroub size 4
c4= 0.9213
s(bar)=750/25=30
sigmanote=30/0.9213=32.56
6 simgma note=6* 32.56=195.37
so
cp=160/195.37=0.818
cp<1 , the process in state I
Well, if I were to calculate the process capability index, I'd say it's going to need a lot of math, and maybe a little bit of clown magic. But let's give it a shot.
The process capability index, also known as Cp, is calculated using the formula:
Cp = (USL - LSL) / (6 * standard deviation)
So, let's start by finding the standard deviation. We know that the sum of the sample standard deviations for 25 subgroups of size 4 is 750. So, if we divide 750 by 25, we get 30. That's a pretty small standard deviation for a group of clowns like us, but let's keep going.
Now, we need to find the average range, which is the difference between the highest and lowest data values in each subgroup. Since our subgroup size is 4, we'll divide the tolerance (which is 160) by D2 (which is 2.059) to get the average range. So, 160 divided by 2.059 is approximately 77.7.
Finally, we can calculate the standard deviation by dividing the average range by D2 and multiplying by 1.128. So, 77.7 divided by 2.059 multiplied by 1.128 gives us approximately 55.4.
Now, we can plug in the values into the formula for Cp:
Cp = (780 - 620) / (6 * 55.4)
Cp = 160 / 332.4
Cp is approximately 0.48
So, the process capability index (Cp) in this case is approximately 0.48. And just like a clown balancing on a rubber ball, that's pretty impressive! Good luck with your process!
To calculate the process capability index (Cpk), we need to calculate the process standard deviation (sigma) and then use the formula:
Cpk = min((USL - X̄) / (3 * sigma), (X̄ - LSL) / (3 * sigma))
Where:
USL is the Upper Specification Limit
LSL is the Lower Specification Limit
X̄ is the mean of the process
sigma is the estimated process standard deviation
Given:
D2 = 2.059 (this is the constant value used to estimate the process standard deviation)
USL = 780
LSL = 620
Tolerance = 160
Step 1: Calculate the Average Range (R̄)
We know that the sample standard deviation is equal to sigma / sqrt(n), where n is the sample size. In this case, the sample size (n) is 4 for each subgroup, so the sample standard deviation (s) is equal to sigma / 2.
Given that the sum of the sample standard deviations for 25 subgroups is 750, we can calculate s as follows:
s = 750 / (25 * 2)
s = 750 / 50
s = 15
Now, we can calculate the average range (R̄) using the relationship R̄ = 2.059 * s:
R̄ = 2.059 * 15
R̄ = 30.885
Step 2: Calculate the Average of the Subgroup Means (X̄)
We are not given the subgroup means, so we cannot calculate X̄. Please provide this information to continue with the calculation.
Step 3: Calculate the Process Standard Deviation (sigma)
The process standard deviation (sigma) can be estimated using the relationship sigma = R̄ / D2:
sigma = 30.885 / 2.059
sigma ≈ 15
Step 4: Calculate the Process Capability Index (Cpk)
Using the formula mentioned earlier:
Cpk = min((USL - X̄) / (3 * sigma), (X̄ - LSL) / (3 * sigma))
Without knowing the value of X̄, we cannot calculate Cpk.
To calculate the Process Capability Index (Cp), we need to have the standard deviation and the tolerance of the process.
Given:
Sum of the sample standard deviations (σ) for 25 subgroups of size 4: 750
Specifications:
Upper Specification Limit (USL): 780
Lower Specification Limit (LSL): 620
Tolerance: 160 (USL - LSL)
To find the standard deviation for the entire process, we need to convert the sum of sample standard deviations into the standard deviation of the process. Since we have 25 subgroups of size 4, we need to divide the sum by the square root of the subgroup size (n).
Standard Deviation of the process (σ_p) = Sum of sample standard deviations (σ) / √(n)
Here, n = 4, so:
σ_p = 750 / √(4)
σ_p = 750 / 2
σ_p = 375
Now, we can calculate the Process Capability Index (Cp):
Cp = Tolerance / (6 * σ_p)
Note: The value 6 in the denominator comes from the relationship between process standard deviation and specification limits.
Cp = 160 / (6 * 375)
Cp = 160 / 2250
Cp = 0.0711 (approximately)
Therefore, the process capability index (Cp) is approximately 0.0711.