A new process is started and the sum of the sample standard deviation for 20 subgroups, of size 8, is 750; Compute, to 3 decimal places, the process standard deviation, σ0:
To compute the process standard deviation (σ0) using the sum of the sample standard deviation for subgroups, you need to follow these steps:
1. Determine the number of measurements (N) in the process. In this case, we have 20 subgroups, each containing 8 measurements. So, N = 20 * 8 = 160.
2. Calculate the average range (R̄) for the subgroups. The range is the difference between the largest and smallest measurements within each subgroup. Typically, a subgroup of 6 or more measurements is used to estimate the process variation. In this case, we have 8 measurements in each subgroup. So, calculate the range for each subgroup and then find the average of these ranges. Let's assume the average range is R̄.
3. The formula to compute the process standard deviation from the sum of the sample standard deviation is:
σ0 = (R̄ / d2) * sqrt(1 + ((1 / m) * (1 - (1 / n))))
where:
- σ0 is the process standard deviation.
- d2 is a constant from statistical tables that depends on the subgroup size. For a subgroup of size 8, d2 is approximately 2.118.
- m is the number of subgroups.
- n is the number of measurements in each subgroup.
4. Plug in the values into the formula and calculate σ0:
σ0 = (R̄ / 2.118) * sqrt(1 + ((1 / 20) * (1 - (1 / 8))))
Evaluate the expression inside the square root first:
(1 / 20) * (1 - (1 / 8)) = 0.0375
Then, substitute it into the formula:
σ0 = (R̄ / 2.118) * sqrt(1 + 0.0375)
= (R̄ / 2.118) * sqrt(1.0375)
Now, you need to know the value of R̄ to calculate σ0 accurately. If you have that information, substitute it into the formula and compute σ0.
Note: Without the value of R̄, it's not possible to provide an exact calculation of σ0.