Calculate acceptance value, mean, standard deviation, sample size, or multiplier from the other four inputs using the AV formula.
Related Calculators
- Percent Nonconforming Calculator
- Bias Percentage Calculator
- Ppk (Process Performance Index) Calculator
- Sigma To Percentage Calculator
- All Statistics Calculators
Acceptance Value Formula
The acceptance value calculator is based on the relationship between the sample mean, standard deviation, sample size, and multiplier:
AV = x̄ + (k*s)/sqrt(n)
- AV = acceptance value
- x̄ = sample mean
- k = multiplier
- s = sample standard deviation
- n = sample size
If one variable is missing, the calculator rearranges the same formula to solve for that value.
x̄ = AV - (k*s)/sqrt(n)
s = abs(AV - x̄)*sqrt(n)/k
n = ((k*s)/(AV - x̄))^2
k = (AV - x̄)*sqrt(n)/s
The calculator functions as follows:
- Calculate acceptance value: enter sample size, mean, standard deviation, and multiplier.
- Calculate mean: enter acceptance value, sample size, standard deviation, and multiplier.
- Calculate standard deviation: enter acceptance value, mean, sample size, and multiplier.
- Calculate sample size: enter acceptance value, mean, standard deviation, and multiplier.
- Calculate multiplier: enter acceptance value, mean, standard deviation, and sample size.
Common Inputs and Result Interpretation
Use these tables to check that your inputs are reasonable before interpreting the result.
| Variable | Meaning | Input notes |
|---|---|---|
| Sample size, n | Number of observations in the sample | Must be greater than 0 |
| Mean, x̄ | Average value of the sample | Use the same units as the acceptance value |
| Standard deviation, s | Spread of the sample values | Cannot be negative |
| Multiplier, k | Factor applied to the standard error | Must be greater than 0 |
| Result pattern | What it indicates |
|---|---|
| Higher standard deviation | Increases the acceptance value because the sample has more variation |
| Larger sample size | Reduces the added term because the standard error decreases |
| Higher multiplier | Increases the acceptance value by giving more weight to variation |
| Standard deviation of 0 | The acceptance value equals the mean, since there is no spread term |
Example Problems
Example 1: Calculate acceptance value
Suppose the sample size is 25, the mean is 98.6, the standard deviation is 2.4, and the multiplier is 1.645.
AV = 98.6 + (1.645*2.4)/sqrt(25)
AV = 98.6 + 3.948/5
AV = 99.3896
The acceptance value is 99.3896.
Example 2: Calculate sample size
Suppose the acceptance value is 51, the mean is 50, the standard deviation is 3, and the multiplier is 1.645.
n = ((1.645*3)/(51 - 50))^2
n = (4.935/1)^2
n = 24.3542
The calculated sample size is 24.3542. If you need a whole-number sample size for a real test plan, you would normally round up to the next whole number.
FAQ
What does the acceptance value represent?
The acceptance value is the sample mean plus an adjustment for variation. The adjustment depends on the standard deviation, the multiplier, and the sample size. A larger standard deviation or multiplier raises the value, while a larger sample size lowers the adjustment.
Why does sample size use the square root of n?
The formula uses s / √n, which is the standard error term. It reflects how uncertainty in the mean decreases as sample size increases. Because of the square root, increasing sample size has a diminishing effect.
Why must AV be greater than the mean when solving for sample size or k?
With a positive multiplier and nonnegative standard deviation, the term added to the mean is zero or positive. That means the acceptance value must be greater than the mean when solving for sample size or multiplier in this formula. If AV is less than or equal to the mean, the calculator cannot solve those variables under the stated assumptions.
