KEVOS® MAKEOS · Quality
Standard Deviation Chi-Square Check (Process Spread Against a Target Sigma, Confidence Interval)
Is my process spread different from (or larger or smaller than) the standard deviation it is meant to hold, judging from a sample, and what range does the true standard deviation lie in?
Chi-square test of a population standard deviation against a target value from one sample: the test statistic, the chi-square critical values for your significance level and test direction, the p-value, a confidence interval for the true standard deviation, and the decision.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Alternative | choice | Two-sided asks whether the spread differs from the target either way; one-sided asks whether it is larger (a problem) or smaller (an improvement). |
| Sample size | values | Number of values in the sample. |
| Sample standard deviation | Standard deviation of the sample (divisor n − 1). | |
| Target standard deviation σ₀ | Standard deviation the process is meant to hold, from a qualification or requirement. | |
| Significance level | % | Chance of a wrong rejection (5 % is common). |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Alternative
- What it is
- Which question the test answers: whether the spread differs from the target either way, is larger than it (a problem) or is smaller (an improvement).
- Units and format
- Choose one of 3 options from the list: Two-sided: spread differs from target; One-sided: spread larger than target; One-sided: spread smaller than target.
- Example
- The form opens with “Two-sided: spread differs from target” selected as a worked example. Choose the option that matches your case.
- Effect on the result
- Changes which tail of the chi-square distribution holds the rejection region.
- Get it right
- To show a process meets a maximum spread, test whether it is smaller; to raise an alarm, test whether it is larger.
What this means: Sample size
- What it is
- The number of values in the sample used to calculate the standard deviation.
- Units and format
- Enter a number in values: values. The form accepts 2 values to 1000000 values. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 100 values as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- More values add degrees of freedom and tighten the confidence interval.
- Get it right
- Use at least 20 to 30 values; with few values the confidence interval for the standard deviation is very wide.
What this means: Sample standard deviation
- What it is
- The standard deviation of the sample, calculated with the divisor n − 1.
- Units and format
- Enter a plain number with no unit. The form accepts 0.000001 or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.0063 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The test statistic rises with its square.
- Get it right
- Calculate it from individual values taken from a stable process.
What this means: Target standard deviation σ₀
- What it is
- The standard deviation the process is meant to hold, from a qualification study, a requirement or a capability target.
- Units and format
- Enter a plain number with no unit. The form accepts 0.000001 or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.1 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The test statistic falls with its square.
- Get it right
- For a capability target, σ₀ = tolerance width ÷ (6 × required Cp).
What this means: Significance level
- What it is
- The significance level: the chance you accept of a wrong rejection when there is no real difference.
- Units and format
- Enter a number in %: percent: the number is read as a percentage, so type 8 for 8 %. The form accepts 0.1% to 20%. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 5% as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A smaller level widens the acceptance region, so a difference must be larger to show.
- Get it right
- Use 5 % unless your procedure says otherwise; 1 % for decisions where a false alarm is costly.
Method and formulas
- T = (N − 1)(s ÷ σ₀)² with N − 1 degrees of freedom (NIST e-Handbook 1.3.5.8).
- Two-sided: reject if T is below the α/2 point or above the 1 − α/2 point of the chi-square distribution; larger: reject if above the 1 − α point; smaller: reject if below the α point.
- Confidence interval for σ: √((N − 1)s² ÷ χ²₁₋α/₂) to √((N − 1)s² ÷ χ²_α/₂).
- Example: T = 0.39293 against critical values 73.361 to 128.42; true standard deviation 0.0055314 to 0.0073186.
Limitations
- The chi-square test for a standard deviation is very sensitive to non-normal data: check normality first.
- Values must be independent and from a stable process; a drifting process inflates s.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Check whether a process still holds its qualified spread
- Show that a new set-up reduced variation below a target
- Put a confidence range on a measured standard deviation
Who it is for
- Quality and process engineers
- Laboratory and calibration staff
- Supplier quality staff
What you get
- Chi-square statistic
- Critical values and p-value
- Confidence interval for the true standard deviation
- Decision against the target
Interpreting the result
- The confidence interval is not symmetric about s: with small samples the upper bound can be far above the sample standard deviation.
References
- NIST/SEMATECH e-Handbook of Statistical Methods, 1.3.5.8 Chi-Square Test for the Standard Deviation — https://www.itl.nist.gov/div898/handbook/eda/section3/eda358.htm
Learn the Concepts Behind This Tool
These KEVOS® Library readings explain the method this tool uses, with worked examples.
Method explained here · KEVOS® original book
Random numbers you can trust
In Simulating Noisy Systems by the Numbers
What it covers: How pseudo-random generators work, seeds and reproducible streams, turning uniform numbers into Gaussian ones with Box–Muller, testing a generator's spread and tails, and the shortcuts that fail.
Useful alongside · KEVOS® original book
Averaging and the square-root rule
In Random Error and Drift by the Numbers
What it covers: Apply the square-root rule to CW-5's 200-sample window, see why its measured scatter is nearly three times what the rule predicts, and use window length to cancel conveyor vibration.
KEVOS® tools are screening and decision-support utilities. Verify important financial, engineering, safety, legal or operational decisions using appropriate professional standards and your organisation’s approved data.