BIZOMAT MAKEOS · Quality
Sample Size to Estimate a Mean
How many measurements or tests are needed to estimate the mean within this margin?
Minimum number of tests or measurements to estimate a mean within a chosen margin of error at a confidence level, with an optional finite-population correction.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Expected standard deviation | Estimate of the standard deviation from earlier data or a pilot study, in the units of the measurement. | |
| Required margin of error (±) | Half-width of the confidence interval you need, in the same units. | |
| Confidence level | choice | Confidence that the true mean lies within the margin. |
| Population size (0 = very large) | items | Total lot or population size for a finite-population correction; 0 if large or ongoing. |
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: Expected standard deviation
- What it is
- Your best estimate of how much individual measurements vary, from earlier data or a small pilot study, in the units of the measurement.
- 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 0.05 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Sample size rises with the square of the standard deviation.
- Get it right
- Use the same units as the margin of error; an underestimate gives too small a sample.
What this means: Required margin of error (±)
- What it is
- How close to the true mean your estimate needs to be: the half-width of the confidence interval, in the same units as the measurement.
- 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 0.02 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Halving the margin roughly quadruples the sample size.
- Get it right
- This is the plus-or-minus half-width, not the full width of the interval.
What this means: Confidence level
- What it is
- How sure you want to be that the true mean lies within the margin of error of your estimate.
- Units and format
- Choose one of 3 options from the list: 90%; 95%; 99%.
- Example
- The form opens with “95%” selected as a worked example. Choose the option that matches your case.
- Get it right
- Higher confidence needs more measurements for the same margin.
What this means: Population size (0 = very large)
- What it is
- The total number of items in a finite lot you are sampling from, used for a finite-population correction.
- Units and format
- Enter a number in items: a count of items. The form accepts 0 items or more. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 items as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A small population reduces the sample size needed.
- Get it right
- 0 means the population is very large or ongoing, and no correction is applied.
Method and formulas
- n₀ = (z σ ÷ E)², with z = 1.645, 1.960 or 2.576 for 90, 95 and 99%.
- Finite population: n = n₀ ÷ (1 + (n₀ − 1) ÷ N).
- Round up to a whole number.
- Example: σ 0.05 and a ±0.02 margin at 95% need 25 measurements.
Limitations
- Uses the normal z value; for small samples with σ estimated from the same data, a t-based size is slightly larger.
- A poor σ estimate gives a poor sample size.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Plan how many test pieces to make
- Size a measurement study
- Justify an inspection sample
Who it is for
- Quality engineers
- Process engineers
- Test engineers
What you get
- Required sample size
- z value used
Interpreting the result
- If the sample is too large to afford, accept a wider margin or reduce variation first.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 130–132. Topic reference; no source tables or recommended settings are reproduced.
Learn the Concepts Behind This Tool
These BIZOMAT Library readings explain the method this tool uses, with worked examples.
Method explained here · BIZOMAT original book
Sampling without fooling yourself
In Blending Bulk Solids by the Numbers
What it covers: Why sampling particulate material adds its own error, the rules for a fair sample from a stream or container, sample dividing, how many samples a decision needs, and a scoop that rejected good premix.
Method explained here · BIZOMAT original book
Segregation in storage
In Bulk Solids Flow by the Numbers
What it covers: How bins and silos sort particles by size during filling and discharge, how the flow pattern decides what comes out when, and how to measure segregation and tell it apart from attrition.
Method explained here · BIZOMAT original book
Measuring conductivity reliably
In Eddy-Current Inspection by the Numbers
What it covers: Edge distance, curvature, lift-off and probe condition in conductivity readings, a Type 1 gauge study on a reference standard, and an uncertainty budget for the acceptance band.
Method explained here · BIZOMAT original book
Phase fractions by point counting
In Microstructures by the Numbers
What it covers: Estimating the volume fraction of a phase by counting grid points, its standard error from field-to-field scatter, the number of fields needed, and image analysis thresholds.
Method explained here · BIZOMAT original book
Measured performance: sample sizes for means and comparisons
In Prototypes and Simulation by the Numbers
What it covers: A 95% interval for the BPU-5's usable energy from 12 DVT units, sample sizes for a mean and for a supplier comparison, and a 3-pack test that nearly cost A$85,000 a year.
Method explained here · Library collection
Sampling: who to ask and how many responses you need
In Research and Evidence for Better Decisions
What it covers: How to define a population and sampling frame, choose probability or non-probability sampling, estimate margins of error and sample sizes, and deal with small populations, subgroups and non-response.
Method explained here · BIZOMAT original book
Planning, measuring and reporting a model test
In Scale Models by the Numbers
What it covers: Turn the questions a model must answer into a test matrix, size the number of readings, carry uncertainties and scale effects into the results, and write a report a designer can act on.
Method explained here · BIZOMAT original book
How many runs
In Simulating Noisy Systems by the Numbers
What it covers: Turning simulated days into estimates with confidence intervals: standard errors of means, intervals for probabilities and rare counts, run counts for a stated precision, and correlated output.
BIZOMAT 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.