Sample Size Calculator: The Number Every Quality Audit Gets Wrong

The Sample Size Calculator gives you the exact number — derived from your actual quality data, your stated confidence level, and your acceptable margin of error — for three distinct sampling situations: Variable sampling for continuous measurements, Attribute sampling for pass/fail classification, and Acceptance sampling for lot inspection decisions.

Most quality teams pick their sample sizes the same way they pick a meeting time — they go with what feels reasonable. Fifty units sounds thorough. A hundred sounds rigorous. Ten sounds like a quick check. None of those numbers came from a formula, and none of them carry a stated confidence level, a margin of error, or a defensible rationale.

That gap — between the number you chose and the number you should have chosen — is where audit findings live. It is where regulators ask uncomfortable questions. It is where certification auditors note that your inspection methodology lacks statistical basis. And it is where you discover, after accepting a lot, that the defect rate you missed was entirely predictable with the right sample size.

 

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Why Sample Size Matters More Than Sampling Frequency

Quality teams spend a lot of time deciding how often to sample. Daily, weekly, per batch, per shift. The frequency question matters, but it is not the most important sampling question. The most important question is: when you do sample, are you taking enough units to reach a reliable conclusion?

A sample that is too small gives you a result that looks precise but carries enormous hidden uncertainty. You measure ten units, find zero defects, and conclude the process is clean. But with ten units and zero defects at 95% confidence, your upper bound on the true defect rate is around 26%. One in four units could be defective, and your sample of ten would still produce zero defects more than 5% of the time. That is not quality assurance. That is luck.

The sample size formulas used in Sample Size Calculator eliminate the luck. The sample size in Sample Size Calculator work backwards from the conclusion you need — a stated confidence level and a maximum acceptable error — to the sample size required to reach that conclusion reliably. The number in Sample Size Calculator is not arbitrary. Every input is explicit and auditable.

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Variable Sampling: When You Measure, Not Just Count

Variable sampling applies when your quality characteristic is a continuous measurement — a dimension, a weight, a temperature, a voltage, a torque value. You are not asking “does it pass?” You are asking “what is the value, and is that value within specification?”

The formula is:

n = (Z × σ / E)²

Z is the Z-score for your confidence level: 1.645 for 90%, 1.960 for 95%, 2.576 for 99%. σ is the population standard deviation — your best estimate of how much the characteristic varies. E is the margin of error you are willing to accept — the maximum distance between your sample mean and the true population mean.

The formula tells you something important: sample size is proportional to the square of the ratio between variation and precision. Halve your acceptable error and you quadruple your required sample size. Double your population standard deviation and you quadruple your required sample size again. Precision is expensive. The Sample Size Calculator makes that cost visible before you commit to a sampling plan.

What Sigma to Use

The standard deviation input is the most common source of confusion in variable sampling. You need an estimate of the population standard deviation before you collect the sample — which means you are using existing data to plan future sampling.

Use your process control data. If you run SPC charts on the characteristic you are auditing, the within-subgroup standard deviation from those charts is your best estimate. When you have historical inspection data, calculate the standard deviation from the last several hundred measurements. If you have neither, run a pilot study of 30 to 50 units specifically to estimate sigma before committing to your full sampling plan.

The worst choice is to guess. A sigma that is too low produces a sample size too small to detect real variation, and a sigma that is too high produces an unnecessarily large sample. The input is worth the time to get right.

What the Margin of Error Means in Practice

The margin of error E defines the precision you need from your sample mean. If your specification has a tolerance of ±0.5mm and you want your sample mean to be within 0.1mm of the true process mean, your E is 0.1mm. If your process target is 50 grams and you want the sample mean within 2 grams, your E is 2 grams.

Set E relative to what matters for the decision you are making. If you are auditing to confirm a process is centred on its target, E should be small relative to the total tolerance — typically 10% to 25% of the tolerance band. If you are doing a rough confirmation that the process is in the right region, E can be larger, but your sample size drops and your conclusion becomes correspondingly less precise.

Finite Population Correction

The standard formula assumes an infinite population — or equivalently, a population so large that sampling a few hundred units does not appreciably change the remaining population’s composition. When your population is small relative to the sample size, that assumption breaks down, and the standard formula overestimates the required sample size.

The finite population correction adjusts for this:

n_adj = n&sub0; × N / (n&sub0; + N − 1)

Enter your population size N in the Sample Size Calculator and the adjustment applies automatically. For a lot of 100 units where the uncorrected formula gives n&sub0; = 16, the corrected sample size drops to 14. For a lot of 500 units with n&sub0; = 16, the corrected size is still 15 — the correction matters most when n&sub0; is large relative to N.

Reading the Variable Sampling Chart

The chart plots required sample size against margin of error for all three confidence levels simultaneously. Three curves appear — 90%, 95%, and 99% — each showing how dramatically sample size grows as you tighten your margin of error.

The curve shape matters as much as the specific number. The relationship is quadratic — the curves are not straight lines. Moving from E=0.5 to E=0.25 does not double your sample size; it quadruples it. Moving from E=0.5 to E=0.1 increases it by a factor of 25. The chart makes this non-linear cost visible in a way that a single calculated number cannot. When you present your sampling plan to a client or auditor, the chart answers the question “why not more precise?” with a clear visual answer about cost.

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Attribute Sampling: When You Count Defects, Not Measure Values

Attribute sampling applies when each unit is either conforming or non-conforming — pass or fail, defective or acceptable, present or missing. You are counting, not measuring. The quality characteristic is binary.

The formula is:

n = Z² × p(1 − p) / E²

Z is the same Z-score as variable sampling. p is the expected proportion defective in the population. E is the margin of error for the proportion estimate — the maximum acceptable distance between your sample proportion and the true population proportion.

Notice the p(1 − p) term. This reaches its maximum at p = 0.5 and decreases as p moves toward 0 or 1. At p = 0.5, you need the largest possible sample because the population is maximally uncertain — half defective, half not. At p = 0.01, the population is mostly conforming and you need a smaller sample to estimate that proportion precisely. This is why low defect rate processes need smaller samples for proportion estimation but can still miss individual defective units.

What Proportion to Enter

Enter your best estimate of the true defect rate in the population. Use recent inspection history, supplier quality data, or internal audit findings. If you have no prior data, enter 0.5 — this gives the worst-case (largest) sample size and ensures your result is conservative. Any actual defect rate below 0.5 means your true required sample size is smaller than the conservative estimate.

Do not confuse the proportion input with your AQL or your quality target. You are estimating what is actually in the population, not what you want to be there. If your process historically runs at 3% defective, enter 0.03, not your target of 0.5%.

Margin of Error for Attribute Sampling

The margin of error in attribute sampling is an absolute margin on the proportion. If you want to estimate the defect rate within ±2 percentage points, E is 0.02. If your expected defect rate is 5% and you want to know whether it is between 3% and 7%, E is 0.02.

A common mistake is entering E larger than p. If your expected defect rate is 2% and your margin of error is 3%, the formula is asking you to estimate a proportion within a range that includes negative values — which is impossible. The Sample Size Calculator flags this situation. Keep E smaller than both p and (1−p).

Reading the Attribute Sampling Chart

The chart plots required sample size against the proportion defective from 1% to 99%, with your chosen margin of error fixed. The curve is an inverted U — sample size rises as p approaches 0.5 and falls as p moves toward 0 or 1.

Your current proportion appears as a reference line on the chart. The three curves — 90%, 95%, and 99% confidence — show how much additional sample size each confidence level adds at your specific proportion. The visual makes the confidence level trade-off concrete: if 95% confidence requires 385 units and 99% requires 664 units, you can see exactly what the additional confidence costs in inspection effort and decide whether that cost is justified for this particular audit.

The chart also reveals an important planning insight: if your defect rate is very low — say 1% or 2% — the sample sizes at the left edge of the chart are much smaller than at p = 0.5. This means that low-defect processes actually need smaller samples for proportion estimation, even though they need large samples to reliably detect individual defective units. These are different problems with different solutions, and the chart keeps them clearly separate.

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Acceptance Sampling: Making Lot Disposition Decisions

Acceptance sampling answers a different question from variable and attribute sampling. Variable and attribute sampling ask: what is the quality level of this process or population? Acceptance sampling asks: should I accept or reject this specific lot?

The Sample Size Calculator uses the c=0 sampling plan — zero acceptance number. You inspect n units from the lot. If zero defectives are found, you accept the lot. If one or more defectives are found, you reject the lot. This is the most common plan used in quality agreements and supplier inspection because it is simple, defensible, and eliminates the ambiguity of plans that allow some defectives.

The sample size comes from two simultaneous constraints:

Consumer (RQL): n = ln(β) / ln(1 − RQL)

Producer (AQL): n = ln(α) / ln(1 − AQL)

The required n is the larger of the two results.

AQL, RQL, and the Two Risks

The AQL — Acceptable Quality Level — is the defect rate you consider acceptable. Lots at or below AQL quality should almost always be accepted. The producer risk α is the probability of incorrectly rejecting an AQL-quality lot. Setting α = 5% means that 5% of lots at exactly AQL quality will be rejected by this plan — a false alarm rate you impose on your supplier.

The RQL — Rejectable Quality Level — is the defect rate you consider clearly unacceptable. Lots at RQL quality should almost always be rejected. The consumer risk β is the probability of incorrectly accepting an RQL-quality lot. Setting β = 10% means that 10% of lots at exactly RQL quality will be accepted by this plan — a false pass rate that exposes you to bad product.

Both constraints must be satisfied simultaneously. The sample size is the maximum of the two because a smaller n would violate at least one constraint. The result shows you separately what n the consumer constraint requires and what n the producer constraint requires, so you can see which risk is driving your sample size.

Reading the OC Curve

The Operating Characteristic curve is the definitive visual for acceptance sampling. It plots the probability of acceptance on the vertical axis against lot quality — percent defective — on the horizontal axis.

An ideal OC curve would be a vertical step: 100% acceptance for all lots at or below AQL and 0% acceptance for all lots above RQL. Real OC curves cannot achieve this step shape with finite sample sizes. Instead, the curve slopes continuously from near 100% at low defect rates to near 0% at high defect rates, passing through the two control points defined by your AQL/alpha and RQL/beta inputs.

The four reference lines on the chart make the plan’s performance explicit. The AQL vertical line shows where acceptable quality lies. The RQL vertical line shows where rejectable quality lies. The 1−α horizontal line shows the minimum acceptance probability at AQL quality. The β horizontal line shows the maximum acceptance probability at RQL quality. The curve passes through or near both intersection points — that is what your sample size guarantees.

Use the OC curve to communicate your sampling plan to suppliers and quality managers. A supplier who sees the OC curve understands immediately that lots well above AQL quality will almost certainly be rejected, while lots well below AQL quality will almost certainly be accepted. The grey zone between AQL and RQL is where quality is ambiguous and inspection results vary — and that zone is visible on the chart.

Use the acceptance table below the curve to identify specific decision points. Lots at AQL quality show their acceptance probability. Lots at RQL quality show theirs. In Between Lots show the interpolated probability that defines the slope of the curve.

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How These Three Charts Change the Quality Audit Conversation

Quality audits often become debates about whether the sampling plan was adequate. The auditor asks how you selected your sample size. The inspector says it followed standard practice or matched the previous audit. The auditor notes it lacks statistical basis. The finding goes into the report.

These three charts end that conversation before it starts.

When a variable sampling chart shows your sample size on the curve at the specific confidence level and margin of error you stated, the rationale is self-documenting. While an attribute sampling chart shows your proportion estimate, your margin of error, and your confidence level, the inspector can confirm all three inputs from your quality records and verify the output independently. When an OC curve shows the AQL and RQL points with their associated acceptance probabilities, the sampling plan is its own evidence.

ISO 9001, ISO 13485, IATF 16949, and virtually every sector-specific quality standard require that sampling plans be statistically valid and that the basis for sample size selection be documented. These charts fulfil that requirement in one image. They turn a number — “we inspected 127 units” — into a statement — “we inspected 127 units, which at 95% confidence and ±3% margin of error on a process with 5% historical defect rate, gives us a result we can defend.”

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Connecting to eAuditor

eAuditor captures the inspection findings, audit results, and process data that feed your sample size decisions. When you complete an audit in eAuditor, the finding records give you the historical defect rates and process variation data you need to set up your next sampling plan correctly.

Pull your defect rate history from eAuditor for the attribute sampling proportion input. Pull your process measurement data for the variable sampling standard deviation input. Run the Sample Size Calculator  to determine your required sample sizes, then use eAuditor to execute the inspection and record the results. The PDF report from the Sample Size Calculator  documents the statistical basis for your sample size alongside the eAuditor inspection record.

Visit eAuditor.app to see how eAuditor supports the full audit cycle — from sampling plan through inspection execution to finding closure.

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Start Calculating Your Sample Sizes Today

The Sample Size Calculator is on this page. Select your sampling mode, enter your parameters, and click Calculate. Your required sample size, the comparison chart, and the detailed breakdown table appear immediately. Download the PDF to document your sampling plan for your quality records or audit file.

eAuditor provides the quality data that makes every input more accurate. Visit eAuditor.app to see how eAuditor connects your audit history to your sampling decisions.

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The Sample Size Calculator processes all data locally in your browser. Nothing is sent to any server. All calculations, chart rendering, and PDF exports happen on your device.

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Frequently Asked Questions

1. What is the difference between variable and attribute sampling?

Variable sampling applies when your quality characteristic is a continuous number — a measurement like length, weight, temperature, or hardness. You collect actual measured values and estimate the population mean or standard deviation. Attribute sampling applies when each unit is simply conforming or non-conforming — a binary pass/fail classification. You count how many units fail and estimate the proportion defective. The key practical difference: variable sampling usually requires smaller sample sizes than attribute sampling to reach the same level of precision, because measurements carry more information than binary classifications.

2. What confidence level should I use for quality audits?

95% is the standard for most quality applications and the default recognised by ISO standards and regulatory bodies. Use 90% when you need a quick screening check and can accept a higher error rate — incoming inspection for low-risk components, for example. Use 99% when the cost of a wrong conclusion is high — safety-critical characteristics, regulatory submissions, final release of high-value products. The Sample Size Calculator shows sample sizes for all three simultaneously so you can see the cost difference and make a deliberate choice.

3. What is a margin of error and how do I choose it?

The margin of error is the maximum distance you will accept between your sample result and the true population value. For variable sampling, it is the maximum difference between your sample mean and the true process mean, expressed in the same units as your measurement. For attribute sampling, it is the maximum difference between your sample proportion and the true defect rate, expressed as a decimal or percentage. Choose it based on what matters for the decision you are making. If a 2% error in defect rate estimate would change your disposition decision, set E to 0.02. If a 5% error is acceptable, set E to 0.05.

4. What standard deviation should I enter for variable sampling?

Use the best available estimate of the true population standard deviation for the characteristic you are auditing. Sources in order of preference: within-subgroup standard deviation from your SPC control charts, standard deviation calculated from a representative sample of recent production measurements, or a pilot study of 30 to 50 units run specifically to estimate sigma. Do not use the specification tolerance as a proxy for standard deviation — they measure completely different things. If you genuinely have no prior data, run a pilot study before committing to a full sampling plan.

5. What is the finite population correction and when does it matter?

The finite population correction (FPC) reduces the required sample size when your population is small relative to the uncorrected sample size. The standard formulas assume the population is large enough that sampling from it does not materially change the remaining composition. When your lot or population is small — say, a batch of 100 units where the formula suggests sampling 40 — the FPC applies a downward adjustment. Enter your population or lot size in the Population Size field and the Sample Size Calculator applies the correction automatically. The adjustment is most significant when the uncorrected sample size exceeds about 5% of the population size.

6. What proportion should I enter for attribute sampling?

Enter your best estimate of the true defect rate in the population you are auditing. Use recent process data, supplier quality records, or historical audit findings. If you have no prior data, enter 0.5 — this gives the maximum (most conservative) sample size because p = 0.5 maximises the p(1−p) term in the formula. Any actual defect rate below 0.5 means your true required sample size is smaller than the conservative estimate. Never enter your quality target as the proportion — you are estimating what is actually in the population, not what you want to be there.

7. Why does sample size peak at p = 0.5 in attribute sampling?

The attribute sampling formula contains the term p(1−p), which reaches its maximum value of 0.25 when p = 0.5. This reflects the statistical reality that a population that is exactly 50% defective is the hardest to characterise precisely — a sample from that population could come back anywhere from mostly conforming to mostly defective. As the true defect rate moves toward 0% or 100%, the population becomes more homogeneous and a smaller sample gives a more precise estimate. The chart makes this clearly visible as the inverted-U curve shape.

8. What is the c=0 acceptance sampling plan?

The c=0 plan — zero acceptance number — means you inspect n units and accept the lot only if you find zero defective units. One defective found in the sample triggers lot rejection. This is the most conservative of all acceptance sampling plans and the most widely used in quality agreements and supplier contracts because it is simple to administer, requires no ambiguous decisions about borderline acceptance numbers, and aligns with zero-defect quality philosophies. The trade-off is that it requires larger sample sizes than plans that allow some defectives (c=1, c=2, etc.) to achieve the same consumer risk protection.

9. What is AQL and how do I set it?

AQL stands for Acceptable Quality Level — the maximum defect percentage that you consider acceptable for routine production. Lots at or below AQL quality should almost always be accepted. Set your AQL based on the quality agreement with your customer or supplier, the criticality of the characteristic, and your process capability. Common values are 0.065%, 0.10%, 0.25%, 0.40%, 0.65%, 1.0%, 1.5%, 2.5%, and 4.0% — these are the values tabulated in ISO 2859 and ANSI/ASQ Z1.4. For critical characteristics, use a lower AQL. For minor characteristics, a higher AQL is often acceptable.

10. What is RQL and how does it relate to AQL?

RQL stands for Rejectable Quality Level — the defect percentage at which you consider a lot clearly unacceptable and want a high probability of rejection. Also called the Lot Tolerance Percent Defective (LTPD) or Unacceptable Quality Level (UQL) in some standards. RQL must be greater than AQL — there must be a quality level that is clearly acceptable and a quality level that is clearly rejectable, with the sampling plan distinguishing between them. A common ratio is RQL = 10 × AQL. Tighter ratios (RQL = 5 × AQL) require larger sample sizes. Wider ratios (RQL = 20 × AQL) allow smaller sample sizes but create a larger grey zone of ambiguous quality.

11. What is producer risk and consumer risk?

Producer risk (α) is the probability that your sampling plan rejects a lot that is actually at or below AQL quality — a false rejection that penalises your supplier for producing acceptable product. Consumer risk (β) is the probability that your sampling plan accepts a lot that is at or above RQL quality — a false acceptance that exposes you to bad product. Both risks are unavoidable with finite sample sizes. The Sample Size Calculator sets the sample size to satisfy both constraints simultaneously: α typically set at 5% and β at 10%, meaning the plan incorrectly rejects 1 in 20 acceptable lots and incorrectly accepts 1 in 10 rejectable lots.

12. How do I read the Operating Characteristic (OC) curve?

The OC curve plots the probability of lot acceptance on the vertical axis against the true lot quality — percent defective — on the horizontal axis. Read it left to right: at low defect rates (good quality), the curve sits near 100% — nearly every lot gets accepted.

As defect rate increases, the curve slopes downward.

At high defect rates (bad quality), the curve approaches 0% — nearly every lot gets rejected. The AQL vertical line shows where acceptable quality sits on the curve, and the 1−α horizontal line shows the minimum acceptance probability there. The RQL vertical line shows where rejectable quality sits, and the β horizontal line shows the maximum acceptance probability there. Your sampling plan guarantees the curve passes through or near both intersection points.

13. Why does the producer constraint sometimes require a larger sample than the consumer constraint?

The relative size of the two sample requirements depends on the AQL/RQL ratio and the relative risk levels. When AQL is very tight (low percentage) and producer risk is also tight (low α), the producer constraint requires a large sample because you need to be very sure you are not rejecting good lots. When RQL is not much larger than AQL, both constraints require large samples because the plan must discriminate between quality levels that are close together. The Sample Size Calculator shows both n values separately so you can see which constraint is binding and adjust your parameters accordingly if the total is too large for your inspection capacity.

14. Can I use this Sample Size Calculator for ISO 2859 / ANSI Z1.4 sampling plans?

The Sample Size Calculator  uses the mathematical framework underlying ISO 2859 and ANSI/ASQ Z1.4 but does not reproduce their exact sampling tables. Those standards use pre-computed tables indexed by lot size and inspection level that round to standard sample sizes and acceptance numbers. The Sample Size Calculator gives you the mathematically exact sample size for your specific AQL, RQL, and risk inputs, which may differ slightly from the nearest tabled value. For formal regulatory submissions or customer contracts that specify compliance with a named standard, use the tables from that standard directly. For internal planning, process capability studies, and audit sampling, the Sample Size Calculator’s exact formula is more flexible and equally defensible.

15. How do I document my sample size for a quality audit?

Document three things: the input parameters (sigma or proportion, margin of error, confidence level, population size), the formula used (variable or attribute, with or without FPC), and the resulting sample size. The PDF report from this Sample Size Calculator  contains all three in a single document. Attach it to your inspection plan, your audit checklist, or your sampling procedure as the basis-of-selection evidence. For acceptance sampling, include the OC curve as a visual demonstration that the plan satisfies both AQL and RQL constraints. Certification auditors reviewing your quality management system will accept statistical documentation at this level of detail.

16. What sample size should I use when I have no historical data at all?

For variable sampling with no historical sigma, run a pilot study of 30 to 50 units, calculate the standard deviation, and use that as your sigma input. When attribute sampling with no historical defect rate, use p = 0.5 — the conservative worst-case that gives the maximum sample size. For acceptance sampling with no quality history, set AQL and RQL based on your quality agreement or product risk assessment and use standard risk levels of α = 5% and β = 10%. These conservative defaults ensure your sample size is large enough to reach valid conclusions even when your prior estimates are uncertain.

17. Is there a minimum sample size I should always use regardless of the formula?

The formula results are mathematically correct, but very small sample sizes carry practical limitations. A formula result of n = 3 or n = 5 is technically valid given the inputs, but such small samples are sensitive to outliers and may not adequately represent the population’s natural variation. Many quality professionals apply a practical minimum of 30 units for variable sampling (to invoke the central limit theorem for non-normal populations) and a practical minimum of 50 to 100 units for attribute sampling when the expected defect rate is very low. When the formula gives a result below your practical minimum, use the larger value and note in your documentation that you exceeded the statistical minimum.

18. How does this Sample Size Calculator  differ from a sample size table?

Published sample size tables — like those in ISO 2859, ANSI/ASQ Z1.4, or MIL-STD-1916 — pre-compute sample sizes for standard combinations of lot size, AQL, and inspection level. They are fast to use but inflexible: if your situation does not match the table’s assumptions, you must accept the nearest tabled value. This Sample Size Calculator  computes exactly from your specific inputs — your actual sigma, your actual defect rate, your specific margin of error, your actual risk tolerances. The result is a sample size tailored to your situation rather than rounded to the nearest standard value. For planning, process design, and quality agreements, the exact computation is more useful. For formal compliance with a named standard’s tables, use the standard’s tables directly.

19. What happens if my required sample size exceeds my lot size?

If the uncorrected formula gives a sample size larger than your lot, apply the finite population correction — enter your lot size in the Population Size field. The FPC will always reduce the required sample size below the population size. In practice, if the FPC result is still very close to the lot size (say, 90% or more of the lot), you should consider 100% inspection rather than sampling. At that point you are inspecting nearly every unit anyway, and 100% inspection eliminates the sampling risk entirely and removes the need for statistical justification.

20. How does this Sample Size Calculator  connect to eAuditor?

eAuditor captures the inspection findings, process measurements, and audit results that provide the historical data for your sample size inputs. Your defect rate history from eAuditor inspections gives you the proportion input for attribute sampling. Your process measurement records from eAuditor audits give you the standard deviation estimate for variable sampling. After calculating your required sample size, use eAuditor to execute the inspection, record every unit’s result, and track findings through to closure. The PDF report from the Sample Size Calculator  documents the statistical basis for your sample size and pairs naturally with the eAuditor inspection record as a complete audit evidence package. Visit eAuditor.app to see how eAuditor supports your full quality audit workflow.