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Variables sampling plan
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<p>In <a href="/facts/Statistics/DNPsKGYU">statistics</a>, a variables sampling plan is an <a href="/facts/Acceptance_sampling/DID8JcyG">acceptance sampling</a> technique. Plans for <a href="/facts/Variable_(mathematics)/VbQlrvKz">variables</a> are intended for quality characteristics that are measured on a continuous scale. This plan requires the knowledge of the <a href="/facts/Statistical_model/jKkT5ftm">statistical model</a> (e.g. <a href="/facts/Normal_distribution/UapjjPyQ">normal distribution</a>). The historical <a href="/facts/Evolution_of_mathematics/18h5rQA5">evolution</a> of this technique dates back to the seminal work of <a href="/facts/W._Allen_Wallis/YkD5HU8A">W. Allen Wallis</a> (1943). The purpose of a plan for variables is to assess whether the process is operating far enough from the specification limit. Plans for variables may produce a similar OC curve to attribute plans with significantly less sample size. </p><p>The decision criterion of these plans are </p> X ¯ + k σ ⩽ U S L {\displaystyle {\bar {X}}+k\sigma \leqslant USL} or X ¯ − k σ ⩾ L S L {\displaystyle {\bar {X}}-k\sigma \geqslant LSL} <p>where X ¯ {\displaystyle {\bar {X}}} and σ {\displaystyle \sigma } are the <a href="/facts/Sample_mean/Ah8VVVDT">sample mean</a> and the <a href="/facts/Standard_deviation/qSug9BRl">standard deviation</a> respectively, k {\displaystyle k} is the <a href="/facts/Critical_distance/pPnS7bcK">critical distance</a>, U S L {\displaystyle USL} and L S L {\displaystyle LSL} are the upper and lower regulatory limits. When the above expression is satisfied the proportion nonconforming is lower than expected and therefore the lot is accepted. </p><p>A variables sampling plan can be designed so that the OC curve passes through two points (AQL, α {\displaystyle \alpha } ) and (LQL, β {\displaystyle \beta } ). AQL and LQL are the <a href="/facts/Acceptable_quality_limit/HADFzpp5">Acceptable quality limit</a> and the limiting quality level respectively. α {\displaystyle \alpha } and β {\displaystyle \beta } are the producer and consumer's risks. The required <a href="/facts/Sample_size_determination/sf8dAFzq">sample size</a> ( n {\displaystyle n} ) and the critical distance ( k {\displaystyle k} ) can be obtained as </p> k = Z L Q L Z α + Z A Q L Z β Z α + Z β {\displaystyle k={\frac {Z_{LQL}Z_{\alpha }+Z_{AQL}Z_{\beta }}{Z_{\alpha }+Z_{\beta }}}} <p>where Z {\displaystyle Z} is the <a href="/facts/Normal_distribution/UapjjPyQ">normal distribution</a> function. </p><p>When the <a href="/facts/Dispersion_(statistics)/xqevRHmz">dispersion</a> is known the required sample size ( n {\displaystyle n} ) is obtained from </p> n = ( Z α + Z β Z A Q L − Z L Q L ) 2 {\displaystyle n=\left({\frac {Z_{\alpha }+Z_{\beta }}{Z_{AQL}-Z_{LQL}}}\right)^{2}} <p>while for unknown σ {\displaystyle \sigma } the sample size is approximately </p> n = ( Z α + Z β Z A Q L − Z L Q L ) 2 ( 1 + k 2 2 ) {\displaystyle n=\left({\frac {Z_{\alpha }+Z_{\beta }}{Z_{AQL}-Z_{LQL}}}\right)^{2}\left(1+{\frac {k^{2}}{2}}\right)} <p>The MIL-STD-414 provides tables to obtain the required sample size and the critical distance according to the type of inspection. </p>