Chi-square　statistics　of　the　uncertainty　weighted　mean,　which　is　a　linear　estimator,　is　widely　used　in　data　analysis　of　inter-laboratory　comparison.　However,　the　chi-square　statistics　of　other　linear　estimators　is　not　investigated.　In　this　study,　a　modified　chi-square　statistics,　which　comprises　the　linear　estimator,　is　proposed　under　the　condition　that　comparison　results　are　Gaussian　distributed　with　a　common　mean.　The　proposed　statistics　is　analyzed　through　Monte　Carlo　simulation　by　combining　the　weights　of　linear　estimator　into　a　multi-dimension　vector.　Simulation　results　show　that　the　proposed　statistics　is　(n-1)　th-order　chi-square　distributed　when　the　weights　vector　of　linear　estimator　is　located　in　a　particular　subspace,　which　is　influenced　by　the　uncertainties　of　participants.　Furthermore,　this　chi-square　statistics　of　arithmetic　mean　is　applied　to　the　common　mean　and　random　effects　models　as　examples.　For　the　common　mean　model,　the　statistics　can　be　applied　to　the　hypothesis　testing　of　arithmetic　mean;　for　the　random　effects　model,　the　statistics　can　be　applied　to　the　variance　estimation　of　random　effects.　(C)　2018　Elsevier　Ltd.　All　rights　reserved.