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The presence of the sigma phase in high-entropy alloys significantly affects their mechanical properties. Therefore, control of sigma phase formation is crucial. Current understanding of the stability of the sigma phase is largely based on the valence electron concentration (VEC) criterion. Here, we show that VEC alone is insufficient to describe its stability. A second criterion for sigma phase formation based on alloy composition is proposed. A new parameter called paired sigma-forming element (PSFE) content is invented to quantify this criterion. If PSFE content is lower than the threshold value, the sigma phase will not form even if the VEC is correct.