An alternative expression for the constant c4[n] with desirable properties

Abstract

The constant c4[n] is commonly used in the construction of control charts and the estimation of process capability indices, where n denotes the sample size. Assuming normal distribution, the unbiased estimator of the population standard deviation is obtained by dividing the sample standard deviation by the constant c4[n]. An alternative expression for c4[n] is proposed and the mathematical induction technique is used to prove its validity in the present paper. Some desirable properties are described. First, the suggested expression provides the exact value of c4[n]. Second, it is not a recursive formula in the sense it does not depend on the previous sample size. Finally, the value of c4[n] can be directly computed for large sample sizes. Such properties suggest that the proposed expression may be a convenient solution in computer programming and it has direct applications in statistical quality control.