To handle the nonnormal data issue, Browne proposed an unbiased distribution free (DF) estimator (ΓˆUDF) and an asymptotically distribution free estimator (ΓˆADF) of the covariance matrix of sample variances/covariances Γ to calculate robust test statistics and robust standard errors. However, ΓˆUDF is ignored in methodological and substantive research, and has not been extended to models with mean structures. To improve robust standard errors and the model fit statistic for nonnormal data with mean structures (e.g., growth curve models), we propose an unbiased distribution free estimator with mean structures considered. In growth curve models, we apply ΓˆUDF to four robust statistics that have relatively simple forms and denote them as TUSB, TUMVA, TUMVA2, and TUCOR1. We compare their performance with 7 robust test statistics that employ ΓˆADF. We find that with the same model fit statistic, ΓˆUDF generally leads to smaller Anderson-Darling distances from the theoretical distribution than ΓˆADF, except TUMVA2 in some skewed cases. Additionally, the p-values from TUMVA2 are distributed closest to the theoretical distribution Uniform(0,1) among the 12 examined statistics. In terms of Type I error rates, TUMVA2 and TMVA2 are the most stable statistics. Additionally, ΓˆUDF provides smaller relative biases of the robust SE estimates than ΓˆADF. Hence, we suggest using ΓˆUDF in both model fit statistics and robust SE calculation. Among the model fit statistics using ΓˆUDF, we suggest TUMVA2.