It is routinely desired to estimate certain statistical properties from the quantized representation of a signal. This article considers the estimators for the local average and variance of the process, and establishes expressions for the mean, variance, and envelopes of the probability density functions of these two estimators. The relationships obtained closely match simulation results, even in abnormal circumstances such as estimation window of a few elements only, or quantization where the step comes close to the signal range. The results give a quantitative justification to the known fact that increasing quantizer resolution brings relatively little improvement in the quality of moment estimates.