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Prior studies on covert communication with noise uncertainty adopted a worst-case approach from the warden's perspective. That is, the worst-case detection performance of the warden is used to assess covertness, which is overly optimistic. Instead of simply considering the worst limit, we take the distribution of noise uncertainty into account to evaluate the overall covertness in a statistical sense. Specifically, we define new metrics for measuring the covertness, which are then adopted to analyze the maximum achievable rate for a given covertness requirement under both bounded and unbounded noise uncertainty models.