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Function-as-a-Service (FaaS) is becoming an increasingly popular cloud-deployment paradigm for serverless computing that frees application developers from managing the infrastructure. At the same time, it allows cloud providers to assert control in workload consolidation, i.e., co-locating multiple containers on the same server, thereby achieving higher server utilization, often at the cost of higher end-to-end function request latency. Interestingly, a key aspect of serverless latency management has not been well studied: the trade-off between application developers' latency goals and the FaaS providers' utilization goals. This paper presents a multi-faceted, measurement-driven study of latency variation in serverless platforms that elucidates this trade-off space. We obtained production measurements by executing FaaS benchmarks on IBM Cloud and a private cloud to study the impact of workload consolidation, queuing delay, and cold starts on the end-to-end function request latency. We draw several conclusions from the characterization results. For example, increasing a container's allocated memory limit from 128 MB to 256 MB reduces the tail latency by 2× but has 1.75× higher power consumption and 59% lower CPU utilization.