Wireless Body Area Networks (WBANs) comprise a network of sensors subcutaneously implanted or placed near the body surface and facilitate continuous monitoring of health parameters of a patient. Research endeavours involving WBAN are directed towards effective transmission of detected parameters to a Local Processing Unit (LPU, usually a mobile device) and analysis of the parameters at the LPU or a back-end cloud. An important concern in WBAN is the lightweight nature of WBAN nodes and the need to conserve their energy. This is especially true for subcutaneously implanted nodes that cannot be recharged or regularly replaced. Work in energy conservation is mostly aimed at optimising the routing of signals to minimise energy expended. In this paper, a simple yet innovative approach to energy conservation and detection of alarming health status is proposed. Energy conservation is ensured through a two-tier approach wherein the first tier eliminates `uninteresting' health parameter readings at the site of a sensing node and prevents these from being transmitted across the WBAN to the LPU. A reading is categorised as uninteresting if it deviates very slightly from its immediately preceding reading and does not provide new insight on the patient's well being. In addition to this, readings that are faulty and emanate from possible sensor malfunctions are also eliminated. These eliminations are done at the site of the sensor using algorithms that are light enough to effectively function in the extremely resource-constrained environments of the sensor nodes. We notice, through experiments, that this eliminates and thus reduces around 90% of the readings that need to be transmitted to the LPU leading to significant energy savings. Furthermore, the proper functioning of these algorithms in such constrained environments is confirmed and validated over a hardware simulation set up. The second tier of assessment includes a proposed anomaly detection model at the LPU that is capable of identifying anomalies from streaming health parameter readings and indicates an adverse medical condition. In addition to being able to handle streaming data, the model works within the resource-constrained environments of an LPU and eliminates the need of transmitting the data to a back-end cloud, ensuring further energy savings. The anomaly detection capability of the model is validated using data available from the critical care units of hospitals and is shown to be superior to other anomaly detection techniques.