Streaming Average Update for Real-Time Estimation

In many real-world applications, data arrives continuously as a stream, and we need to maintain statistical summaries without storing every data point. One of the most essential streaming statistics is the running average (also known as the online mean or incremental mean).

Consider a scenario where you're monitoring sensor readings, user engagement metrics, or reward signals in a reinforcement learning agent. Storing millions or billions of historical observations is often impractical due to memory constraints. Instead, we can maintain an accurate running average using only three pieces of information:

• The current mean estimate (Q)
• The count of observations (n)
• The new observation (x)

The Incremental Update Formula:

When a new observation arrives, the updated mean can be computed using:

$$Q_{new} = Q_{old} + \frac{1}{n} \cdot (x - Q_{old})$$

This elegant formula has an intuitive interpretation: we adjust the old estimate toward the new value by a fraction that decreases as more samples are accumulated. Early observations have more influence on the estimate, while later observations make smaller adjustments—exactly the behavior we'd expect from a proper averaging process.

Why This Formula Works:

The term (x - Q_old) represents the prediction error—how far the new observation deviates from our current estimate. We then scale this error by 1/n to determine the appropriate adjustment. This is mathematically equivalent to recomputing the mean from scratch, but requires O(1) space and O(1) time per update instead of O(n) for both.

Your Task: Write a function that computes the updated running average given:

• The previous mean estimate
• The current sample count (including the new observation)
• The new observed value

Return the updated mean after incorporating the new observation.