Rolling Statistics

Rolling Statistics

A rolling window calculation computes a statistic over a sliding fixed-size window of data. This is essential for detecting trends, smoothing noise, and measuring recent volatility in time series.

For a window of size $w$ applied to a series $x_1, x_2, ..., x_n$:

• Positions 0 through $w-2$ have fewer than $w$ values available → output None
• Position $i \geq w-1$ uses values $x_{i-w+1}$ through $x_i$

Rolling Mean at position $i$: $\bar{x}_i = \frac{1}{w} \sum_{j=i-w+1}^{i} x_j$

Rolling Std at position $i$ (sample standard deviation): $s_i = \sqrt{\frac{1}{w-1} \sum_{j=i-w+1}^{i} (x_j - \bar{x}_i)^2}$

Your Task

Implement:

• rolling_mean(xs, window) — list of rolling means, None for first window-1 positions
• rolling_std(xs, window) — list of rolling sample stds, None for first window-1 positions