The mean is the arithmetic average of a set of values — the sum divided by the count. For values :
The mean is the first moment of the distribution. It tells us where the values are centered, in the same units as the data itself.
In Feature extraction from a signal, the rolling mean within a window is one of the most basic features. For an ECG window containing one heartbeat, the mean is the average voltage — close to zero for a centered signal, a small positive number for an uncentered one. The mean captures the baseline level of the signal in that window.
In Normalization, the mean is subtracted from each value to produce a zero-mean signal — the first step in computing the z-score.
For higher moments that describe more of the distribution’s shape:
- Variance (second moment) — how spread out the values are. Its square root is the Standard deviation.
- Skewness (third moment) — asymmetry. Positive skew means a long right tail; negative skew means a long left tail.
- Kurtosis (fourth moment) — tail heaviness. Tells us how prone the distribution is to extreme values.
Together, the first four moments are a remarkably compact summary of a distribution’s shape. They’re the standard feature set for windowed analysis of signals.
The mean is computed in Pandas with df['col'].mean() (over the whole column) or df['col'].rolling(N).mean() (over a sliding window). In NumPy it’s np.mean(arr).
A subtle point: the arithmetic mean is one of several notions of average. The median (middle value) and mode (most common value) are alternatives that behave differently for skewed distributions. For a perfectly symmetric distribution, mean = median = mode. For a positively skewed distribution, mean > median > mode. For a negatively skewed distribution, the order reverses.