Descriptive Statistics: Mean, Variance & Std

Overview

Before modelling anything, you describe it. The mean is the center of mass of your data; the median is the middle value (robust to outliers); the variance measures how spread out the values are (average squared distance from the mean); and the standard deviation is the square root of variance, back in the original units. These are not just summary trivia — they drive real ML practice. Feature normalization (subtract the mean, divide by the std) rescales every feature to a comparable range so gradient descent converges faster and no single large-scale feature dominates. Understanding spread also underlies detecting outliers, comparing distributions, and reasoning about model confidence. When a data scientist "looks at the data first", these numbers are what they look at.

Center and spread in one glance

import numpy as np

data = np.array([10, 12, 11, 13, 12, 100])   # one outlier (100)

print("mean  ", np.mean(data))     # 26.33 -> pulled up by the outlier
print("median", np.median(data))   # 12.0  -> robust to it
print("var   ", round(np.var(data), 1))
print("std   ", round(np.std(data), 1))

Standardization: the everyday use in ML

import numpy as np

feature = np.array([50.0, 60.0, 70.0, 80.0, 90.0])
z = (feature - feature.mean()) / feature.std()   # standardize
print(np.round(z, 3))              # mean 0, std 1
print(round(z.mean(), 6), round(z.std(), 6))     # 0.0 1.0

Key Points to Remember

• 1Mean = center of mass; median = middle value (robust to outliers)
• 2Variance = average squared distance from the mean; std = its square root (same units)
• 3Standardization (z-score) rescales features to mean 0, std 1
• 4Normalized features help gradient descent converge and prevent scale dominance