Variance and Standard Deviation: How Spread Out?

Both classes have an average test score of 80. Class A: everyone got between 78 and 82. Class B: half got 60 and half got 100. Same mean, very different stories. Variance and standard deviation capture this.

The formula

import numpy as np

class_a = [78, 79, 80, 80, 81, 82]
class_b = [60, 60, 60, 100, 100, 100]

print('Class A mean:', np.mean(class_a))       # 80
print('Class A std:', np.std(class_a))          # ~1.3
print('Class B mean:', np.mean(class_b))       # 80
print('Class B std:', np.std(class_b))          # ~20

# Same mean, very different spreads
# Class B's standard deviation is ~15x Class A's

The 68-95-99.7 rule

For a normal (bell-shaped) distribution, about 68 percent of values fall within one standard deviation of the mean, 95 percent within two, and 99.7 percent within three. This is the empirical rule, and it lets you eyeball whether a single value is normal or unusual.

Why ML cares about variance

• Feature scaling: algorithms like SVMs and neural nets expect features with similar variance
• Model evaluation: reporting mean accuracy across runs without standard deviation is misleading
• Z-scores: (x - mean) / std is a common normalization step
• Detecting drift: if a feature's variance shifts over time, data is changing

Standard deviation in the wild

• IQ scores: mean 100, std 15. A score of 130 is two standard deviations above average.
• US adult male height: mean 175 cm, std 7 cm. A 190 cm man is about two std above average.
• Daily stock returns: often a std of 1 to 2 percent for broad indices

The big idea: a dataset is a cloud, not a point. Mean locates the center; standard deviation describes the cloud's fuzziness. Always report both.