What is Standard Deviation?

The Story That Explains Standard Deviation

You are a quality control manager at a biscuit factory. Every biscuit should weigh exactly 20 grams. At the end of the day two machines both produced biscuits with a mean weight of 20g. Your boss is happy. But you check the individual weights.

Machine A: mean = 20g, std dev = 0.15g — nearly perfect every time.
Machine B: mean = 20g, std dev = 4.36g — all over the place.

What Standard Deviation Actually Means

Standard deviation answers one simple question: "On average, how far is each value from the mean?"

Step-by-Step Calculation

Dataset — exam scores of 8 students: [52, 74, 68, 90, 61, 85, 72, 78]

Python Implementation

Manual calculation

scores = [52, 74, 68, 90, 61, 85, 72, 78]

n    = len(scores)
mean = sum(scores) / n

squared_diffs = [(x - mean) ** 2 for x in scores]
variance      = sum(squared_diffs) / (n - 1)
std_dev       = variance ** 0.5

print(f"Mean:     {mean}")           # 72.5
print(f"Variance: {variance:.2f}")   # 152.57
print(f"Std Dev:  {std_dev:.2f}")    # 12.35

Using the statistics module

import statistics

scores = [52, 74, 68, 90, 61, 85, 72, 78]

print(statistics.mean(scores))       # 72.5
print(statistics.variance(scores))   # 152.57   (sample)
print(statistics.stdev(scores))      # 12.35    (sample std dev)
print(statistics.pstdev(scores))     # 11.57    (population std dev)

Using NumPy

import numpy as np

scores = [52, 74, 68, 90, 61, 85, 72, 78]

print(np.mean(scores))               # 72.5
print(np.std(scores, ddof=1))        # 12.35   (sample)
print(np.std(scores, ddof=0))        # 11.57   (population)
print(np.var(scores, ddof=1))        # 152.57  (sample variance)

Complete comparison

import numpy as np

machine_a = [19.8, 20.1, 20.0, 19.9, 20.2]
machine_b = [14.0, 26.0, 18.0, 22.0, 20.0]

for name, data in [("Machine A", machine_a), ("Machine B", machine_b)]:
    mean    = np.mean(data)
    std_dev = np.std(data, ddof=1)
    print(f"{name}: mean={mean:.1f}g  std_dev={std_dev:.2f}g")

# Machine A: mean=20.0g  std_dev=0.15g
# Machine B: mean=20.0g  std_dev=4.36g

The 68 – 95 – 99.7 Rule

When data follows a normal distribution (bell curve), standard deviation lets you predict exactly what percentage of values fall within any range. This is called the Empirical Rule.

Real Example — Adult Male Heights

Adult male heights follow a normal distribution with mean = 175 cm and std dev = 7 cm.

import numpy as np

mean    = 175   # cm
std_dev = 7     # cm

print(f"68% range: {mean - std_dev} – {mean + std_dev} cm")
# 68% range: 168 – 182 cm

print(f"95% range: {mean - 2*std_dev} – {mean + 2*std_dev} cm")
# 95% range: 161 – 189 cm

print(f"99.7% range: {mean - 3*std_dev} – {mean + 3*std_dev} cm")
# 99.7% range: 154 – 196 cm

Standard Deviation in Machine Learning

Standard deviation is not just a statistics concept — it powers several core machine learning techniques you will use every day.

Feature Scaling — StandardScaler

Many ML algorithms (SVM, KNN, Neural Networks) are sensitive to the scale of features. StandardScaler transforms each feature so that it has mean = 0 and std dev = 1. This is called Z-score normalisation.

from sklearn.preprocessing import StandardScaler
import numpy as np

# Raw features — very different scales
heights = [[160], [175], [180], [155], [190]]   # cm
weights = [[55],  [70],  [80],  [50],  [95]]    # kg

scaler = StandardScaler()
scaled = scaler.fit_transform(heights)

print("Original heights:", [h[0] for h in heights])
print("Scaled heights:  ", [round(s[0], 2) for s in scaled])
# Original heights: [160, 175, 180, 155, 190]
# Scaled heights:   [-0.91, 0.15, 0.54, -1.3, 1.52]

Anomaly Detection

import numpy as np

# Server response times (ms)
response_times = [120, 118, 122, 119, 121, 120, 118, 500, 123, 119]

mean    = np.mean(response_times)
std_dev = np.std(response_times, ddof=1)

print(f"Mean:    {mean:.1f} ms")
print(f"Std Dev: {std_dev:.1f} ms")

# Any value more than 3 std devs from mean is an anomaly
threshold = mean + 3 * std_dev

anomalies = [t for t in response_times if t > threshold]
print(f"Anomalies detected: {anomalies}")
# Anomalies detected: [500]

Variance vs Standard Deviation — Summary

Golden Rules