Descriptive Statistics — Summarizing Data You Can Trust

Why Statistics Still Matters in the Age of AI

Large language models can generate analysis, but they can't replace the judgment to know when the numbers are lying. Statistics is that judgment. It's the difference between a data scientist who ships accurate models and one who ships confident-sounding wrong answers.

This course teaches statistics the way it connects to real data work — not as abstract math, but as thinking tools.

Measures of Central Tendency

The three ways to describe "the middle" of a dataset each capture something different:

• Mean: Sum divided by count. Sensitive to outliers. Use when the distribution is symmetric.
• Median: Middle value when sorted. Robust to outliers. Use for skewed data (income, prices).
• Mode: Most frequent value. Useful for categorical data.

import numpy as np
import pandas as pd

data = [12, 15, 14, 10, 13, 14, 100]  # 100 is an outlier

print(f"Mean:   {np.mean(data):.1f}")    # 25.4 — pulled by outlier
print(f"Median: {np.median(data):.1f}")  # 14.0 — robust
print(f"Mode:   {pd.Series(data).mode()[0]}")  # 14

# With pandas describe()
s = pd.Series(data)
print(s.describe())
# count    7.000000
# mean    25.428571
# std     32.307...
# min     10.000000
# 25%     12.500000
# 50%     14.000000  <- median
# 75%     14.500000
# max    100.000000

Measures of Spread

Variance measures average squared distance from the mean. Standard deviation is its square root — same units as the data. Interquartile range (IQR) is the range from 25th to 75th percentile — robust to outliers.

data = np.array([12, 15, 14, 10, 13, 14, 100])

print(f"Variance: {np.var(data, ddof=1):.1f}")  # ddof=1 for sample
print(f"Std dev:  {np.std(data, ddof=1):.1f}")
print(f"IQR:      {np.percentile(data, 75) - np.percentile(data, 25):.1f}")

# Rule of thumb: data more than 1.5*IQR from Q1 or Q3 is an outlier
q1, q3 = np.percentile(data, [25, 75])
iqr = q3 - q1
outliers = data[(data < q1 - 1.5*iqr) | (data > q3 + 1.5*iqr)]
print(f"Outliers: {outliers}")  # [100]

When to Use Each Statistic