Descriptive Statistics (mean, median, variance)

Central tendency

Mean

• Sensitive to outliers
• Good for symmetric distributions

Mean

import numpy as np
 
x = np.array([10, 12, 12, 13, 12, 11, 100])
print(np.mean(x))

Median

• Robust to outliers

Median

import numpy as np
 
x = np.array([10, 12, 12, 13, 12, 11, 100])
print(np.median(x))

Mode

Useful for categorical data.

Mode (SciPy)

import numpy as np
from scipy import stats
 
x = np.array([1, 1, 2, 2, 2, 3])
print(stats.mode(x, keepdims=True))

Spread (variability)

• Range: max - min (very sensitive)
• Variance: average squared distance from mean
• Standard deviation (std): sqrt(variance)

Variance / Std

import numpy as np
 
x = np.array([10, 12, 12, 13, 12, 11, 100])
print("var:", np.var(x, ddof=1))
print("std:", np.std(x, ddof=1))

IQR (interquartile range)

Robust measure of spread.

IQR

import numpy as np
 
x = np.array([10, 12, 12, 13, 12, 11, 100])
q1 = np.percentile(x, 25)
q3 = np.percentile(x, 75)
print("IQR:", q3 - q1)

Quick checklist

• Use median/IQR when outliers exist
• Use mean/std when distribution is roughly symmetric
• Always visualize (histogram/boxplot) before trusting summary stats

How mean and median relate

diagram Choosing a central tendency measure mermaid

  flowchart TD
  A["Do you have outliers
or a skewed shape?"] -->|"Yes"| B["Use median
(robust)"]
  A -->|"No"| C["Use mean
(uses every value)"]
  D["Is the data categorical?"] --> E["Use mode"]

A quick decision path for picking mean, median, or mode.

From the book

Pandas’ own summary tool, describe(), prints count, mean, std, min, quartiles, and max in one call — McKinney recommends running it as a first pass on any new DataFrame before writing custom aggregation code, precisely because it surfaces skew and outliers at a glance.

Visualize it

The mean is pulled toward extreme values because it uses every number in the calculation. The median only cares about the middle position, so it barely moves when one point is far away. Watch the outlier drag the mean marker to the right while the median stays put:

sketch Mean vs. median on a number line p5.js Pause Restart

The mean shifts toward the outlier; the median stays anchored near the middle of the data.

🧪 Try It Yourself

Exercise 1 – Mean vs. median with an outlier

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Exercise 2 – Variance and standard deviation

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Exercise 3 – IQR to spot spread

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Next

Continue to Probability Basics to build the foundation for confidence intervals and hypothesis tests.