Descriptive Statistics: Mean, Median, Variance & SD

Interpret summary statistics before reaching for a formula

If most customer orders are between $20 and $80 but a few enterprise orders exceed $20,000, median order value may represent the typical purchase better than the mean.

• Mean summarizes the arithmetic center, median resists extreme values, and mode identifies the most frequent value.
• Variance and standard deviation quantify spread around the mean, but their interpretation depends on units and distribution shape.
• Percentiles and the interquartile range are often more informative than mean and standard deviation for skewed data.
• A summary table should be paired with a plot because different distributions can share similar summary statistics.

Calculate a small case and verify it in Python

For summary statistics, predict one visible result before each snippet runs—a row count, shape, statistic, ranking or error pattern—then compare it with the printed checkpoint.

# summary statistics · example 1
import pandas as pd
s = pd.Series([20, 25, 25, 30, 35, 80, 20000])
print('mean', s.mean())
print('median', s.median())
print('std', s.std())
print('IQR', s.quantile(.75) - s.quantile(.25))

summary statistics checkpoint 1: Run the cell and inspect the printed values before changing anything.

# summary statistics · example 2
import pandas as pd
import numpy as np

df = pd.read_csv('practice.csv')
x = df.select_dtypes('number').iloc[:,0].dropna()
print('n=', len(x), 'mean=', round(x.mean(),2), 'median=', round(x.median(),2))

summary statistics checkpoint 2: n, mean and median are printed from the first numeric column.

# summary statistics · example 3
q1, q3 = x.quantile([0.25,0.75])
iqr = q3 - q1
print('std=', round(x.std(ddof=1),2), 'IQR=', round(iqr,2))

summary statistics checkpoint 3: Standard deviation and IQR provide two different views of spread.

# summary statistics · example 4
rng = np.random.default_rng(42)
boot = [rng.choice(x, size=len(x), replace=True).mean() for _ in range(1000)]
print(np.percentile(boot,[2.5,50,97.5]).round(2))

summary statistics checkpoint 4: The three values are the bootstrap 2.5th, 50th and 97.5th percentiles.

# summary statistics · example 5
x2 = pd.concat([x, pd.Series([x.max()*5])], ignore_index=True)
print('mean change:', round(x2.mean()-x.mean(),2))
print('median change:', round(x2.median()-x.median(),2))

summary statistics checkpoint 5: The mean usually moves more than the median when one extreme value is added.

Stress-test the result against assumptions and outliers

Self-check with answers

Practise interpretation, not only arithmetic