Python Statistics with Python

The three common measures of centre answer slightly different questions:

The mean uses every value, so one unusual observation pulls it strongly: the 48-minute repair lifts the mean to 20 even though four of five repairs took 16 minutes or less. The median only cares about the middle position, so it barely moves. When data is skewed, as durations, prices and incomes often are, the median usually describes a "typical" case better, and reporting both reveals the skew.

With an even number of values, median() averages the two middle ones; median_low() and median_high() return an actual data value instead. mode() suits categories such as the most common repair type; multimode() returns every value tied for most frequent.

Spread tells you how consistent the values are. Two workshops can both average 20 minutes, one with every repair between 18 and 22, the other ranging from 5 to 60.

• Range: max(x) - min(x); simple, but decided entirely by the two extremes.
• Variance: the average squared distance from the mean.
• Standard deviation: the square root of the variance, in the same units as the data (minutes), so it's easier to interpret.

from statistics import stdev, pstdev, variance
minutes = [10, 12, 14, 16, 48]
print(round(stdev(minutes), 2), round(pstdev(minutes), 2))   # 15.81 14.14

stdev() and variance() are sample statistics: they divide by n − 1, which corrects for estimating the spread of a larger process from a sample. pstdev() and pvariance() are population statistics: they divide by n, appropriate when you have every value of interest. With these five repairs as a sample of all repairs, stdev() is the right choice. The difference shrinks as n grows, but matters for small datasets.

Every statistic describes the data you have, which may not represent what you want to know about. Ask how the observations were collected:

• Were repairs timed only on quiet weekday mornings? Then weekend rush times are missing.
• Did mechanics only record repairs that went smoothly? Then difficult jobs are under-represented.
• Are five repairs enough to say anything general at all?

Data collected because it was convenient can differ systematically from the whole process, and no formula corrects that. Larger samples reduce random noise but not systematic bias.

Write down the selection: the period, which repairs were included or excluded, and how many observations there are. A statistic quoted without its sample size and collection method is hard for anyone to judge. When the question needs a fair sample, choose observations randomly, for example with random.sample(all_job_ids, 30), rather than taking whichever were easiest to record.

A number on its own isn't a finding. Report statistics with:

• units: "median repair time 14 minutes", not "median 14";
• the count: "n = 5 repairs";
• centre and spread together: "median 14 min, ranging from 10 to 48";
• known limitations: "one repair needed a part ordered in; it accounts for the 48".

from statistics import median, quantiles
minutes = [10, 12, 14, 16, 48, 11, 13, 15, 17, 19]
q1, q2, q3 = quantiles(minutes, n=4)
print(f"n={len(minutes)}, median {median(minutes)} min, middle half {q1}-{q3} min")

Be careful with relationships. If repairs on older bikes take longer, the data shows an association; it doesn't prove that age causes the delay. Older bikes might simply be booked for bigger jobs. statistics.correlation() (Python 3.10+) measures how closely two variables move together, but deciding why needs knowledge of the process, not just the numbers.

Task: summarise repair durations with units, sample size, location and spread, explaining an unusually long observation.

from statistics import mean, median, stdev

minutes = [22, 18, 25, 20, 19, 75, 21]
note = "The 75-minute repair waited for a replacement derailleur."

print(f"n = {len(minutes)} repairs")
print(f"median {median(minutes)} min, mean {mean(minutes):.1f} min")
print(f"sample sd {stdev(minutes):.1f} min, range {min(minutes)}-{max(minutes)} min")
without = [m for m in minutes if m != 75]
print(f"without the outlier: median {median(without)} min, mean {mean(without):.1f} min")
print(note)

The median moves from 21 to 20.5 minutes when the outlier is set aside, while the mean drops from 28.6 to 20.8, showing which summary is robust. The outlier is reported and explained, not silently deleted. For the worked check, the statistic generally less affected by one extreme duration is the median.

The statistics module documentation lists every function with examples. Continue with data visualization and charts and distributions and percentiles. Practise with the Python exercises and Python quiz.