Types of data, and the centre: mean, median, mode

Why statistics?

A column of 10,000 numbers means nothing to a human. Statistics is a set of tools that squeezes a column into a few honest numbers: what is typical, how spread out the values are, and what is unusual. Descriptive statistics, the subject of this module, describes the data you have. Module 3 adds inferential statistics, which uses a sample to make careful claims about a bigger group.

Kinds of data

Before choosing a summary, know what kind of data a column holds. The kind decides which summaries make sense.

Population and sample

The population is the whole group you care about, such as all voters in India. A sample is the part you actually measured, such as 2,000 voters in a survey. Usually you cannot measure everyone, so you summarise a sample and use it to estimate the population. Keep the difference in mind: it matters for the spread in the next topic.

The mean

The mean (the ordinary average) is the total divided by how many values there are. In words: add everything up, then share it out equally. It uses every value, which is its strength and its weakness, because one extreme value pulls it a long way.

The median

The median is the middle value after sorting. With an odd number of values it is the one in the middle; with an even number it is the average of the two middle values. Half the data is below it and half above, and extreme values barely move it.

salaries = [25, 28, 30, 32, 35, 40, 400]   # monthly salary, ₹ thousand, in a small company

mean = sum(salaries) / len(salaries)
print(round(mean, 2))   # 84.29

s = sorted(salaries)
print(s[len(s) // 2])   # 32

def median(values):
    s = sorted(values)
    n = len(s)
    mid = n // 2
    if n % 2 == 1:
        return s[mid]
    return (s[mid - 1] + s[mid]) / 2

print(median([7, 1, 5]))         # 5
print(median([40, 10, 30, 20]))  # 25.0

Skew and outliers

An outlier is a value far away from the rest. A distribution is skewed when one side has a long tail. Salaries, house prices and delivery times are usually right-skewed: most values are modest and a few are very large. In right-skewed data the mean is bigger than the median. That is why reports on income usually give the median.

The mode

The mode is the value that appears most often. It is the only one of the three that works for nominal data: the most common payment method, the most sold shoe size. A column can have two modes (bimodal) or more. Python's statistics.mode returns the first one it meets in the data, and statistics.multimode returns all of them as a list.

import statistics

sizes = [8, 9, 8, 10, 7, 8, 9]
print(statistics.mean(sizes), statistics.median(sizes), statistics.mode(sizes))   # 8.428571428571429 8 8

print(statistics.multimode(["UPI", "cash", "UPI", "cash", "card"]))   # ['UPI', 'cash']

Which one to use

• Mean: numbers without extreme values, or when the total matters (average bill, average marks).
• Median: skewed numbers or data with outliers (salaries, house prices, delivery times).
• Mode: categories, or the most common exact value (popular size, top payment method).

Common mistakes

• Reporting only the mean of skewed data. Fix: report the median too, and say why they differ.
• Forgetting to sort before taking the middle value. Fix: always use sorted(values) first.
• Averaging labels such as PIN codes or ratings coded 1–5 without thinking. Fix: check the kind of data first; for ordinal ratings the median or the share of each rating is often clearer.
• Treating 0 as missing, or missing as 0. Fix: remove or mark missing values before averaging.