Statistics: Mean, Median, Mode, SD

Why statistics?

A list of 500 marks is hard to understand. Statistical measures summarise a whole data set in a few numbers: what is a typical value (central tendency — mean, median, mode) and how spread out the values are (variability — range, standard deviation). We will use the heights (cm) of 8 students:

Mean

Mean (average) = sum of all values ÷ number of values.

Sum = 150 + 155 + 160 + 160 + 162 + 165 + 170 + 175 = 1297; Mean = 1297 ÷ 8 = 162.125 cm.

Median

Median is the middle value after sorting. With an odd count it is the centre value; with an even count it is the average of the two middle values.

Mode

Mode is the value that occurs most often. Here 160 appears twice, so mode = 160. Mode is the only measure that also works for non-numeric data, e.g. the most popular shoe size or favourite colour.

Range

Range = maximum − minimum = 175 − 150 = 25 cm. It is quick but depends only on two extreme values.

Standard deviation

Standard deviation (σ) tells how far values are, on average, from the mean. Steps: (1) find the mean, (2) subtract the mean from each value, (3) square the differences, (4) average the squares — this is the variance, (5) take the square root — this is the SD.

data = [2, 4, 4, 4, 5, 5, 7, 9]
mean = sum(data) / len(data)
squares = [(x - mean) ** 2 for x in data]
variance = sum(squares) / len(data)
sd = variance ** 0.5
print('Mean:', mean)
print('Squared deviations:', squares)
print('Variance:', variance, '| SD:', sd)

All measures in Python

Method 1 — write the logic yourself (good for exams):

heights = [150, 155, 160, 160, 162, 165, 170, 175]
n = len(heights)
mean = sum(heights) / n
s = sorted(heights)
if n % 2 == 0:
    median = (s[n // 2 - 1] + s[n // 2]) / 2
else:
    median = s[n // 2]
mode = max(set(heights), key=heights.count)
rng = max(heights) - min(heights)
print('Mean  :', mean)
print('Median:', median)
print('Mode  :', mode)
print('Range :', rng)

Method 2 — use the statistics module:

import statistics as st
heights = [150, 155, 160, 160, 162, 165, 170, 175]
print(st.mean(heights), st.median(heights), st.mode(heights))
print('Population SD:', round(st.pstdev(heights), 2))
print('Variance     :', round(st.pvariance(heights), 2))

Effect of outliers

An outlier is a value far away from the rest. It pulls the mean strongly but hardly moves the median.

import statistics as st
salary = [25, 28, 30, 32, 35, 250]      # thousand rupees; one CEO salary
print('Mean  :', round(st.mean(salary), 2))
print('Median:', st.median(salary))

Two classes can have the same mean but very different spread — SD shows the difference:

import statistics as st
class_a = [60, 62, 64, 66, 68]
class_b = [40, 50, 64, 78, 88]
print('Means:', st.mean(class_a), st.mean(class_b))
print('SD A :', round(st.pstdev(class_a), 2))
print('SD B :', round(st.pstdev(class_b), 2))

Which measure to use?

Summary & practice

• Mean = Σx ÷ n; median = middle of sorted data; mode = most frequent.
• Range = max − min; SD = √(Σ(x − mean)² ÷ n).
• Outliers affect the mean, not much the median.
• Python: statistics.mean, median, mode, pstdev, pvariance.