Measures of Central Tendency: Mean, Median, Mode and Applications

Overview

Measures of central tendency (averages) provide a single representative value that summarizes a data set. In business, averages are used to report average sales, average wages, average price, and typical demand. The most common averages are Arithmetic Mean, Median and Mode. Each has advantages and limitations, and the correct choice depends on the type of data and the presence of extreme values.

Central tendency: meaning

Central tendency refers to the tendency of observations to cluster around a central/typical value. Measures of central tendency represent this typical value.

Arithmetic mean (AM): meaning and formulas

Arithmetic Mean (Mean) is the sum of observations divided by the number of observations.

Formulas

• Individual series: $\\bar{x} = \\frac{\\sum x}{n}$
• Discrete series (with frequency $f$): $\\bar{x} = \\frac{\\sum fx}{\\sum f}$
• Continuous series (use class midpoints $m$): $\\bar{x} = \\frac{\\sum fm}{\\sum f}$

Weighted mean: meaning and formula

When items have different importance, we use weighted mean: $\\bar{x}_w = \\frac{\\sum wx}{\\sum w}$ Examples: average marks with credit weights, average price with quantities.

Median: meaning and formulas

Median is the middle value that divides the data into two equal halves when arranged in order.

• Individual series:

  • If $n$ is odd: median is $(n+1)/2$th item.
  • If $n$ is even: median is average of $n/2$th and $(n/2 + 1)$th items.

• Continuous series: $\\text{Median} = L + \\left(\\frac{\\frac{N}{2} - c.f}{f}\\right) \\times h$ Where:

• $L$ = lower boundary of median class

• $N$ = total frequency

• $c.f$ = cumulative frequency before median class

• $f$ = frequency of median class

• $h$ = class width

Mode: meaning and formulas

Mode is the value (or class) with the highest frequency.

• Discrete series: the value with maximum frequency.
• Continuous series: $\\text{Mode} = L + \\left(\\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\\right) \\times h$ Where:
• $L$ = lower boundary of modal class
• $f_1$ = frequency of modal class
• $f_0$ = frequency of preceding class
• $f_2$ = frequency of succeeding class
• $h$ = class width

Empirical relation (approx.) when distribution is moderately skewed: $\\text{Mode} \\approx 3\\,\\text{Median} - 2\\,\\text{Mean}$

Merits and demerits of mean, median, mode (table)

Choosing the best average (when to use what)

• Use Mean when data is symmetric and you need further calculations (variance, regression).
• Use Median when data is skewed or has outliers (income, wages).
• Use Mode when you need the most common size/choice (popular brand, shoe size).

Short worked examples (exam style)

Example 1 (Mean – individual series)

Data: 10, 12, 8, 15, 5
$\sum x = 50$, $n = 5$
$\bar{x} = 50/5 = 10$

Example 2 (Median – individual series)

Arrange: 5, 8, 10, 12, 15
Median = 3rd item = 10

Example 3 (Weighted mean)

Prices: 10, 12; Quantities: 2, 3
$\sum wx = 10×2 + 12×3 = 56$, $\sum w = 5$
Weighted mean = $56/5 = 11.2$

Common exam mistakes

• not arranging data before finding median
• using class limits instead of boundaries (for continuous median/mode)
• forgetting $N/2$ in median formula
• mixing up $f_0, f_1, f_2$ in mode formula

Exam tips (how to write 3/5 marks)

• For 3 marks: definition + formula + one small example.
• For 5 marks: explain mean/median/mode + merits/demerits table + example + conclusion.

Key Terms

• Mean — average using all values.
• Median — middle value.
• Mode — most frequent value.
• Weighted mean — mean using weights.
• Outlier — extreme value far from others.

Quick Recap (1-minute revision)

• Mean uses all values but is affected by extremes.
• Median is best for skewed data/outliers.
• Mode is most common value; useful for typical choice.