Measures of Central Tendency • Topic 2 of 3

Median and Mode

What is the Median?

The median is the middle value of a data set when arranged in ascending or descending order. It divides the data into two equal halves.

How to Find the Median:

Number of values (\(n\))Median Position
Odd (\(n\))Middle value at position \(\frac{n+1}{2}\)
Even (\(n\))Average of the two middle values at positions \(\frac{n}{2}\) and \(\frac{n}{2}+1\)

What is the Mode?

The mode is the value that appears most frequently in a data set. A data set can have:

  • One mode (unimodal)
  • Two modes (bimodal)
  • More than two modes (multimodal)
  • No mode (if all values occur with same frequency)

Comparison of Measures:

MeasureBest used whenAdvantageDisadvantage
MeanData is symmetricalUses all dataAffected by outliers
MedianData has outliersNot affected by outliersDoesn't use all data
ModeData is categoricalEasy to understandMay not be unique
Median and ModeDataset: 3, 5, 5, 7, 8, 8, 8, 10, 12 (sorted)31525374858687108129↑ Median = 8 (middle value, position 5 of 9)Mode = Most Frequent Value35781012Mode = 8 (appears 3 times — most frequent)
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Worked Example

Find the median of: 22, 18, 25, 20, 24, 19, 21.

Solution
  • Arrange in ascending order: 18, 19, 20, 21, 22, 24, 25
  • Number of values = 7 (odd)
  • Median = value at position \(\frac{7+1}{2} = 4\)th position
  • 4th value = 21

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Answer: 21 **Example 2:** Find the median and mode of: 15, 12, 18, 12, 15, 20, 12, 15. *Solution:* - Arrange: 12, 12, 12, 15, 15, 15, 18, 20 - n = 8 (even), middle positions: 4th and 5th = 15 and 15 - Median = $\frac{15+15}{2} = 15$ - Mode = 12 and 15 (both appear 3 times) → bimodal - **Answer:** Median = 15, Mode = 12 and 15 **Example 3:** Find the median of the first 10 natural numbers. *Solution:* - First 10 natural numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 - n = 10 (even), middle positions: 5th and 6th = 5 and 6 - Median = $\frac{5+6}{2} = 5.5$ - **Answer:** 5.5

Key Points

  • Median = middle value of sorted data
  • For odd \(n\): median at position \(\frac{n+1}{2}\)
  • For even \(n\): median = average of two middle values
  • Mode = most frequent value(s)
  • Median is not affected by outliers; mode works for non-numeric data
Tap an option to check your answer0 / 4
Q1.The median is the ___ value when data is ordered.
Explanation: Middle value.
Q2.The mode is the value that occurs:
Explanation: Most frequent.
Q3.The median of $3, 5, 7$ is:
Explanation: Middle value $5$.
Q4.The mode of $2, 2, 3, 4$ is:
Explanation: $2$ occurs most.