11.2.2 Median

Introduction

The median is a measure of central tendency that represents the middle value of a dataset. Unlike the mean, the median is not affected by extreme values (outliers). It divides the data into two equal parts — one half has values less than the median, and the other half has values greater than the median. This topic explains the definition, formula, and calculation of the median for both ungrouped and grouped data.

Definition

Median: The median is the middle value in a dataset that are arranged in ascending or descending order of magnitude. Median divides the data set into two equal parts.

• One part of the dataset has the values less than the middle value.
• The other part of the dataset has values greater than the middle item.

Symbol: It is denoted by x (read as "x tilde" or "x snake").

Important: It is necessary to arrange the values of dataset in ascending or descending order.

Main Idea

The median is the middle value in a dataset arranged in ascending or descending order. The value of median is not affected by change in extreme values of dataset.

Core Concept

Formula for Median

Detailed Explanation

1. Median for Ungrouped Data

If the number of values is odd:

The median is the middle value.

Median (x) = Value of (n+12)^th item in the dataset

where n = number of items or data.

If the number of values is even:

The median is the average of the two middle values.

Median (x) = (n2)^th value + (n2 + 1)^th value2

Simple explanation:

• Arrange the data in order.
• If odd number of values, find the middle one.
• If even number of values, find the average of the two middle ones.

2. Median for Grouped Data

(i) Discontinuous or Unconnected values in dataset:

To calculate the value of median for discontinuous grouped data, first of all the cumulative frequency (cf) of complete dataset is obtained.

For odd number of data:

Median (x) = Value of data against (n+12)^th cumulative frequency

For even number of data:

Median (x) = Class values against n2^th + (n2 + 1)^th cumulative frequencies2

Simple explanation: For grouped data, use cumulative frequency to find the median.

Solved Examples

Example 1: Median (Ungrouped Data, Odd Number)

Dataset: The measured heights of plants after arranging in ascending order:

46, 49, 55, 58, 62, 63, 64, 65, 66, 67, 68, 70, 72

Solution:

Given:

- Number of values in the dataset = 13 (odd)

Required: Median

Formula:

Median (x) = Value of (n+12)^th item

Substitution:

Median (x) = 13+12 = 142 = 7

Calculation:

The 7th value in the dataset is 64.

Final Answer: Median = 64

Example 2: Median (Ungrouped Data, Even Number)

Dataset: The number of fruits produced on each of the 16 plants of experimental group:

3, 5, 18, 21, 15, 10, 8, 12, 13, 7, 11, 14, 9, 16, 20, 24

Solution:

Given:

- Number of values = 16 (even)

Required: Median

Step 1: Arrange in ascending order:

3, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 21, 24

Formula:

Median (x) = (n2)^th value + (n2 + 1)^th value2

Substitution:

Median (x) = (162)^th value + (162 + 1)^th value2

= 8^th value + 9^th value2

Calculation:

8th value = 12; 9th value = 13

Median (x) = 12 + 132 = 252 = 12.5

Final Answer: Median = 12.5 fruits per plant

Example 3: Median (Grouped Data, Odd Number)

Dataset: Calculate the median of the following dataset obtained by counting the number of flowers on 21 rose plants.

Solution:

Step 1: Calculate cumulative frequency.

Given:

- Number of plants (n) = 21 (odd)

Formula:

Median (x) = Value of data against (n+12)^th cumulative frequency

Substitution:

Median (x) = 21+12 = 222 = 11

Calculation:

Since cumulative frequency 11 is included in 14, which represents the class value 3, therefore, for cumulative frequency 11, the class value will be 3.

Final Answer: Median = 3 flowers per plant

Example 4: Median (Grouped Data, Even Number)

Dataset: Calculate the median for the following data recorded for height (in cm) of 80 plants.

Solution:

Step 1: Calculate cumulative frequency.

Given:

- Number of plants (n) = 80 (even)

Formula:

Median (x) = Class values against n2^th + (n2 + 1)^th cumulative frequencies2

Substitution:

Median (x) = Class values against 802^th + (802 + 1)^th cumulative frequencies2

= Class values against 40^th + 41^th cumulative frequencies2

Calculation:

The class values for cumulative frequencies 40 and 41 are included in the class value of cumulative frequency 45, which is 122.

Median (x) = 122 + 1222 = 122

Final Answer: Median = 122 cm

Activity (To Be Solved by Students)

Calculate the median of the following values of data set obtained by measuring the heights of 22 plants in an experimental group to assess the effects of less water availability:

8, 9, 10, 18, 10, 8, 9, 7, 8, 11, 23, 19, 17, 15, 7, 12, 14, 12, 11, 14, 15, 13

Advantages of Median

• It is easy to calculate and is located exactly.
• It is not affected by abnormally large and small values in dataset.
• It can be used in quantitative measurements.

Disadvantages of Median

• The median of two or more datasets cannot be calculated by using/adding the median of the individual datasets.
• Median value may not be necessarily represented in central data.
• It cannot be used where specific values e.g. weight & age used.

Summary Table

Intermediate Level

The median is the middle value in a dataset arranged in ascending or descending order.

Formula for ungrouped data:

• Odd: (n+12)^th value
• Even: (n2)^th value + (n2 + 1)^th value2

Formula for grouped data:

• Odd: Value against (n+12)^th cumulative frequency
• Even: Class values against n2^th + (n2 + 1)^th cumulative frequencies2

Key points:

• Median is not affected by extreme values.
• Data must be arranged in order.
• Median divides data into two equal parts.
• Symbol: x (x tilde).

Important Points

• - The median is the middle value in a dataset that are arranged in ascending or descending order of magnitude.
• - Median divides the data set into two equal parts.
• - One part of the dataset have the values less than the middle value and the other part of the dataset have values greater than the middle item.
• - It is denoted by x (read as "x tilde" or "x snake").
• - It is necessary to arrange the values of dataset in ascending or descending order.
• - The value of median is not affected by change in extreme values of dataset.
• - If the number of values is odd, the median is the middle value.
• - If the number of values is even, the median is the average of the two middle values.
• - Advantages of Median:
• 1. It is easy to calculate and is located exactly.
• 2. It is not affected by abnormally large and small values in dataset.
• 3. It can be used in quantitative measurements.
• - Disadvantages of Median:
• 1. The median of two or more datasets cannot be calculated by using/adding the median of the individual datasets.
• 2. Median value may not be necessarily represented in central data.
• 3. It cannot be used where specific values e.g. weight & age used.

Remember

• - Median = middle value of ordered data
• - Symbol = x (x tilde)
• - Arrange data in ascending or descending order
• - Odd: (n+12)^th value
• - Even: Average of n2 and n2+1 values
• - Not affected by extreme values
• - Divides data into two equal parts
• - Can be used for skewed data

Common Mistakes

• 1. Forgetting to arrange data in order.
• - Data must be arranged in ascending or descending order before finding the median.
• 2. Confusing odd and even formulas.
• - Odd: (n+12)^th value
• - Even: Average of n2 and n2+1 values
• 3. Thinking the median is always in the dataset.
• - For even number of values, the median is the average of two middle values and may not be in the dataset.
• 4. Mixing up median and mean.
• - Median = middle value.
• - Mean = average.
• 5. Forgetting to use cumulative frequency for grouped data.
• - For grouped data, use cumulative frequency to find the median.
• 6. Believing the median is affected by outliers.
• - The median is not affected by extreme values.
• 7. Thinking the median can be used for all types of data.
• - The median cannot be used where specific values (e.g., weight, age) are used.

Quick Revision

60-Second Revision

• Median = middle value of ordered data
• Symbol = x (x tilde)
• Arrange data in ascending or descending order
• Odd: (n+12)^th value
• Even: Average of n2 and n2+1 values
• Not affected by extreme values
• Divides data into two equal parts
• Can be used for skewed data
• Warning: Do not forget to arrange the data in order before finding the median. For even number of values, the median is the average of two middle values.