Difference between revisions of "Median"

Revision as of 19:11, 10 March 2024

Median represents the middle value for any group. Median helps to represent a large number of data points with a single data point. The median is the easiest statistical measure to calculate. For calculation of median, the data has to be arranged in ascending order, and then the middlemost data point represents the median of the data.

Further, the calculation of the median depends on the number of data points. For an odd number of data, the median is the middlemost data, and for an even number of data, the median is the average of the two middle values.

Definition

Median is one of the three measures of central tendency. When describing a set of data, the central position of the data set is identified. This is known as the measure of central tendency. The three most common measures of central tendency are mean, median, and mode.

Median Definition

The value of the middle-most observation obtained after arranging the data in ascending order is called the median of the data. Many an instance, it is difficult to consider the complete data for representation, and here median is useful. Among the statistical summary metrics, the median is an easy metric to calculate.

Example

$4,4,6,3,2$

Find the median for a above set of data.

Step 1: Arrange the given data in ascending order: 2, 3, 4, 4, 6.
Step 2: Count the number of values. There are 5 values.
Step 3: Look for the middle value. The middle value is the median. Thus, median = 4.

Median Formula

Using the median formula, the middle value of the arranged set of numbers can be calculated. For finding this measure of central tendency, it is necessary to write the components of the group in increasing order. The median formula varies based on the number of observations and whether they are odd or even. The following set of formulas would help in finding the median of the given data.

Median Formula for Ungrouped Data

The following are the steps helpful while applying the median formula for ungrouped data.

Step 1: Arrange the data in ascending or descending order.
Step 2: Count the total number of observations ' $n$ '
Step 3: Check if the number of observations ' $n$ ' is even or odd.

Median Formula When n is Odd

The median formula of a given set of numbers having ' $n$ ' odd number of observations, can be expressed as:

Median = $\left[{\frac {n+1}{2}}\right]$ ^th term

Median Formula When n is Even

The median formula of a given set of numbers having ' $n$ ' even number of observations, can be expressed as:

Median = [(n/2)^th term + ((n/2) + 1)^th term]/2

Example: The age of the members of a team has been listed below. Find the median of the above set.

$42,40,50,60,35,58,32$

Solution:

Step 1: Arrange the data items in ascending order.

Original data: $42,40,50,60,35,58,32$

Ordered data: $32,35,40,42,50,58,60$

Step 2: Count the number of observations ( $n=7$ , odd). If the number of observations is odd, then we will use the following formula

Median = $\left[{\frac {n+1}{2}}\right]$ ^th term

Median = $\left[{\frac {7+1}{2}}\right]$ ^th term

Median = $4$ ^th term = $42$

@@ Line 14: / Line 14: @@
 Find the  median for a above set of data.
-* Step 1: Arrange this data in ascending order: 2, 3, 4, 4, 6.
+* Step 1: Arrange the given data in ascending order: 2, 3, 4, 4, 6.
 * Step 2: Count the number of values. There are 5 values.
 * Step 3: Look for the middle value. The middle value is the median. Thus, median = 4.
@@ Line 22: / Line 22: @@
 == Median Formula for Ungrouped Data ==
-The following steps are helpful while applying the median formula for ungrouped data.
+The following are the steps helpful while applying the median formula for ungrouped data.
 * Step 1: Arrange the data in ascending or descending order.
-* Step 2: Count the total number of observations 'n'.
+* Step 2: Count the total number of observations '<math> n</math>'
-* Step 3: Check if the number of observations 'n' is even or odd.
+* Step 3: Check if the number of observations '<math> n</math>' is even or odd.
 === Median Formula When n is Odd ===
-The median formula of a given set of numbers, say having 'n' odd number of observations, can be expressed as:
+The median formula of a given set of numbers having '<math> n</math>' odd number of observations, can be expressed as:
-'''Median = [(n + 1)/2]<sup>th</sup> term''' <math> \left [ \frac{n+1}{2} \right ]</math>
+Median = <math> \left [ \frac{n+1}{2} \right ]</math><sup>th</sup> term
 === Median Formula When n is Even ===
-The median formula of a given set of numbers say having 'n' even number of observations, can be expressed as:
+The median formula of a given set of numbers having '<math> n</math>' even number of observations, can be expressed as:
 '''Median = [(n/2)<sup>th</sup> term + ((n/2) + 1)<sup>th</sup> term]/2'''
-'''Example:''' The age of the members of a weekend poker team has been listed below. Find the median of the above set.
+Example''':''' The age of the members of a team has been listed below. Find the median of the above set.
-{42, 40, 50, 60, 35, 58, 32}
+<math>42,40,50,60,35,58,32</math>
-'''Solution:'''
+Solution:
 Step 1: Arrange the data items in ascending order.
-Original set: {42, 40, 50, 60, 35, 58, 32}
+Original data: <math>42,40,50,60,35,58,32</math>
-Ordered Set: {32, 35, 40, 42, 50, 58, 60}
+Ordered data: <math>32,35,40,42,50,58,60</math>
-Step 2: Count the number of observations. If the number of observations is odd, then we will use the following formula: Median = [(n + 1)/2]<sup>th</sup> term
+Step 2: Count the number of observations (<math>n=7</math> , odd). If the number of observations is odd, then we will use the following formula
-Step 3: Calculate the median using the formula.
+Median = <math> \left [ \frac{n+1}{2} \right ]</math><sup>th</sup> term
-Median = [(n + 1)/2]<sup>th</sup> term
+Median = <math> \left [ \frac{7+1}{2} \right ]</math><sup>th</sup> term
-= (7 + 1)/2<sup>th</sup> term = 4<sup>th</sup> term = 42
+Median = <math> 4</math><sup>th</sup> term = <math>42</math>
-Median = 42