Examples of The Median for Ungrouped Data

THE MEDIAN

The median is the middle value of an ordered data set.

An ordered data set is obtained by listing the data from smallest to largest value.

The median splits the data in halves. Half of the data are less than or equal to the median, and half are greater than or equal to it.

For example, if the median mark for a test is 73% then you know that half the class scored less than or equal to 73% and half scored greater than or equal to 73%.

For an odd number of data, the median is one of the original data values.

For an even number of data, the median is the average of the two middle values, and hence may not be in the original data set.

If there are n data values listed in order from smallest to largest, the median is the ½(n+1) th data Value.

For example:
If n=13, ½(13+1)=7, so the median is the 7th ordered data value.
If n=14, ½(14+1)=7.5, so the median is the average of the 7th and 8th ordered data values.

Median

Definition: Median
The median of a data set is the value in the central position, when the data set has been arranged from the lowest to the highest value
.
Note that exactly half of the values from the data set are less than the median and the other half are greater than the median.

To calculate the median of a quantitative data set, first sort the data from the smallest to the largest value and then find the value in the middle. If there is an odd number of values in the data set, the median will be equal to one of the values in the data set. If there is an even number of values in the data set, the median will lie halfway between two values in the data set.

How to work out the Median:
1. Place all your data values in ascending order (biggest to smallest)
2. The piece of data in the middle is your median

Worked example 1: Median for an odd number of values.
QUESTION
What is the median of {10; 14; 86; 2; 68; 99; 1}?
SOLUTION
Step 1: Sort the values
The values in the data set, arranged from the smallest to the largest, are

1; 2; 10; 14; 68; 86; 99

Step 2: Find the number in the middle
There are 7 values in the data set. Since there are an odd number of values, the median will be equal to the value in the middle, namely, in the fourth position. Therefore the median of the data set is 14.

NOTE: If you have an EVEN number of data values, there will be TWO pieces of data in the middle. No problem, just add them together and divide by two to find the number halfway between them… and this is your median!

Worked example 2: Median for an even number of values.
QUESTION
What is the median of {11; 10; 14; 86; 2; 68; 99; 1}?
SOLUTION
Step 1: Sort the values
The values in the data set, arranged from the smallest to the largest, are

1; 2; 10; 11; 14; 68; 86; 99

Step 2: Find the number in the middle
There are 8 values in the data set. Since there are an even number of values, the median will be halfway between the two values in the middle, namely, between the fourth and fifth positions. The value in the fourth position is 11 and the value in the fifth position is 14. The median lies halfway between these two values. Therefore


Worked example 3: Medians
Question:
In a Mathematics class; 23 learners completed a test out of 25 marks. Here is a list of their results:
14 10 23 21 11 19 13 13 20 21 9 13 17 17 18 14 19 13 24 21 9 16 6

Calculate the median of this data.
Solution
● First put the data in order, from lowest to highest.


There are 23 numbers, so the middle number is the 12th number out of 23 numbers. So 16 is the median, the number in the middle of the data.
● When there is an even number of values in the data set, the median lies halfway between the middle two values.
● We can add these two values and divide by 2.
For example, what if another learner wrote the test and her result was 7? We can add this to the ordered data set.


Now there are 24 numbers and the middle two numbers are the 12th and 13th numbers. The middle two numbers are 14 and 16. Add 14 and 16 to get 30 and divide by 2 to get a median of 15.


More Related Questions and Solutions

Q1. Calculate the median of the following data set:
{4; 13; 10; 13; 13; 4; 2; 13; 13; 13}

solution:
We first need to order the data set:
{2;4;4;10;13;13;13;13;13;13}

Since there are an even number of values in this data set (10) the median lies between the fifth and sixth place:


The median is: 13.

Q2. Calculate the median of the following data set:

{5; 5; 10; 7; 10; 2; 16; 10; 10; 10; 7}

solution:
We first need to order the data set:
{2; 5; 5; 7; 7; 10; 10; 10; 10; 10; 16}

Since there are an odd number of values in this data set (11) the median lies at the sixth place. The median is: 10.

Q3. In a park, the tallest 7 trees have heights (in metres):

{41; 60; 47; 42; 44; 42; 47}

Find the median of their heights.
solution:
We first need to order the data set:
{41; 42; 42; 44; 47; 47; 60}

Since there are an odd number of values in this data set (7) the median lies at the fourth place. The median is: 44.

Q4. A student got the following marks in 9 questions of a question paper.

3, 5, 7, 3, 8, 0, 1, 4 and 6.

Find the median of these marks.
solution:
Arranging the given data in descending order: 8, 7, 6, 5, 4, 3, 3, 1, 0
The middle term is 4 which is the 5th term.
Median=4

Q5. The weights (in kg) of 10 students of a class are given below:

21, 28.5, 20.5, 24, 25.5, 22, 27.5, 28, 21 and 24.

Find the median of their weights.
Solution:
Arranging the given data in descending order:
28.5. 28, 27.5. 25.5, 24, 24. 22. 21. 21. 20.5

The middle terms are 24 and 24. 5th and 6th terms


Q6. For the following set of data, find the median:
10, 75, 3, 81, 17, 27, 4, 48, 12, 47, 9 and 15.

solution:
Arrange the given terms in ascending order:
3, 4, 9, 10, 12, 15, 17, 27, 47, 48, 75, 81

Number of terms =12
Median


Median =16

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