Median

How to find the median of the given data: 4 examples and their solutions.

Example 1

Example

Solution

The median is the middle value of a data.

The median is one of the numbers
that represents a data.

Mean

Mode

To find the median of a data,
cancel the least value and the greatest value.

So cancel the least value, 60,
and the greatest value, 100.

Repeat this cancelling.

Cancel the least value, 70,
and the greatest value, 90.

Then the middle value, 80, remains.

So the median is 80.

Just like this case,
when there are odd number of values,
then you'll get one middle value:
the median.

So 80 is the answer.

Example 2

Example

Solution

Cancel the least value and the greatest value.

Cancel the least value, 50,
and the greatest value, 100.

Repeat this cancelling.

Cancel the least value, 60,
and the greatest value, 90.

Then there are two middle values:
70 and 80.

Then the median is
the mean of these two values:
[70 + 80]/2.

Just like this case,
when there are even number of values,
then you'll get two middle values.
And the median is
the mean of the two values.

70 + 80 = 150

150/2 = 75

So 75 is the median of the given data.

Example 3

Example

Solution

To find the median from a frequency table,
first find the number of the values n.

To find n,
add the frequencies.

n = 1 + 2 + 4 + 7 + 5 + 2

1 + 2 = 3
4 + 7 = 11
5 + 2 = 7

3 + 7 = 10

10 + 11 = 21

n = 21

n is an odd number.

Then find [n + 1]/2:
[21 + 1]/2.

21 + 1 = 22

22/2 = 11

This 11 means
the median is the 11th value.

To find the 11th value,
make a cumulative frequency table.

To make a cumulative frequency table,
draw a 3 column table like this.

Name the titles
Score, Frequency, and Cumulative Frequency.

Fill in the Cumulative Frequency column.

For the first row,
the frequency is 1.

So the cumulative frequency is
1.

This 1 is not greater than or equal to 11(th).

So find the next cumulative frequency.

The previous cumulative frequency is 1.
The frequency of this row is 2.

Then the cumulative frequency is
1 + 2 = 3.

3 is not greater than or equal to 11(th).

So find the next cumulative frequency.

The previous cumulative frequency is 3.
The frequency of this row is 4.

Then the cumulative frequency is
3 + 4 = 7.

7 is not greater than or equal to 11(th).

So find the next cumulative frequency.

The previous cumulative frequency is 7.
The frequency of this row is 7.

Then the cumulative frequency is
7 + 7 = 14.

14 is greater than (or equal to) 11(th).

Then stop finding the next cumulative frequency.

The cumulative frequency 14 at 3 points
means
the 14th value is 3 points.

Then it's obvious that
the 11th value, the median, is 3 points.

So the median of the data is 3.

So 3 is the answer.

Example 4

Example

Solution

To find the median from a frequency table,
first find the number of the values n.

To find n,
add the frequencies.

n = 1 + 6 + 8 + 9 + 4 + 2

1 + 9 = 10
6 + 4 = 10
8 + 2 = 10

10 + 10 + 10 = 30

n = 30

n is an even number.

Then find n/2:
30/2 = 15.

This 15 means
the median is
the mean of the 15th and 16th values.

To find the 15th and 16th values,
make a cumulative frequency table.

Draw a 3 column table like this.

Name the titles
Score, Frequency, and Cumulative Frequency.

Fill in the Cumulative Frequency column.

For the first row,
the frequency is 1.

So the cumulative frequency is
1.

This 1 is not greater than or equal to
either 15(th) or 16(th).

So find the next cumulative frequency.

The previous cumulative frequency is 1.
The frequency of this row is 6.

Then the cumulative frequency is
1 + 6 = 7.

7 is not greater than or equal to
either 15(th) or 16(th).

So find the next cumulative frequency.

The previous cumulative frequency is 7.
The frequency of this row is 8.

Then the cumulative frequency is
7 + 8 = 15.

15 is (greater than or) equal to 15(th).

Then stop finding the next cumulative frequency.

The cumulative frequency 15 at 2 points
means
the 15th value is 2 points.

Then the 16th value is 3 points.

The 15th value is 2 points.
The 16th value is 3 points.

Then the median is,
the mean of these two values,
[2 + 3]/2.

2 + 3 = 5

So the median of the data is 5/2.

So 5/2 is the answer.