# Median

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

## Example 1

The median is the middle value of the data.

If *n*, the number of the data, is an odd number,

then the middle value is the median.

So 80 is the median of the given test scores.

## Example 2

If *n* is an even number,

then there are two middle numbers.

Then the median is

the mean of the two middle values.

(blue arrow)

For the given test scores,

the two middle values are 70 and 80.

So (median) = [70 + 80]/2.

So 75 is the median.

## Formula

If *n* gets quite large,

it's tricky to pick the exact middle value.

Then you can use these formulas.

1) If *n* is an odd number.

The middle value, which is the median, is

[(*n* + 1)/2] th value.

2) If *n* is an even number.

The middle values are

[*n*/2] th and [*n*/2 + 1] th values.

So the median is

the mean of these two values.

## Example 3

Find *n* by adding the frequencies.*n* = 1 + 2 + 4 + 7 + 5 + 2

= 21

*n* = 21 is an odd number.

So (21 + 1)/2 = 11,

11th value is the median.

To find out where 11th value, the median, is,

make a column for the Cumulative Frequency

right next to the given table.

Add up the frequencies from 0 points

until the cumulative frequency becomes

greater than or equal to 11.

For 0 points,

the cumulative frequency is 1.

For 1 point,

the cumulative frequency is

1 + 2 = 3.

For 2 points,

the cumulative frequency is

3 + 4 = 7.

For 3 points,

the cumulative frequency is

7 + 7 = 14.

This means

14th value is 3 points.

So the 11th value is 3 points.

So 3 is the median.

## Example 4

Find *n* by adding the frequencies.*n* = 1 + 6 + 8 + 9 + 4 + 2

= 30

*n* = 30 is an even number.

So 30/2 = 15,

the mean of 15th and 16th values

is the median.

To find out where the 15th and 16th values are,

make a column for the Cumulative Frequency

right next to the given table.

Add up the frequencies from 0 points

until the cumulative frequency becomes

greater than or equal to 15 or 16.

For 0 points,

the cumulative frequency is 1.

For 1 point,

the cumulative frequency is

1 + 6 = 7.

For 2 points,

the cumulative frequency is

8 + 7 = 15.

This means

15th value is 2 points.

Then 16th value is 3 points.

So (median) = [2 + 3]/2.

2 + 3 = 5

So (median) = 5/2.