Median is the "middle" or mid value of a data set.
To find the median, you have to first order the numbers - from smallest to largest, and then find the middle number.
So, in a set of 10, 20, 30, the middle number is 20.
So, in a set of 10, 20, 30, 40, 50, the middle number is 30.
1. When you put the numbers in numerical order, it will look like this:
10, 15, 25, 45, 50, 60,90
2. There are seven numbers in the series, and out of which the middle number is the fourth from the start. This divides the set into two equal parts - three on the left side and three on the right side of the fourth number.
3. So, the median is 45.
The formulas for calculating the median of an ungrouped data, which has "n" number of observations are:
If the number of observations is odd:
`x_m` = `x_((n+1)/2)`
If the number of observations is even:
`x_m` = `(x_(n/2)+x_((n+2)/2))/2`
First of all, we need to arrange the data in the ascending order. The arranged data is:
Number of observations = 7 (odd)
Since the number of observations is odd, we can find the median using the following formula:
`x_m` = `x_((n+1)/2)`
`bar(x)` = `(( 7+1 )/2)th` observation = `4^(th)` observation = 70
Given data needs to be arranged in ascending order as shown in the table below :
Number of observations = 10 (even)
Since the number of observations is even, we can find the median using the following formula:
`x_m` = `(x_(n/2)+x_((n+2)/2))/2`
`bar(x)= 1/2 [ n/( 2 ) th` observation+`( n/( 2 ) +1)th` observation `]`
` = 1/2 [ 10/( 2 ) th` observation+`( 10/( 2 ) +1)th` observation `]`
`bar(x)= 1/2 [ 5th` observation+`6th` observation `]`
`bar(x)= 1/2 [30+34 ] = ( 64 )/2 = 32`
To calculate median of a grouped data (which has class and frequency distribution) we need to,
| Number on the Dice (X) | Frequency (f) |
|---|---|
| 1 | 4 |
| 2 | 6 |
| 3 | 2 |
| 4 | 2 |
| 5 | 5 |
| 6 | 1 |
Make a new column of cumulative frequency by multiplying frequency with class.
| Number on the Dice (X) | Frequency (f) | Cumulative Frequency |
|---|---|---|
| 1 | 4 | 4 |
| 2 | 6 | 10 |
| 3 | 2 | 12 |
| 4 | 2 | 14 |
| 5 | 5 | 19 |
| 6 | 1 | 20 |
Number of observation (n) = 20
Median = class containing `(n/2)^(th)` observation
Median = class containing `(20/2)^(th)` observation
Median = class containing 2nd observation = 2
That is, median of given grouped data is 2.
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