Find the median of the following sets of data.
| Value | Frequency |
|---|---|
| 1 | 11 |
| 2 | 2 |
| 3 | 16 |
| 4 | 7 |
| 5 | 3 |
| 6 | 4 |
| 7 | 24 |
The number of elements in this set of data is:
11 + 2 + 16 + 7 + 3 + 4 + 24 = 67
Because there are an odd number of elements, there's a single middle element that has an equal number of elements above it as below it.
If the set is arranged in ascending order the 34th element will be the median, having 33 elements below it and 33 elements above it.
To find the value of this element, count up along the frequency table:
- The first 11 elements (elements 1 to 11) have a value of 1.
- The next 2 elements (elements 12 and 13) have a value of 2.
- The next 16 elements (elements 14 to 29) have a value of 3.
- The next 7 elements (elements 30 to 36) have a value of 4.
The 34th element is 4.
The median of this dataset is 4.
| Value | Frequency |
|---|---|
| 2 | 8 |
| 4 | 2 |
| 5 | 5 |
| 8 | 1 |
| 10 | 3 |
| 14 | 3 |
| 15 | 4 |
The number of elements in this set of data is:
8 + 2 + 5 + 1 + 3 + 3 + 4 = 26
Because there are an even number of elements, the exact middle of the set is not a single element but is between two elements :
- \left(\dfrac{N}{2}\right)
- \left(\dfrac{N}{2} + 1\right)
Their average is the median, when the set is arranged in order.
For this set, the two middle elements are the 13th and 14th elements. To find the value of this element, count up along the frequency table:
- The first 8 elements (elements 1-8) have a value of 2.
- The next 2 elements (elements 8-9) have a value of 4.
- The next 5 elements (elements 10-14) have a value of 5.
So, the 13th and 14th elements both have a value of 5, and their average is 5.
The median of this dataset is 5.
| Value | Frequency |
|---|---|
| 4 | 3 |
| 6 | 7 |
| 13 | 11 |
| 14 | 4 |
| 15 | 1 |
| 17 | 9 |
| 22 | 4 |
| 27 | 3 |
The number of elements in this set of data is:
3 + 7 + 11 + 4 + 1 + 9 + 4 + 3 = 42
Because there are an even number of elements, the exact middle of the set is not a single element but is between two elements :
- \left(\dfrac{N}{2}\right)
- \left(\dfrac{N}{2} + 1\right)
Their average is the median, when the set is arranged in order.
For this set, the two middle elements are the 21st and 22nd elements. To find the value of this elements, count up along the frequency table:
- The first 3 elements (elements 1-3) have a value of 4.
- The next 7 elements (elements 4-10) have a value of 6.
- The next 11 elements (elements 11-21) have a value of 13.
- The next 4 elements (elements 22-25) have a value of 14.
So, the 21st and 22nd elements are 13 and 14, respectively, and their average is:
\dfrac{13+14}{2} = 13.5
The median of this dataset is 13.5.
| Value | Frequency |
|---|---|
| 1 | 3 |
| 2 | 6 |
| 4 | 1 |
| 6 | 3 |
| 7 | 8 |
| 9 | 9 |
| 12 | 6 |
The number of elements in this set of data is:
3 + 6 + 1 + 3 + 8 + 9 + 6 = 36
Because there are an even number of elements, the exact middle of the set is not a single element but is between two elements :
- \left(\dfrac{N}{2}\right)
- \left(\dfrac{N}{2} + 1\right)
Their average is the median, when the set is arranged in order.
For this set, the two middle elements are the 18th and 19th elements. To find the value of this elements, count up along the frequency table:
- The first 3 elements (elements 1-3) have a value of 1.
- The next 6 elements (elements 4-9) have a value of 2.
- The next 1 element (element 10) has a value of 4.
- The next 3 elements (elements 11-13) have a value of 6.
- The next 8 elements (elements 14-21) have a value of 7.
So, the 18th and 19th elements both have a value of 7, and their average is 7.
The median of this dataset is 7.
| Value | Frequency |
|---|---|
| 2 | 3 |
| 7 | 7 |
| 8 | 14 |
| 9 | 1 |
| 11 | 4 |
| 15 | 2 |
| 23 | 3 |
| 26 | 9 |
The number of elements in this set of data is:
3 + 7 + 14 + 1 + 4 + 2 + 3 + 9 = 43
Because there are an odd number of elements, there's a single middle element that has an equal number of elements above it as below it. If the set is arranged in order, the 22nd element will be exactly in the middle (is the median), having 21 elements below it and 21 elements above it.
To find the value of this element, count up along the frequency table:
- The first 3 elements (elements 1-3) have a value of 2.
- The next 7 elements (elements 4-10) have a value of 7.
- The next 14 elements (elements 11-24) have a value of 8.
The 22nd element is 8.
The median of this dataset is 8.
\left\{ 2{,}3{,}6{,}6{,}7{,}5{,}12{,}15{,}14{,}17{,}5\right\}
In this set, there are 11 elements. Because there are an odd number of elements, there's a single middle element that has an equal number of elements above it as below it. If the set is arranged in order, the 6th element will be exactly in the middle (is the median), having 5 elements below it and 5 elements above it.
Arranging the set in order gives :
\left\{ 2{,}3{,}5{,}5{,}6{,}6{,}7{,}12{,}14{,}15{,}17 \right\}
The 6th element is 6.
The median of this dataset is 6.
\left\{ 2{,}3{,}9{,}15{,}16{,}1{,}12{,}7{,}18{,}6{,}10{,}22\right\}
In this set, there are 12 elements. Because there are an even number of elements, the exact middle of the set is not a single element but is between two elements:
- \left(\dfrac{N}{2}\right)
- \left(\dfrac{N}{2} + 1\right)
Their average is the median when the set is arranged in order.
Arranging the set in order gives:
\left\{ 1{,}2{,}3{,}6{,}7{,}9{,}10{,}12{,}15{,}16{,}18{,}22 \right\}
For this set, the two middle elements are the 6th and 7th elements, which are respectively 9 and 10. Their average is:
\dfrac{9+10}{2}=9.5
The median of this dataset is 9.5.