To calculate the arithmetic mean and median for the given list of numbers [tex]\(5, 7, 3, 5, 7, 3, 9, 5, 7, 9, 3, 6, 5, 0, 7, 3\)[/tex], we can follow these steps:
### Arithmetic Mean
The arithmetic mean (average) is calculated by summing up all the numbers in the list and then dividing by the count of the numbers.
1. Sum the Numbers:
[tex]\[
5 + 7 + 3 + 5 + 7 + 3 + 9 + 5 + 7 + 9 + 3 + 6 + 5 + 0 + 7 + 3 = 84
\][/tex]
2. Count the Numbers:
There are 16 numbers in the list.
3. Calculate the Mean:
[tex]\[
\text{Mean} = \frac{\text{Sum of the numbers}}{\text{Count of the numbers}} = \frac{84}{16} = 5.25
\][/tex]
### Median
The median is the middle number in a sorted list of numbers. If the list has an even number of observations, the median is the average of the two middle numbers.
1. Sort the List:
[tex]\[
0, 3, 3, 3, 3, 5, 5, 5, 5, 6, 7, 7, 7, 7, 9, 9
\][/tex]
2. Find the Middle Position(s):
For 16 numbers, the median will be the average of the 8th and 9th numbers in the sorted list.
3. Locate the Middle Numbers:
[tex]\[
8^{\text{th}}\, \text{Number} = 5
\][/tex]
[tex]\[
9^{\text{th}}\, \text{Number} = 5
\][/tex]
4. Calculate the Median:
[tex]\[
\text{Median} = \frac{8^{\text{th}}\, \text{Number} + 9^{\text{th}}\, \text{Number}}{2} = \frac{5+5}{2} = 5.0
\][/tex]
### Conclusion
The arithmetic mean of the list is [tex]\(5.25\)[/tex], and the median of the list is [tex]\(5.0\)[/tex].
