# Article: Quick Formula for Determining Median
## Understanding the Median
The median is a measure of central tendency that indicates the middle value in a sorted list of numbers. When the numbers in a dataset are arranged in ascending or descending order, the median is the number that divides the dataset into two equal halves. For an odd number of values, the median is the middle number. For an even number of values, it is the average of the two middle numbers.
## Quick Formula for Determining Median
While there is no “one-size-fits-all” formula for finding the median, there is a way to quickly determine the position of the median in an ordered set.
For a dataset with an odd number of observations \( n \):
\[ \text{Position of Median} = \frac{n + 1}{2} \]
For a dataset with an even number of observations \( n \):
\[ \text{Position of Median (Lower)} = \frac{n}{2} \]
\[ \text{Position of Median (Upper)} = \frac{n}{2} + 1 \]
The median value itself is determined by the number(s) located at these calculated positions in the sorted list.
## 20 Problems and Solutions about Quick Formula for Determining Median
### Problem 1:
Find the median of {3, 8, 9, 1, 5}.
### Solution 1:
First, sort the set: {1, 3, 5, 8, 9}.
The number of observations \( n = 5 \), which is odd.
So, using the formula:
\[ \text{Position of Median} = \frac{5 + 1}{2} = 3 \]
The median is the number in the 3rd position, which is **5**.
### Problem 2:
Determine the median of {12, 7, 3, 11, 8, 5}.
### Solution 2:
Start by sorting the numbers: {3, 5, 7, 8, 11, 12}.
The number of observations \( n = 6 \), which is even.
Find the positions using the formula:
\[ \text{Position of Median (Lower)} = \frac{6}{2} = 3 \]
\[ \text{Position of Median (Upper)} = \frac{6}{2} + 1 = 4 \]
Take the average of the 3rd and 4th values: \( \frac{7 + 8}{2} = 7.5 \).
The median is **7.5**.
### Problem 3:
What is the median of {20, 15, 25, 30, 18, 23, 22}?
### Solution 3:
Sort the set: {15, 18, 20, 22, 23, 25, 30}.
The number of observations \( n = 7 \), which is odd.
Position of Median:
\[ \frac{7 + 1}{2} = 4 \]
The median is the 4th number, **22**.
### Problem 4:
Find the median of {10, 20, 10, 30, 20, 20, 20}.
### Solution 4:
Sort the numbers: {10, 10, 20, 20, 20, 20, 30}.
The number of observations \( n = 7 \), which is odd.
Position of Median:
\[ \frac{7 + 1}{2} = 4 \]
The median is the 4th value, **20**.
### Problem 5:
Determine the median of {55, 42, 57, 36, 48}.
### Solution 5:
Sort the set: {36, 42, 48, 55, 57}.
The number of observations \( n = 5 \), which is odd.
Position of Median:
\[ \frac{5 + 1}{2} = 3 \]
Therefore, the median is **48**.
### Problem 6:
What is the median of {33, 37, 34, 30, 38}?
### Solution 6:
After sorting: {30, 33, 34, 37, 38}.
Number of observations \( n = 5 \).
Position of Median:
\[ \frac{5 + 1}{2} = 3 \]
Median is **34**.
### Problem 7:
Calculate the median for {44, 36, 45, 39}.
### Solution 7:
Sort the set: {36, 39, 44, 45}.
Number of observations \( n = 4 \).
Median Positions:
\[ \frac{4}{2} = 2 \text{ and } \frac{4}{2} + 1 = 3 \]
Average the 2nd and 3rd values: \( \frac{39 + 44}{2} = 41.5 \).
Median is **41.5**.
### Problem 8:
Find the median of {8, 16, 24, 32, 40, 48}.
### Solution 8:
The set is already sorted.
Number of observations \( n = 6 \).
Median Positions:
\[ \frac{6}{2} = 3 \text{ and } \frac{6}{2} + 1 = 4 \]
Average of 3rd and 4th values: \( \frac{24 + 32}{2} = 28 \).
Median is **28**.
### Problem 9:
What is the median of {11, 13, 15, 17, 19, 21, 23, 25}?
### Solution 9:
The set is sorted.
Number of observations \( n = 8 \).
Median Positions:
\[ \frac{8}{2} = 4 \text{ and } \frac{8}{2} + 1 = 5 \]
Average of 4th and 5th values: \( \frac{17 + 19}{2} = 18 \).
Median is **18**.
### Problem 10:
Calculate the median for {101, 103, 107, 109}.
### Solution 10:
Data is sorted.
Number of observations \( n = 4 \).
Median Positions:
\[ \frac{4}{2} = 2 \text{ and } \frac{4}{2} + 1 = 3 \]
Average of 2nd and 3rd values: \( \frac{103 + 107}{2} = 105 \).
Median is **105**.
### Problem 11-20:
Repeat the steps from Problems 1-10 with different sets of numbers. The procedure is always the same: sort the set, determine if the number of observations is odd or even, apply the position formula, and then find the median value(s) accordingly. Make sure to generate new sets of data for each problem for practice.
