Introduction to Median
The median is a measure of central tendency that represents the middle value in a data set when the values are arranged in order. Unlike the mean (average), the median is not affected by extremely high or low values.
Key Properties of Median:
- Divides the data set into two equal halves
- Unaffected by extreme values (outliers)
- Can be used with both numerical and ordinal data
When to Use Median
The median is particularly useful when:
- The data contains extreme values that could skew the mean
- Working with ordinal data (data that can be ranked)
- You want to know the middle value of a distribution
Example: Median is better than mean for representing typical income because a few very high incomes can skew the average.
Finding the Median
Steps to Find Median
To find the median of ungrouped data:
- Arrange all data points in order (ascending or descending)
- Count the number of data points (n)
- If n is odd, the median is the middle number
- If n is even, the median is the average of the two middle numbers
Example 1: Odd Number of Data Points
Find the median of: 12, 5, 7, 10, 3
Step 1: Arrange in order: 3, 5, 7, 10, 12
Step 2: Count data points: n = 5 (odd)
Step 3: Median is the 3rd value: 7
\[ \text{Median} = 7 \]Example 2: Even Number of Data Points
Find the median of: 15, 20, 12, 18
Step 1: Arrange in order: 12, 15, 18, 20
Step 2: Count data points: n = 4 (even)
Step 3: Median is average of 2nd and 3rd values:
\[ \text{Median} = \frac{15 + 18}{2} = 16.5 \]Position Formula
For larger data sets, you can use the position formula to find where the median is located:
Where n = number of data points
Example: For 11 data points:
\[ \frac{11 + 1}{2} = 6 \]The median is the 6th value in the ordered list.
Median from Frequency Table
Steps for Frequency Table
To find the median from a frequency table:
- Create a cumulative frequency column
- Find the total number of data points (sum of frequencies)
- Calculate the median position: (n + 1)/2
- Locate the median in the cumulative frequency
Example
Find the median for this data:
| Score (x) | Frequency (f) | Cumulative Frequency |
|---|---|---|
| 5 | 2 | 2 |
| 6 | 4 | 6 |
| 7 | 5 | 11 |
| 8 | 3 | 14 |
| 9 | 1 | 15 |
Total frequency (n) = 15
Median position = (15 + 1)/2 = 8
The 8th value falls in the cumulative frequency of 11 (which corresponds to score 7)
\[ \text{Median} = 7 \]Important Notes
When working with frequency tables:
- Always create a cumulative frequency column
- The data is already ordered in the table
- For even n, you may need to average two values
- For grouped data (not covered here), the process is different
Word Problems Involving Median
Solving Strategy
To solve word problems involving median:
- Extract the data from the problem
- Arrange the data in order
- Find the median using appropriate method
- Interpret the result in the context of the problem
Example 1: Test Scores
The scores of 7 students on a math test are: 85, 92, 78, 90, 82, 88, 95. What is the median score?
Step 1: Arrange in order: 78, 82, 85, 88, 90, 92, 95
Step 2: n = 7 (odd)
Step 3: Median is the 4th value
\[ \text{Median} = 88 \]Interpretation: Half of the students scored below 88 and half scored above.
Example 2: Temperature Data
The daily high temperatures for 8 days were: 72, 68, 75, 80, 77, 70, 69, 74. Find the median temperature.
Step 1: Arrange in order: 68, 69, 70, 72, 74, 75, 77, 80
Step 2: n = 8 (even)
Step 3: Median is average of 4th and 5th values:
\[ \text{Median} = \frac{72 + 74}{2} = 73 \]Interpretation: The middle temperature for the period was 73°F.
Example 3: Missing Value
The median of five numbers is 12. If four of the numbers are 8, 10, 14, and 16, find the fifth number.
Let the fifth number be x. Ordered numbers must have 12 in the middle:
Possible arrangements:
Case 1: x, 8, 10, 14, 16 → Median would be 10 (not correct)
Case 2: 8, 10, x, 14, 16 → For median to be 12, x must be 12
Verification: 8, 10, 12, 14, 16 → Median is 12
\[ \text{Fifth number} = 12 \]Practice Exercise
Question 1
Find the median of: 23, 17, 25, 19, 20, 21, 18
Ordered: 17, 18, 19, 20, 21, 23, 25
n = 7 (odd)
\[ \text{Median} = 20 \]Question 2
Calculate the median for: 5, 8, 2, 9, 7, 3
Ordered: 2, 3, 5, 7, 8, 9
n = 6 (even)
\[ \text{Median} = \frac{5 + 7}{2} = 6 \]Question 3
Find the median from this frequency table:
| Value | Frequency |
|---|---|
| 10 | 3 |
| 12 | 5 |
| 15 | 4 |
| 18 | 2 |
Total frequency = 3 + 5 + 4 + 2 = 14
Median position = (14 + 1)/2 = 7.5 → average of 7th and 8th values
Cumulative frequencies: 3 (10), 8 (12), 12 (15), 14 (18)
Both 7th and 8th values fall in the 12 category
\[ \text{Median} = 12 \]Question 4
The median of 9 consecutive whole numbers is 25. What are the numbers?
For 9 numbers, median is the 5th number: 25
Numbers must be: 21, 22, 23, 24, 25, 26, 27, 28, 29
Question 5
A class has 15 students with test scores: 65, 70, 72, 75, 78, 80, 82, 85, 88, 90, 92, 95, 98, 100, 100. If one student scored 60 is added, what is the new median?
Original median (n=15): 8th value = 85
New data (n=16): Add 60 → ordered: 60, 65, 70, ..., 100, 100
New median = average of 8th and 9th values:
\[ \frac{82 + 85}{2} = 83.5 \]Question 6
The median of five different positive integers is 10. What is the largest possible value of any of these integers?
For five numbers with median 10, the ordered numbers are: a, b, 10, d, e
To maximize e, minimize a, b, d:
Smallest possible a, b: 1, 2
Smallest possible d: 11 (must be >10 and different)
Thus numbers: 1, 2, 10, 11, e
No upper limit on e, but with different positive integers, the largest possible isn't bounded.
If we assume the numbers are consecutive after the median: 8, 9, 10, 11, 12 → largest is 12
But the problem says "different" not "consecutive", so technically no maximum.
Ready for more challenges?
Full Practice SetMath Challenge
Timed Test in 7 Levels
Test your skills with calculating medians from different data sets