Finding the Median from a Table
The median is another important measure of central tendency in GCSE Maths. It gives us the middle value of a dataset when the values are ordered from smallest to largest. Unlike the mean, the median is not affected by extreme values (outliers), making it a useful measure when the data contains unusually high or low numbers.
This guide will walk you through how to find the median from a frequency table.
What is the Median?
The median is the middle value in a data set when it is ordered. If the number of values is odd, the median is the middle value. If the number of values is even, the median is the average of the two middle values.
This video explains more about calculating the Mean, Median and Mode
Step-by-Step Guide to Finding the Median from a Frequency Table
When data is given in a table with frequencies, we need to first understand the cumulative frequency. The cumulative frequency is the running total of the frequencies as we move through the values in the table.
Example Table:
| Value (x) | Frequency (f) |
|---|---|
| 3 | 4 |
| 6 | 3 |
| 8 | 2 |
| 10 | 1 |
Steps:
Calculate the Cumulative Frequency: Start by calculating the cumulative frequency. This is the total number of data points up to and including each value. Add the frequency for each row to the cumulative total from the previous rows.
| Value (x) | Frequency (f) | Cumulative Frequency (CF) |
|---|---|---|
| 3 | 4 | 4 |
| 6 | 3 | 7 |
| 8 | 2 | 9 |
| 10 | 1 | 10 |
Find the Position of the Median: To find the median, you need to determine the position of the middle value. If there are NNN total values in the dataset, the position of the median is given by:
$$\text{Median position} = \frac{N + 1}{2}$$
In this example, the total number of data points is the final cumulative frequency, which is 10.
So, the median position is:
$$\frac{10 + 1}{2} = 5.5$$
The median will be the value at position 5.5 in the ordered list.
Identify the Median Value: Look at the cumulative frequency column and find the value that corresponds to the position of the median. Since position 5.5 falls between the 5th and 6th values, we check where the cumulative frequency reaches or exceeds 5.5.
From the table:
- The cumulative frequency reaches 4 for the value 3.
- The cumulative frequency reaches 7 for the value 6.
The median will therefore fall between the value 3 and 6. Since position 5.5 is in this range, 6 is the median.
If the Number of Data Points is Even
If the total number of data points is even, the median will be the average of the two middle values. For example, if the cumulative frequencies were such that the median position was between two values, you would take the average of those two values.
Example for Even Total:
| Value (x) | Frequency (f) |
|---|---|
| 1 | 2 |
| 4 | 4 |
| 7 | 3 |
| 10 | 1 |
Calculate the cumulative frequency:
| Value (x) | Frequency (f) | Cumulative Frequency (CF) |
|---|---|---|
| 1 | 2 | 2 |
| 4 | 4 | 6 |
| 7 | 3 | 9 |
| 10 | 1 | 10 |
The total number of data points is 10 (as the cumulative frequency is 10). The median position is:
$$\frac{10 + 1}{2} = 5.5$$
- The median lies between the 5th and 6th values. From the cumulative frequency, we see:
- The 4th value (with cumulative frequency 4) is 1.
- The 5th and 6th values (with cumulative frequency 6) are 4.
Therefore, the median will be the average of 4 and 7:
$$\frac{4 + 7}{2} = 5.5$$
So, in this case, the median is 5.5.
Key Tips:
- Cumulative Frequency: Always calculate the cumulative frequency first, as this helps you find the position of the median.
- Position of the Median: Use the formula N+12\frac{N + 1}{2}2N+1 to find the median position. If the position is a whole number, the median is the value at that position. If it is a decimal, the median is between the two surrounding values.
- Data Order: Make sure the data values are ordered correctly before you begin calculating the cumulative frequency and median.
- Even Total: When there’s an even number of data points, the median is the average of the two middle values.
Practice Example:
Here’s another table for you to try:
| Value (x) | Frequency (f) |
|---|---|
| 2 | 5 |
| 5 | 3 |
| 8 | 2 |
Calculate the cumulative frequency:
| Value (x) | Frequency (f) | Cumulative Frequency (CF) |
|---|---|---|
| 2 | 5 | 5 |
| 5 | 3 | 8 |
| 8 | 2 | 10 |
The total number of data points is 10, so the median position is:
$$\frac{10 + 1}{2} = 5.5$$
From the cumulative frequency:
- The value at position 5 is 5.
- The value at position 6 is 5.
The median is therefore 5.
Finding the median from a table involves calculating the cumulative frequency and identifying the position of the middle value. By following the steps and practicing with different tables, you'll soon become proficient in determining the median, whether for an odd or even number of data points.
