Mode and median (Edexcel GCSE Statistics): Revision Notes
Mode and median
What is the mode?
The mode represents the most frequently occurring value in a dataset. When working with frequency tables, you need to identify which data value appears most often, not which frequency is the largest. This is a crucial distinction that many students get confused about.
Critical Distinction: The mode is the actual data value that appears most frequently, NOT the frequency number itself. This is one of the most common mistakes students make when working with frequency tables.
To find the mode from a frequency table, look through the frequency column to find the highest number. Once you've identified the highest frequency, look across to see what data value corresponds to it. That data value is your mode.
For example, if you have a table showing the number of cars parked in different areas, and you see that 1 car appears 16 times (the highest frequency), then the mode is 1 car, not 16.
What is the median?
The median is the middle value when all data points are arranged in ascending order. In frequency tables, the data is already organised, but you need to account for the fact that some values appear multiple times according to their frequencies.
Since frequency tables show how many times each value occurs, you cannot simply pick the middle row. Instead, you must work out which position the middle value occupies when all individual data points are considered.
Finding the median using cumulative frequency
The most efficient method for finding the median from a frequency table involves creating a cumulative frequency column. This running total helps you track how many data points you've counted as you move through the table.
Here's the step-by-step process:
- Add a cumulative frequency column to your table
- Start with the first frequency and write it in the cumulative frequency column
- Add each subsequent frequency to the running total
- Calculate the median position using the formula: , where is the total frequency
- Find which data value corresponds to this position using your cumulative frequency column
Worked Example: Cars in car parks
Let's work through an example with data about cars parked in different areas:
| Number of cars | Frequency | Cumulative frequency |
|---|---|---|
| 0 | 5 | 5 |
| 1 | 16 | 21 (5 + 16) |
| 2 | 12 | 33 (21 + 12) |
| 3 | 10 | 43 (33 + 10) |
| 4 | 7 | 50 (43 + 7) |
| 5 | 3 | 53 (50 + 3) |
Finding the mode: The highest frequency is 16, which corresponds to 1 car. Therefore, the mode is 1.
Finding the median:
- Total frequency
- Median position = th value
- Looking at the cumulative frequency column, the 27th value falls within the range where cumulative frequency goes from 21 to 33
- This corresponds to 2 cars, so the median is 2
Worked Example: Children in families
Here's another example using data about the number of children in families:
| Number of children | Frequency | Cumulative frequency |
|---|---|---|
| 0 | 8 | 8 |
| 1 | 11 | 19 (8 + 11) |
| 2 | 10 | 29 (19 + 10) |
| 3 | 7 | 36 (29 + 7) |
| 4 | 4 | 40 (36 + 4) |
Finding the mode: The highest frequency is 11, which corresponds to 1 child. Therefore, the mode is 1.
Finding the median:
- Total frequency
- Median position = th value
- Since we need the 20.5th position, we find the mean of the 20th and 21st values
- The 20th value corresponds to 1 child (cumulative frequency reaches 19, then jumps to 29)
- The 21st value also corresponds to 2 children
- Mean of 20th and 21st values = , but since we're dealing with discrete data, the median is 2
Common exam tips and traps
Understanding these common pitfalls will help you avoid losing marks in examinations and develop a stronger grasp of the concepts.
Watch out for these common mistakes:
- Don't confuse the mode with the highest frequency - the mode is the data value, not how often it appears
- Remember to use cumulative frequency for the median, not just the middle row of the table
- When the median position is not a whole number (like 20.5), you may need to find the mean of two adjacent values
- Always double-check that your final answer makes sense within the context of the data
Key exam techniques:
- Always add a cumulative frequency column when finding the median
- Show your calculation of clearly
- State your final answers clearly, using appropriate units where given
- Check your cumulative frequency column adds up to the total frequency
Key Points to Remember:
- The mode is the data value with the highest frequency, not the frequency itself
- The median requires a cumulative frequency column and the formula to find its position
- Cumulative frequency is a running total that helps locate the median position
- Always double-check that your mode corresponds to the highest frequency and your median position makes sense
- When dealing with discrete data, your final answers should be appropriate whole numbers or categories