Mode and median (Edexcel GCSE Statistics): Revision Notes
Mode and median
Understanding grouped data limitations
When working with grouped data, we face an important limitation that affects how we calculate averages. We cannot find exact values for the mode and median because we only know the class intervals where the data values lie, not the precise individual data values themselves.
This is different from ungrouped data where we can identify exact values for both mode and median. With grouped data, we work with estimates and class intervals instead.
This means we need to use different approaches:
- For the mode: we identify the modal class (the class interval with the highest frequency)
- For the median: we find the class containing the median and then estimate its value using a special formula
Modal class
The modal class is simply the class interval that contains the most data values - in other words, the interval with the highest frequency.
To identify the modal class:
- Look at the frequency column in your grouped data table
- Find the class interval with the largest frequency value
- This interval is your modal class
For example, if you have a frequency table showing time intervals, and the interval "0 < T ≤ 5" has a frequency of 12 (which is the highest), then "0 < T ≤ 5" is the modal class.
Finding the median class
The median represents the middle value when data is arranged in order. For grouped data, we need to:
- Calculate the median position: Use the formula (n + 1) ÷ 2, where n is the total frequency
- Use cumulative frequency: Add up frequencies as you go down the table
- Identify the median class: Find which class interval contains the median position
Cumulative frequency is a running total - you add each frequency to all the frequencies above it in the table. This helps you see where specific data positions fall within the class intervals.
The process works like this: if your median position is the 22nd data value, look at the cumulative frequency column to see which class interval contains the 22nd value.
Estimating the median value
Once you've found the median class, you can estimate the actual median value using this formula:
Where:
- L = lower boundary of the median class interval
- n = total number of values (total frequency)
- F = cumulative frequency of all intervals before the median class
- f = frequency of the median class interval
- w = width of the median class interval
Worked example step-by-step
Worked Example: Calculating Median from Grouped Age Data
Let's work through a complete example using data about people's ages on an outing:
Step 1: Identify the median position
- Total frequency (n) = 91
- Median position = (91 + 1) ÷ 2 = 46th data value
Step 2: Find the median class using cumulative frequency
- Looking at cumulative frequencies: 18, 38, 54, 75, 91
- The 46th value falls in the class where cumulative frequency first exceeds 46
- This is the "25 ≤ N < 35" class (cumulative frequency = 54)
Step 3: Apply the estimation formula
- L = 25 (lower boundary of median class)
- n = 91 (total frequency)
- F = 38 (cumulative frequency before median class)
- f = 16 (frequency of median class)
- w = 10 (width of class interval: 35 - 25 = 10)
Step 4: Calculate Estimated median = 25 + ((½ × 91 - 38) ÷ 16) × 10 = 25 + ((45.5 - 38) ÷ 16) × 10 = 25 + (7.5 ÷ 16) × 10 = 25 + 4.69 = 29.7 (to 3 significant figures)
Common exam tips
Critical Points to Avoid Common Mistakes:
- Always check your cumulative frequency calculations are correct - they should add up to the total frequency
- Remember that the median position uses (n + 1) ÷ 2, but in the estimation formula you use ½n
- Class width is always the difference between the upper and lower boundaries of the interval
- Round your final answer to the number of significant figures requested in the question
Remember!
Key Points to Remember:
- You cannot find exact mode and median values from grouped data - only estimates and class intervals
- The modal class is simply the class with the highest frequency
- Find the median class by calculating the (n + 1) ÷ 2 position and using cumulative frequency
- The estimated median formula has five components: L, n, F, f, and w - make sure you identify each one correctly
- Always double-check your cumulative frequency calculations before using the estimation formula