Frequency Distributions (Leaving Cert Mathematics): Revision Notes
Frequency Distributions
What is a frequency distribution?
A frequency distribution is a way of organising data that shows how often each value appears in a dataset. The data is presented in a table format with two rows:
- The first row shows the variable (the different values or categories)
- The second row shows the frequency (how many times each value occurs)
Understanding the Structure
In frequency distribution tables, we use specific terminology:
- Variable (x): The different values being measured
- Frequency (f): How many times each value appears in the data
For example, if we surveyed 31 people about how many emails they received in a day, we might get this frequency distribution table:
| Number of emails | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|---|
| Frequency (number of people) | 4 | 11 | 8 | 6 | 1 | 0 | 0 | 1 |
This table tells us that 4 people received 0 emails, 11 people received 1 email, 8 people received 2 emails, and so on.
Finding the mode
The mode is the value that appears most frequently in the distribution. To find the mode, simply look for the value with the highest frequency.
Definition of Mode
The mode is the value that occurs most often in a frequency distribution.
Looking at a word length distribution:
| No. of letters in word | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|
| Frequency | 3 | 4 | 9 | 5 | 2 |
The mode is 5 letters because it has the highest frequency (9).
Common Mistake Alert
The mode is always a value from the original data, not the frequency number. In the example above, the mode is 5 (letters), not 9 (the frequency).
Finding the median
The median is the middle value when all the data is arranged in order from smallest to largest.
Definition of Median
The median is the middle value in a frequency distribution when the data is arranged in order.
To find the median:
- Calculate the total frequency by adding all the frequencies: Σf
- Find the median position using: ½(n + 1) where n is the total frequency
- Count through the frequencies until you reach the median position
Worked Example: Finding the Median
Using the word length data above:
- Total frequency = 3 + 4 + 9 + 5 + 2 = 23
- Median position = ½(23 + 1) = 12th value
- Count through frequencies: 3 + 4 = 7, then 7 + 9 = 16
- The 12th value falls in the group with 5 letters
- Therefore, median = 5
Quick Method
You can find the median without listing all values by adding frequencies from left to right until you reach the median position.
Calculating the mean
The mean of a frequency distribution is the average value, taking into account how often each value appears.
Definition of Mean
The mean is the sum of all values divided by the total number of values in a frequency distribution.
Formula: Mean =
Where:
- Σfx = sum of (frequency × variable) for each value
- Σf = sum of all frequencies
- x = variable (the values)
- f = frequency
Worked Example: Calculating the Mean
Calculate the mean for this distribution:
| Marks | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|
| No. of pupils | 1 | 1 | 1 | 3 | 5 | 3 | 2 | 2 | 1 | 1 |
Step 1: Calculate Σfx
- 1(1) + 2(1) + 3(1) + 4(3) + 5(5) + 6(3) + 7(2) + 8(2) + 9(1) + 10(1)
- = 1 + 2 + 3 + 12 + 25 + 18 + 14 + 16 + 9 + 10 = 110
Step 2: Calculate Σf
- 1 + 1 + 1 + 3 + 5 + 3 + 2 + 2 + 1 + 1 = 20
Step 3: Apply the formula
- Mean = = 5.5 marks
Worked Example: Finding an Unknown Value
Problem: If the mean of this frequency distribution is 3, find the value of x and write down the mode.
| Goals scored | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| Number of matches | 7 | 8 | 4 | 4 | 3 | x |
Step 1: Set up the mean formula Mean =
Step 2: Calculate Σfx and Σf
- Σfx = 7(1) + 8(2) + 4(3) + 4(4) + 3(5) + x(6)
- = 7 + 16 + 12 + 16 + 15 + 6x = 66 + 6x
- Σf = 7 + 8 + 4 + 4 + 3 + x = 26 + x
Step 3: Substitute into the mean formula
Step 4: Solve for x
- 3(26 + x) = 66 + 6x
- 78 + 3x = 66 + 6x
- 78 - 66 = 6x - 3x
- 12 = 3x
- x = 4
Step 5: Find the mode Looking at the frequencies: 7, 8, 4, 4, 3, 4 The highest frequency is 8, so the mode = 2 goals.
Exam tips and common mistakes
Common Exam Mistakes to Avoid
- Confusing the mode with the highest frequency (the mode is the value, not how often it appears)
- Forgetting to multiply each variable by its frequency when calculating the mean
- Not showing working clearly for the mean calculation
Exam Techniques
- Always check your total frequency adds up correctly
- When finding the median, clearly show the median position calculation
- For mean calculations, set up a clear table showing x, f, and fx columns
- Double-check your arithmetic, especially in Σfx calculations
Time-Saving Tips
- For the median, you can often find it without listing all values by using cumulative frequencies
- When calculating means, organise your work in columns to avoid errors
- Always state your final answers clearly with appropriate units
Key Points to Remember
- Frequency distributions organise data to show how often each value appears
- Mode = the value that occurs most frequently (highest frequency)
- Median = the middle value; find using position
- Mean = where you multiply each value by its frequency
- Always show clear working in exam questions, especially for mean calculations