How does an extreme outlier affect the mean compared to the median?

The mean is significantly pulled toward the outlier because it incorporates the magnitude of every value into its sum. The median remains unchanged or moves only slightly because it only considers the middle position of the data.

When is the mode a more appropriate measure of average than the mean or median?

The mode is most appropriate for non-numerical (categorical) data where mathematical calculations are impossible. It is also preferred when the goal is to identify the most common or popular item in a set.

What is the difference between the 'mode' and the 'modal class'?

The mode is a specific single value that occurs most often in discrete data. The modal class is an interval in grouped data that contains the highest frequency of observations.

What is a common mistake when identifying the mode from a frequency table?

A common error is stating the highest frequency itself as the mode. The mode is actually the data value (or category) that corresponds to that highest frequency.

Why is it an error to calculate the median of a data set without sorting it first?

The median is defined as the middle value of an ordered sequence. Without sorting the data in ascending or descending order, the middle position does not represent the 50th percentile of the data.

What happens if you use the class width instead of the midpoint to estimate the mean of grouped data?

Using class width results in a meaningless value. The midpoint is required because it acts as the best mathematical estimate for the average value of all data points within that specific interval.

Define the 'Mean' in terms of a mathematical process.

The mean is the sum of all observations in a data set divided by the total number of observations ($n$). It is represented by the formula $\bar{x} = \frac{\sum x}{n}$.

What does the 'Median' represent in a distribution?

The median represents the 50th percentile, meaning exactly half of the data values are less than or equal to it, and half are greater than or equal to it.

What is the formula for the position of the median in a data set of size $n$?

The position is found using $\frac{n+1}{2}$. This tells you which ranked term to select (e.g., if the result is 5, you take the 5th value; if 5.5, you average the 5th and 6th values).

Write the formula for the estimated mean of grouped data.

The estimated mean is $\frac{\sum (f \cdot m)}{\sum f}$, where $f$ is the frequency of each class and $m$ is the midpoint of that class.

Mean, Median & Mode | WJEC GCSE Maths

Maths And Numeracy Double Award / Foundation

Statistics & Probability

Mean, Median & Mode

Summary

Averages are statistical measures of central tendency used to represent a whole data set with a single value. The mean, median, and mode each provide a different perspective on the 'center' of data, with varying levels of sensitivity to extreme values and suitability for different data types.

1. Definition & Core Concepts

Mean: The arithmetic average calculated by summing all values in a data set and dividing by the total count of those values. It represents the 'balance point' of the data.
Median: The middle value of a data set when all observations are arranged in ascending or descending order. If the set has an even number of values, it is the numerical average of the two central points.
Mode: The value that appears with the highest frequency in a data set. A set can be unimodal (one mode), bimodal (two modes), or multimodal, and it is the only average applicable to non-numerical (categorical) data.

A diagram showing a right-skewed distribution curve where the Mode is at the peak, the Median is to the right of the peak, and the Mean is pulled furthest toward the tail.

2. Underlying Principles

Central Tendency: This principle suggests that data points in a distribution tend to cluster around a central value. Each average attempts to locate this cluster using different mathematical logic.
Sensitivity to Outliers: The mean is highly sensitive to extreme values (outliers) because every value contributes to the sum. In contrast, the median is 'resistant' because it only depends on the relative order of values, not their specific magnitudes.
Data Aggregation: When data is presented in frequency tables, the averages are calculated by weighting each value by its frequency, effectively treating the table as a compressed list of individual points.

3. Methods & Techniques

4. Key Distinctions

Mean vs. Median: The mean is best for symmetric data without outliers, while the median is superior for skewed data (like house prices or salaries) where a few high values would distort the mean.
Mode vs. Others: The mode is the only measure that can be used for qualitative data (e.g., 'favorite color'). It is also useful in business for identifying the most popular product size.

Feature	Mean	Median	Mode
Calculation	Sum / Count	Middle Value	Most Frequent
Outlier Impact	High	Low	Low
Data Type	Numerical	Numerical	Any

5. Exam Strategy & Tips

6. Common Pitfalls & Misconceptions