How does an extreme high outlier affect the mean versus the median?

An extreme high outlier will pull the mean upward, making it higher than the majority of the data. The median remains largely unchanged because it only depends on the middle position of the ordered data.

When is the mode the only appropriate measure of central tendency to use?

The mode is the only appropriate measure when dealing with qualitative (categorical) data, such as eye color or brand names, where numerical calculations like sums or ordering are not possible.

In a right-skewed distribution, what is the typical relationship between the mean and the median?

In a right-skewed distribution (positive skew), the mean is typically greater than the median because the long tail of high values pulls the mean toward the right.

What is a common error when calculating the median of a data set?

A common error is failing to arrange the data in ascending or descending order before identifying the middle value. The median is defined by its position in a sorted list.

What mistake is made when a student says the mode of {1, 2, 3, 4} is 0?

The mistake is confusing 'no mode' with a value of zero. If no value repeats, the set has no mode; saying the mode is 0 implies that the number 0 appeared most frequently.

Why is it risky to use the mean for a data set representing house prices in a city?

House prices are often skewed by a few extremely expensive mansions (outliers). Using the mean would overstate the 'average' price, while the median would better represent what a typical buyer might pay.

Define the 'Mean' in terms of its mathematical property regarding deviations.

The mean is the value such that the sum of the deviations (differences) of all data points from that value is exactly zero. It is the arithmetic balance point of the data.

What does it mean for a data set to be 'bimodal'?

A bimodal data set has two distinct values that share the highest frequency. This often suggests that the data set might be composed of two different sub-groups.

How is the median calculated for a data set with an even number of values ($n$)?

For an even $n$, the median is the arithmetic mean of the two middle values (the values at positions $\frac{n}{2}$ and $\frac{n}{2} + 1$).

What is the formula for the mean of a sample?

The sample mean is calculated as $\bar{x} = \frac{\sum x_i}{n}$, where $\sum x_i$ is the sum of all individual observations and $n$ is the total number of observations.

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Processing, Representing & Analysing Data

Using Measures of Central Tendency

Summary

Measures of central tendency are statistical tools used to identify a single central value that represents an entire data set. By summarizing complex data into a single 'average,' researchers can compare different groups and identify typical characteristics within a population.

1. Definition & Core Concepts

Mean (Arithmetic Average): The sum of all values in a data set divided by the total number of values. It represents the 'balance point' of the data where the sum of distances to all points is zero.
Median: The middle value in a data set when the values are arranged in ascending or descending order. If the data set has an even number of observations, the median is the average of the two central values.
Mode: The value that appears most frequently in a data set. A data set can have one mode (unimodal), two modes (bimodal), multiple modes (multimodal), or no mode at all if all values are unique.

A right-skewed distribution curve showing the relative positions of the Mode (at the peak), Median (to the right of the mode), and Mean (furthest to the right).

2. Underlying Principles

Sensitivity to Values: The Mean is highly sensitive to every value in the set; changing a single number changes the mean. This makes it a powerful but potentially misleading measure if extreme values are present.
Positional Stability: The Median is a positional measure, meaning it only cares about the order of values. It is 'robust' because extreme values (outliers) do not pull it away from the center of the majority of the data.
Frequency Focus: The Mode is the only measure that can be used for Qualitative (Categorical) Data, such as favorite colors or car brands, where numerical calculations are impossible.

3. Methods & Techniques

4. Key Distinctions

Feature	Mean	Median	Mode
Data Type	Quantitative only	Quantitative / Ordinal	All types
Outlier Impact	High (Sensitive)	Low (Robust)	Low (Robust)
Uniqueness	Always one	Always one	Can be multiple or none
Best Use	Symmetric data	Skewed data	Categorical data

Mean vs. Median: In a perfectly symmetric distribution, the mean and median are equal. In a skewed distribution, the mean is pulled toward the 'tail' (the direction of the outliers).

5. Exam Strategy & Tips

6. Common Pitfalls & Misconceptions