Classifying Data: Median, Mode, and Range

Expert reviewed • 08 January 2025 • 7 minute read

HSC Maths Advanced Syllabus

classify data relating to a single random variable

The following information is explored in the year 11 course. We must review these terms before starting the following module.

Data Terminology

Scores: The results of an experiment are noted as scores. Multiple scores combined together in a categorical manner create a ‘sample’
Frequency: The frequency of a score, is the number of times it occurs.
Random Variables: A random experiment is the source of randomness or uncertainty, thus a random variable provides a framework to quantify the outcomes of a random experiment (This is generally denoted by the variable $X$ $X$ ).
- A random experiment is an experiment with various outcomes
- A deterministic experiment is an experiment with only one result

A discrete random variable has a countable number of possible values. This means the values can be listed, like whole numbers. Alternatively, a continuous random variable can take any numeric value within a range or interval, which can be finite or infinite. These can be written down or explained in a sequence.

What is Cumulative Frequency?

Cumulative Frequency is the sum of the frequencies of all values that are less than or equal to a specific value in a dataset. To determine the cumulative frequency of a function from a frequency distribution table, we can follow the steps listed below.

List the data in ascending order
Tally the frequency of each individual value or class interval if the data is grouped.
Add the frequency of each value to the sum of the frequencies of all preceding values.

The Median

The median score of a set of data is noted as the middle score. When a dataset has an odd number of scores (values found during the experiment), the median is the direct middle score, when the dataset is arranged in ascending order. When there is an even number of scores, the median is the average of the two most middle scores.

The Mode and the Range

The mode of a dataset, refers to the score with the highest frequency. If a dataset has no repeating values, there is no mode.
The range of a dataset refers to the difference between the maximum and minimum scores.

Practice Question 1

Determine the median and mode of the following scores: $2, 46 ,31, 151, 150, 150, 8, 97$

Our first step is to arrange the dataset into ascending order:

1,8,31,46,97,150,150,151

As we can see there is an even amount of scores in this dataset (8 scores). Thus, to find the median we must find the average of the two middle terms. These terms are $46$ and $97$ .

Median=\frac{46+97}{2}\\=\frac{143}{2}\\=71.5

In this case, the mode of the dataset is simple to find. From looking at the small dataset above, we can see that the value with the highest frequency is $150$ .

$\therefore$ Median $=71.5$ , Mode $=150$

What are Two-Way Tables?

Two-way tables, known as contingency tables, provide a method to display data that compares two different categorical variables. A comprehensive understanding of two-way tables is important, as they relate to coming chapters regarding correlation and regression. Below is an example of a two-way table, whereby the two variables, height $(X)$ and weight $(Y)$ , are being compared.

Height (X)	Weight (Y)
160	60
170	70
180	80
190	90

Up Next

Calculating and Graphing Grouped Data: Frequency Tables, Histograms, & More

Return to Module 9: Displaying and Interpreting Data