Displaying Data (HSC SSCE Mathematics Advanced): Revision Notes

Displaying Data

Introduction to data analysis

When you collect data, whether small amounts or huge datasets, you usually end up with unsorted lists of numbers or categories. Just scanning through this raw data rarely gives you useful insights. The field of statistics provides tools to help analyse and make sense of this information.

After this analysis, you might suggest further experiments to collect more data and test your findings.

Review of random variables

Before working with data displays, let's review some fundamental concepts about random variables and experiments.

Experiments and random variables

An experiment in statistics can be either deterministic or random:

• A deterministic experiment has only one possible outcome
• A random experiment has more than one possible outcome

A random variable is the outcome when you run a random experiment. We usually denote it with an uppercase letter like $X$. The various possible outcomes of the experiment are called the values of the random variable.

Scores and frequency

When you run an experiment many times, the outcomes are called scores. The complete (finite) list of all the scores is called a sample.

The frequency of an outcome or value is the number of times it occurs in your sample.

Types of random variables

Random variables can be classified in different ways:

• A random variable may be numeric (if its values are numbers) or categorical (if its values are categories or labels)
• A numeric random variable is called discrete if its values can be listed in a sequence like $x_1, x_2, x_3, ...$
• A numeric random variable is called continuous if its values cannot be listed (like measuring height as a real number rather than a rounded measurement)

Frequency tables and cumulative frequency tables

What is a frequency table?

The most basic tool for organising raw data is a frequency table. You can create one using a spreadsheet or database, or by hand using tally marks.

When working with numeric data, you can extend this to a cumulative frequency table, which shows the number of scores less than or equal to each given value.

Cumulative frequency for categorical data

If the values in a categorical dataset have been sorted into a meaningful order, you can also create a cumulative frequency table for them (this is particularly useful for Pareto charts, which we'll discuss later).

Understanding cumulative frequency

For numeric data, the cumulative frequency tells you the number of scores that are less than or equal to a given score.

Finding the median from cumulative frequencies

When analysing single variable data, two main questions guide our investigation:

• Measures of location: Where is the centre of the distribution?
• Measures of spread: How spread out are the data?

The median is an important measure of location.

What is the median?

The median (symbol $Q_2$, meaning second quartile) is the middle score when you arrange all scores in ascending order. More specifically:

• For an odd number of scores, the median is the single middle score
• For an even number of scores, the median is the average of the two middle scores

A cumulative frequency table makes it easy to identify the median.

Using cumulative frequency to find the median

Let's use the spelling test data from earlier to find the median.

There are $40$ scores, so the median is the average of the 20th and 21st scores.

Displaying categorical data

There are many different ways to display data using tables and graphs. Good displays should be easy to read, even for people who aren't statistics experts. They should help you gain insights and communicate findings clearly.

Pareto charts

A Pareto chart is a powerful tool for displaying categorical or discrete data. While it can be used for any categorical dataset, its main purpose is to identify which problems in a business are most urgent and need attention first. It's classified as one of the 'seven basic tools of quality' in business management.

Constructing a Pareto chart

To create a Pareto chart, follow these steps:

Step 1: Arrange the categories in descending order of frequency (highest first)

Step 2: Calculate the cumulative frequency for this new order

Step 3: Draw two graphs together:

• A frequency histogram with columns in descending order
• A cumulative frequency polygon (line graph)

The chart typically has two vertical axes:

• Left axis: actual frequencies
• Right axis: percentage frequencies

Here's the reorganised table and the Pareto chart:

Interpreting the Pareto chart

With this chart, the manager can work through issues from left to right, tackling the most serious problems first:

• Rain ($88$ occurrences): The manager might decide to only schedule external roof repairs three days ahead when weather forecasts are more reliable
• Owner not home ($64$ occurrences): The manager could personally ring each owner two days ahead with a friendly reminder, followed by an SMS the evening before
• Truck breakdown ($22$ occurrences): Perhaps a new truck has been budgeted for next year
• Other issues (illness, absence, blackout): The manager may have little control over these

Conventions for Pareto charts

There are no universal standards for drawing Pareto charts, but common features include:

• The histogram rectangles join up with each other
• The cumulative frequency polygon starts at the left-hand bottom corner (because the initial sum is zero)
• Each point is plotted at the right-hand top corner of its corresponding rectangle

Two-way tables (contingency tables)

A two-way table (also called a contingency table) combines two or more related frequency tables. It allows you to investigate whether two variables are related.

The importance of marginal frequencies

To analyse this properly, we need to add row and column totals (called marginal frequencies):

Analysing proportions

To answer the question "Do men prefer dark colours more than women do?", we need to calculate proportions:

For men:

• Proportion preferring dark covers: $\frac{38}{50} = 76\%$
• Proportion preferring coloured covers: $\frac{12}{50} = 24\%$

For women:

• Proportion preferring dark covers: $\frac{56}{150} \approx 37\%$
• Proportion preferring coloured covers: $\frac{94}{150} \approx 63\%$

The analysis clearly shows that men prefer dark covers much more than women do ($76\%$ versus $37\%$).

Understanding the table structure

Each row and each column in the extended table is actually a frequency table:

• The last row and last column are called marginal distributions
• The inner rows and columns are called conditional distributions

This terminology relates to conditional probability concepts.

Conditional probability in two-way tables

The proportions we calculated are actually probabilities. If we choose a person from the survey at random:

$P(\text{person prefers dark covers}) = \frac{94}{200} = 0.47$

We can also calculate conditional probabilities. These tell us the probability of one event given that another event has occurred.

To find the probability that a person prefers dark covers given that they are a man (or woman), we use a reduced sample space:

$$P(\text{prefers dark} \mid \text{man}) = \frac{38}{50} = 0.76 P(\text{prefers dark} \mid \text{woman}) = \frac{56}{150} \approx 0.37$$

We can also work in the opposite direction:

$$P(\text{person is a man}) = \frac{50}{200} = 0.25$$ $$P(\text{man} \mid \text{prefers dark}) = \frac{38}{94} \approx 0.40 P(\text{man} \mid \text{prefers coloured}) = \frac{12}{106} \approx 0.11$$

These conditional probabilities give us deeper insights into the relationships between variables.

The mode and the range

Besides the median, there are two other simple but useful statistics: the mode and the range.

The mode

The mode is the most popular score in a dataset. It's the score with the greatest frequency.

Multiple modes

Some frequency tables have two or more scores with the same maximum frequency. These are called:

• Bimodal: two scores with equal highest frequency
• Trimodal: three scores with equal highest frequency
• Multimodal: several scores with equal highest frequency

The range

The range is only defined for numeric data. It measures how spread out the data are.

Definition: The range is the difference between the minimum and maximum scores.

Summary: measures of location and spread

• The mode is a measure of location (tells you where the centre or most common value is)
• The range is a measure of spread (tells you how dispersed the data are)