Frequency Tables & Mean, Median, and Mode (HSC SSCE Mathematics Standard): Revision Notes

Frequency Tables & Mean, Median, and Mode

What is a frequency table?

A frequency table is a systematic way to organise data. It displays each unique score or outcome in your dataset alongside how many times that score appears. This type of table is also known as a frequency distribution.

A frequency table typically consists of three columns:

• Score column: Lists all the different scores or outcomes in ascending order (from smallest to largest)
• Tally column: Records the count using tally marks, which are grouped in sets of five for easy counting
• Frequency column: Shows the final count of how many times each score occurred

The tally column helps you keep track as you count through your data. However, if you're given a frequency table that has already been created, the tally column might be omitted.

In the example above, you can see how the table clearly identifies:

• The lowest score (17) and highest score (21) in the dataset
• The lowest frequency (1 occurrence) and highest frequency (7 occurrences)
• How tally marks are grouped in fives with a diagonal line through each group

How to construct a frequency table

Building a frequency table from raw data is straightforward when you follow these steps:

1. Draw a table with three columns labelled "Score", "Tally", and "Frequency"
2. In the score column, list all the unique values from your dataset in ascending order, starting with the lowest and ending with the highest
3. Go through your data one value at a time, making a tally mark in the appropriate row each time you encounter that score. Remember to group your tally marks in fives
4. Count the tally marks for each score and record the total in the frequency column
5. Add up all the values in the frequency column. This total should match the number of data points in your original dataset - this is a useful way to check your work

Understanding measures of central tendency

When you have a dataset, you often want to describe it using a single value that represents the centre or "typical" value of the data. Three common measures help us do this: the mean, median, and mode. Each measure gives us different information about our data.

Mean

The mean represents the arithmetic average of your dataset. It's calculated by adding all the scores together and dividing by the number of scores you have.

The formula for the mean is:

$\bar{x} = \frac{\sum x}{n}$

Where:

• $\bar{x}$ (read as "x bar") represents the mean
• $\sum x$ means the sum of all the scores
• $n$ is the total number of scores

The mean is what most people refer to when they talk about "the average". For example, if you have the scores 1, 6, 3, and 2, you would calculate:

$\bar{x} = \frac{1 + 6 + 3 + 2}{4} = \frac{12}{4} = 3$

The mean is particularly useful because it takes into account every value in your dataset. However, it can be affected by extreme values (outliers).

Median

The median is the middle value in your dataset when all the scores are arranged in order from smallest to largest. It divides your data in half - 50% of the scores are below the median and 50% are above it.

To find the median:

1. Arrange all scores in ascending order (from smallest to largest)
2. If you have an odd number of scores, the median is simply the middle valueFor example: In the dataset 1, 4, 5, 7, 8, the median is 5
3. If you have an even number of scores, the median is the average of the two middle valuesFor example: In the dataset 1, 1, 4, 5, 7, 8, there are two middle values (4 and 5)The median is $\frac{4 + 5}{2} = 4.5$

The median is less affected by extreme values than the mean, making it useful when your data contains outliers.

Mode

The mode is the score that appears most frequently in your dataset. In a frequency table, it's the score with the highest frequency.

To find the mode:

1. Count how many times each score occurs (or look at the frequency column in a frequency table)
2. The score that occurs most often is the mode

The mode is particularly useful for categorical data where you can't perform numerical calculations. For example, if you're collecting data about favourite colours, the mode tells you which colour was chosen most often.

Note that a dataset can have more than one mode if multiple scores appear with the same highest frequency. Modes can occur at any position in the range of values - at the beginning, middle, or end.

Worked example: Calculating mean, median, and mode

Let's work through a complete example that brings together all three measures.

The table below shows the number of rainy days for the first six months of the year: