Describing and Comparing Distributions (HSC SSCE Mathematics Standard): Revision Notes

Describing and Comparing Distributions

Understanding distribution shapes

When working with data, we need to describe the overall shape and pattern of how values are distributed. The general shape of a distribution can be described using three main characteristics: smoothness, symmetry, and the number of modes.

A smooth distribution is one where data changes gradually without sudden breaks or irregular jumps. The pattern flows continuously from one value to the next.

Symmetry refers to whether the distribution is balanced evenly on both sides of a central vertical line. If you could fold the distribution along this centre line, both halves would match up like a mirror image.

Types of distribution shapes

Distributions can be classified based on how many peaks or modes they contain:

Unimodal distributions have a single peak or mode. Most of the data clusters around one central value, creating one high point in the distribution.

Bimodal distributions have two distinct peaks or modes. This often happens when you have two different groups mixed together in your dataset, each with its own typical value.

Multimodal distributions have many peaks or modes. This suggests the data comes from several different groups or sources, each contributing its own cluster of values.

Symmetry and skewness

Data is symmetric when it forms a mirror image of itself around a central vertical line. If you were to fold the distribution in half at the middle, both sides would match up perfectly. In symmetric distributions, the data is evenly balanced on both sides.

However, many real-world datasets are not symmetric. When data has more values concentrated on one side, we say it is skewed. The direction of the skew is determined by where the long tail of the distribution points.

Types of skewness

There are three main categories of skewness:

No skew (symmetric): The data is evenly balanced on both sides of a vertical centre line. The left and right halves are mirror images of each other.

Positively skewed: There is more data concentrated on the left side of the distribution, with a long tail extending toward the right (positive) side. The tail points in the positive direction along the number line. This often occurs with data like income or house prices, where most values are lower but a few are extremely high.

Negatively skewed: There is more data concentrated on the right side of the distribution, with a long tail extending toward the left (negative) side. The tail points in the negative direction along the number line. This might occur with test scores where most students score highly but a few score very low.

Measures of location

Measures of location (also called measures of central tendency) tell us about the typical or central value in a dataset. The three main measures are the mean, median, and mode. Each has its own strengths and weaknesses, making them suitable for different situations.

Mean

The mean is calculated by adding up all the values and dividing by the number of values:

$\bar{x} = \frac{\text{sum of all scores}}{\text{number of scores}}$

Advantages:

• Simple to understand and calculate
• Takes into account every single value in the dataset
• Most consistent across different samples from the same population

Disadvantages:

• Strongly influenced by extreme values (outliers)
• Cannot be used with categorical data (data in categories rather than numbers)

Median

The median is the middle value when all data values are arranged in order from smallest to largest.

Advantages:

• Simple to understand
• Not affected by outliers or extreme values

Disadvantages:

• May not represent the centre well in some distributions
• Data must be sorted before calculating
• Varies more than the mean when taking different samples from the same population

Mode

The mode is the value that occurs most frequently in the dataset.

Advantages:

• Very simple to identify
• Not affected by outliers
• Can be used with categorical data (like favourite colours or types of pets)

Disadvantages:

• There may be no mode at all, or multiple modes
• May not represent the centre of the data well

Measures of spread

Measures of spread tell us how much the data values vary or are dispersed around the centre. The three main measures are the range, interquartile range, and standard deviation.

Range

The range is the difference between the highest and lowest values:

$\text{Range} = \text{highest score} - \text{lowest score}$

Advantages:

• Very simple to understand
• Quick and easy to calculate

Disadvantages:

• Only depends on the two extreme values (smallest and largest)
• Can be heavily distorted by a single outlier

Interquartile range (IQR)

The interquartile range is the difference between the upper quartile ($Q_3$) and the lower quartile ($Q_1$):

$\text{IQR} = Q_3 - Q_1$

This represents the spread of the middle 50% of the data.

Advantages:

• Straightforward to calculate for small datasets
• Easy to understand (focuses on middle 50% of data)
• Not affected by outliers

Disadvantages:

• More difficult to calculate for large datasets
• Only depends on the two quartile values
• Data must be sorted before calculating

Standard deviation

The standard deviation measures how far values typically deviate from the mean. It takes into account every value in the dataset.

Advantages:

• Uses every single data value
• Not significantly affected by outliers (compared to range)

Disadvantages:

• Complex to calculate without a calculator
• More difficult concept to understand

Comparison table

Measure	Advantages	Disadvantages
Mean	- Easy to understand and calculate - Uses every score - Most consistent across samples	- Distorted by outliers - Not suitable for categorical data
Median	- Easy to understand - Not affected by outliers	- May not be central - Data needs sorting - Varies more than mean in samples
Mode	- Easy to determine - Not affected by outliers - Suitable for categorical data	- May be no mode or multiple modes - May not be central
Range	- Easy to understand - Easy to calculate	- Depends only on extreme values - Distorted by outliers
IQR	- Easy for small datasets - Easy to understand - Not affected by outliers	- Difficult for large datasets - Depends on quartiles only - Data needs sorting
Standard deviation	- Uses every score - Not significantly affected by outliers	- Difficult to calculate without calculator - Difficult to understand

Worked example: Comparing statistics for two sets of data