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‘Unbounded’ data refers to data that can take any value along the real number line, without upper or lower limits.

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For example, temperature measured in Kelvin is bounded at 0 (cannot go below absolute zero), but in Celsius or Fahrenheit, temperatures can theoretically be negative or positive without any upper limit.

• Surface temperature of the Sun: ~5,500°C
• Deep space temperature: ~-270°C
• In Celsius or Fahrenheit, this represents ‘unbounded’ data.

Handling unbounded data in standardisation involves strategies to manage extreme values since these can skew the mean and standard deviation.

Methods such as logarithmic transformations or trimming extremes might be used before standardising to mitigate the impact of outliers.

Example

In athletics, if we consider the dataset of marathon finishing times where elite athletes might have exceptionally low times, applying a logarithmic transformation before Z-score standardisation can reduce the influence of these outliers, leading to a more representative comparison across all runners' performances.

Here's a visualisation of how logarithmic transformation affects unbounded data:

graph LR;
    A["Raw Data
    (Unbounded)"] --> B["Log
    Transform"];
    B --> C["Standardise"];
    
    style A fill:#ffcccc
    style B fill:#ccffcc
    style C fill:#cce5ff
    
    %% Example values shown below
    subgraph "Data Distribution"
    D["Original:
    1, 10, 100, 1000"] --> E["Log Transform:
    0, 1, 2, 3"] --> F["Standardised:
    -1.5, -0.5, 0.5, 1.5"]
    end

This figure shows how logarithmic transformation compresses large values while preserving relative relationships, making the data more suitable for standardisation.

Next element:

Normalisation