Understanding Color Tolerance: A Deep Dive into MacAdam Ellipses

Introduction to MacAdam Ellipses and Color Tolerance

The study of color tolerance and the role of MacAdam ellipses in this field is a fascinating journey into the realm of color science. MacAdam ellipses were developed to address a critical question that arose after the establishment of the CIE 1931 standard: what should be the range of difference between the color coordinates of two objects for them to be perceived as having no color Tolerance? This question is at the heart of understanding color tolerance, a concept that refers to the degree to which we can perceive differences in color. MacAdam, a scientist specializing in color tolerance, conducted a series of experiments to answer this question. His findings revealed that if the chromaticity coordinate of a color sample diverges beyond the boundary of a MacAdam ellipse centered around a specific point (the target color), more than 99% of people will notice a color Tolerance. This discovery provided a new way to measure and understand color Tolerance, paving the way for further advancements in color science. These circles in the picture are called “MacAdam circles,” also known as MacAdam ellipses because they are elliptical in shape.

MacAdam Ellipses

CIE 1931 Color Space

The CIE 1931 color space, while a significant advancement in color science, has its limitations, most notably its non-uniformity. This non-uniformity is evident in the varying sizes of MacAdam ellipses across different color regions. For instance, the ellipse in the green section is larger than that in the blue section, indicating that the color tolerance in the green area is higher. In other words, even if there are significant differences in the Δx and Δy coordinates in the green area, they may still appear as similar shades of green. Conversely, even slight differences in the Δx and Δy coordinates in the blue area can result in perceptibly different shades of blue. Furthermore, the elliptical shape of the MacAdam circles suggests that color tolerance varies in the x and y directions. This non-uniformity presents a challenge in determining color differences, as knowing the color coordinates of two colors does not necessarily allow us to discern whether they will appear different to the human eye. This realization underscores the complexity of color perception and the need for more sophisticated tools and methods to accurately measure color differences.

Efforts to Improve Color Space Uniformity

Transforming MacAdam Ellipses into Circles

The non-uniformity of the CIE 1931 color space presented a significant challenge in the field of color science. To address this, subsequent developments in the CIE series aimed to transform the MacAdam ellipses into circles, thereby improving the uniformity of the color space. This transformation was a significant step forward, as it made it easier to determine color differences based on the distance between color coordinates.

Introduction to the Lab Color Space

One of the most notable advancements in this regard was the introduction of the Lab* color space, which is more uniform than the CIE 1931 color space. In the Lab* color space, the color difference ΔE*ab between two colors can be calculated simply by measuring the distance between their coordinates. This calculation is straightforward and intuitive, making it easier for scientists and practitioners to understand and apply.

ΔΕ ab = | (L₁₂ — L₁)² + (a₂ — a₁)² + (bb)²

The Importance of ΔE in Color Difference Calculation

The concept of ΔE, or Delta E, is central to understanding color differences. Derived from the German word “Empfindung,” meaning “perception,” ΔE represents the perceptible difference between two colors. However, it’s important to note that the calculation of ΔE has evolved over time, with different formulas being developed to improve its accuracy and applicability. This evolution underscores the ongoing efforts to improve our understanding and measurement of color differences.

Evolution of Color Difference Calculation

The calculation of color difference has seen significant evolution over the years. Starting with the ΔEab formula in the Lab color space, scientists recognized that the MacAdam ellipse in this space was still elliptical. This meant that even if the ΔE value was 1 (NBS unit), the perceived color difference could still vary slightly. To address this, the ΔE**ab formula was upgraded to ΔE*cmc, which introduced a weight function representing the size and eccentricity of the ellipse.

Further improvements were made with the introduction of the ΔE**94 (CIE94 Published 1995)formula, which considered different observation conditions that could affect the subjective perception of color difference. The most recent upgrade is the ΔE00((CIE2000 Published 2000)* formula, which considers a wider range of parameters to provide a more accurate calculation of color difference.

Understanding the Specific Algorithm Used

These developments highlight the importance of understanding the specific algorithm used when interpreting ΔE color difference parameters. Different formulas have different units, and color difference values in different systems cannot be easily converted. Therefore, when dealing with ΔE color difference parameters, it’s crucial to know which algorithm has been used to ensure accurate interpretation and application.

The Randomness of Color Perception

The Concept of ‘Order’ in MacAdam Ellipses

Understanding human color perception requires acknowledging its inherent randomness. This randomness is reflected in the concept of ‘step’ in MacAdam ellipses, which relates to standard deviation in statistics. The ‘first step’ refers to +/-1 standard deviation, indicating that once the color coordinate exceeds this first-order circle, there’s a probability of about 68.3% of being judged as having a color difference.

Normal Distribution and Its Relevance to Color Difference Judgment

This concept is rooted in the normal distribution, also known as the Gaussian distribution, which is widely used in statistics. The normal distribution curve is bell-shaped, and the area under the curve can be used to estimate the cumulative probability of a specific event occurring. In the context of color difference judgment, this means that as the error between the sample value and the mean increases, the probability of occurrence decreases rapidly.

Implications for Color Difference Perception

These statistical insights have significant implications for understanding color difference perception. They highlight the fact that judging color differences with the naked eye is not a deterministic event but a random one that follows a probability distribution. This realization has led to a shift in perspective, viewing color differences from a statistical standpoint rather than a deterministic one.

MacAdam’s Experiment

The Experimental Process

MacAdam’s experiment was a groundbreaking study in the field of color science. The experiment involved choosing a standard color for comparison and then slightly altering the mixing conditions of two other colors to generate a new color that deviated slightly from the standard. Observers would then readjust the mixing conditions to make the new color return to the standard. By comparing the color coordinates of the standard color and the adjusted color, MacAdam was able to learn about the color tolerance range around the standard color.

Findings and Implications

The data resulting from MacAdam’s experiment revealed that there is no clear boundary for color tolerance. Instead, the data suggested that judging color differences is a random event that should be viewed from a statistical perspective. This discovery was a significant shift in understanding color differences, moving away from the idea of a clear threshold to a statistical understanding based on standard deviation.

The Role of Perley G. Nutting

A notable figure in MacAdam’s experiment was Perley G. Nutting, the main observer whose observations formed the bulk of the experiment’s data. Nutting’s significant contribution to the experiment underscores the importance of meticulous observation in advancing our understanding of color science.

Practical Implications

Color Difference Standards in Different Situations

The findings from MacAdam’s experiment have significant practical implications. One of the key takeaways is that the control of color differences should be analyzed based on specific situations. There is no universal threshold to determine whether there is a color difference. Instead, a reasonable color difference standard should be set according to specific needs. This understanding allows for more nuanced and accurate color difference management in various contexts.

Application in LED Light Source Management

One practical application of MacAdam ellipse knowledge is in the management of color differences in LED light sources. During the production of LEDs, they are divided into bins according to color and luminous intensity. Each manufacturer has a different method for binning, but generally, they follow ANSI standards and use a 4th-step MacAdam ellipse for binning. Understanding the order of the MacAdam ellipse used can provide a rough idea of the color control level in LED manufacturing.

Monocular Testing Device

An interesting fact from MacAdam’s experiment is that the left and right eyes of a person do not observe color synchronously, which can cause fluctuations in color difference judgments. Therefore, a monocular testing device was used in the experiment. This insight could be useful in practical applications where precise color difference judgment is required.

Fun Facts and Additional Information

Asynchronous Color Observation of Human Eyes

One fascinating fact that emerged from MacAdam’s experiment is that the left and right eyes of a person do not observe color synchronously. This means that when both eyes are observing, it can cause fluctuations in color difference judgments. This insight could be useful in practical applications where precise color difference judgment is required. For instance, one could try closing one eye when judging the color difference of a sample.

The Device Used in MacAdam’s Experiment

MacAdam’s experiment required a complex device that could automatically maintain a stable brightness level when adjusting the color coordinates. This was necessary to avoid system errors caused by the testing device. The device, which was quite intricate and advanced for its time, is a testament to the effort and resources invested in the experiment.

The Shift in Understanding Color Tolerance

Before MacAdam’s experiment, people understood the tolerance for color differences based on “just-noticeable differences” and studied color differences from that perspective. MacAdam’s experiment led to a shift in this understanding, with the standard deviation replacing the threshold as the best way to express color tolerance. This shift marked a significant advancement in the field of color science.


MacAdam’s experiment was a significant milestone in the field of color science. It marked a shift in understanding color tolerance from a deterministic concept to a statistical one. The experiment required a significant investment of time, resources, and effort, with the main tester, Perley G. Nutting, conducting approximately 25,000 observations in total. This hard-won data has greatly contributed to our understanding of color differences and color tolerance.

The experiment’s findings have practical implications in various fields, including LED light source management. They also provide valuable insights for individuals and professionals who need to make precise color difference judgments. The experiment’s findings remind us that color perception is not a deterministic event but a random one that follows a probability distribution.

In conclusion, MacAdam’s experiment underscores the complexity of color perception and the importance of a statistical approach to understanding color differences. It also highlights the ongoing efforts to improve our understanding and measurement of color differences. As we continue to delve deeper into the realm of color science, we can look forward to further advancements and discoveries that will enhance our understanding of this fascinating field.

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