U-Shaped Distribution

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A U-shaped distribution is characterized by two peaks at both ends of the range of possible values and a minimum in the middle. As a result, values at the extremes are more likely than those in the middle. With frequencies that steadily fall and then steadily rise, a U-shaped distribution can be considered a bimodal distribution.

u-shaped arcsin distribution
A U-shaped distribution.

Typically, measurements with cyclical or sinusoidal patterns follow a U-shape distribution. Although sometimes referred to as an “upside-down bell curve,” it cannot be transformed into a bell curve even if the distribution is perfectly symmetric. This was established by Iversen and Norpoth in 1987 [1]. However, the observation of U-shaped distributions dates back over a century. Pearson pointed out in 1895 [2] that such distributions, characterized by the prevalence of extreme values, could be found in meteorological events and competitive examinations. In the latter case, he noted that individuals with mediocre abilities might sometimes wisely choose not to participate.

U-shaped probability distributions

Many distributions can take on u-shapes, depending on the value of their parameters. They include:

u-shaped distribution
The beta distribution takes on a wide variety of shapes depending on its parameters α and β; this can include a U-shape (red)when α and β are both less than 1.

Properties of the symmetric U-shaped distribution

A symmetric U-shaped distribution exhibits these properties:

  1. Both tails are identical on either side of the distribution.
  2. The coefficient of skewness is zero.
  3. The mean is smaller than the highest mode.
  4. The mean and median values are equal.
  5. Quartiles can be expressed as (Q3 – Md) = Md – Q1), meaning the difference between the third quartile and the median is equal to the difference between the first quartile and the median.

Examples of the U-shaped distribution

While the specific distribution of data in real-life scenarios can vary depending on various factors, some natural phenomena can result in a U-shaped distribution, such as:

  1. Meteorological phenomena: Extreme weather conditions such as very high or very low temperatures may follow a U-shaped distribution, with fewer occurrences of moderate temperatures. For example, deserts are known for their extreme temperature fluctuations between day and night. During the daytime, temperatures can soar to extremely high levels, while at night, they can plummet drastically. This daily cycle of temperature extremes may create a U-shaped distribution.
  2. Competitive exams: In some cases, the score distribution for competitive exams might be U-shaped if a significant number of participants score either very high or very low, while fewer people achieve average scores. This could be because individuals with mediocre abilities may choose not to participate, as Pearson (1895) suggested, but there other examples. For instance, consider a highly specialized professional certification exam, such as one for advanced technical skills or industry-specific knowledge. The candidates taking this test are likely to fall into two main categories: Well-prepared candidates who have invested substantial time and effort in mastering the subject matter, either through formal education, on-the-job training, or rigorous self-study and unprepared candidates who might be attempting the exam without sufficient preparation, perhaps due to overconfidence, misinformation about the exam’s difficulty, or external pressure from an employer. In this scenario, there may be fewer candidates who achieve average scores because the exam’s difficulty level and specialized nature make it challenging for unprepared individuals to score moderately well.
  3. Age at marriage: The age at which people marry might follow a U-shaped distribution, with peaks at younger and older ages and a dip in the middle. This pattern could arise due to cultural, social, or economic factors that influence people’s decisions about when to marry.
  4. Product pricing: The pricing of certain products or services may exhibit a U-shaped distribution, with more items priced at either the lower or upper ends of the market and fewer in the middle range. This can be seen in markets where there are both budget and premium offerings, with fewer options in the mid-price range. For example, airlines often exhibit a U-shaped pricing distribution. Low-cost carriers cater to cost-conscious travelers, while premium airline classes target affluent customers seeking comfort and exclusivity. Mid-tier options may be fewer in comparison.
  5. Job satisfaction: Employee job satisfaction levels could also follow a U-shaped distribution, with many employees reporting either high or low satisfaction and fewer reporting moderate satisfaction. This might be due to a combination of factors, such as company culture, job roles, and individual expectations.
  6. Survey results: The Beginning-Ending list bias exhibits a U-shaped pattern. When presented with numerous options, individuals tend to select items at the start or end of a list [4]. This type of bias can be particularly noticeable in rating scales ranging from 1 to 5 or 1 to 10, such as product reviews, where people either love or hate the item.
  7. Disease manifestation: Francis Galton’s curve of consumptivity displays a U-shape. Consumptive diseases, like tuberculosis, disproportionately affect the very young and the very old, while those in between experience comparatively lower rates.
  8. The Barthel Index is U-shaped; indexes like this are sometimes called bounded scores in the medical literature [5]. The Barthel Index is a widely used clinical tool in medicine that measures a patient’s functional independence and ability to perform activities of daily living (ADLs). It was first introduced by Mahoney and Barthel in 1965 [6]. The index assesses an individual’s level of independence in ten basic ADLs, which are essential for self-care and mobility.
  9. The Bathtub curve, which models failure rates, is U-shaped. The Bathtub curve is a U-shaped graph that represents the failure rates of a system or product over time. It is commonly used in the field of reliability engineering to illustrate the life cycle of a product, consisting of three distinct phases: infant mortality, normal life, and wear-out. A real-life example of the Bathtub curve can be found in the electronics industry, particularly with electronic devices like smartphones: Early failures may occur in some smartphones due to manufacturing defects, design issues, or other problems that become apparent shortly after the device is put into use. During the normal life phase, the failure rate is low and relatively constant. Smartphones are expected to function reliably throughout their normal life cycle, which may last for several years. Finally, as smartphones age and reach the end of their useful life, the failure rate starts to increase. Components may wear out, batteries may lose their ability to hold a charge, and performance may degrade due to accumulated wear and tear. At this stage, users are more likely to experience failures and may need to replace their devices.


[1] Iversen, G. & Norpoth, H. (1987). Analysis of Variance 1st Edition. SAGE.

[2] Pearson, K. (1895). Contributions to the mathematical theory of evolution. In Philosophical Transactions of the Royal Society of London. Vol. 180A. p. 174.

[3] Siegrist, K. 5.26: The U Power distribution. Retrieved December 31, 2021 from: https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_(Siegrist)/05%3A_Special_Distributions/5.26%3A_The_U-Power_Distribution

[4] Mangione, T. (1995). Mail Surveys: Improving the Quality. Book 40 in Applied Social Research Methods. SAGE Publications, Inc; 1 edition.

[5] Everitt, B. (2005). U-Shaped Distribution. Retrieved June 3, 2023 from: http://onlinelibrary.wiley.com/doi/10.1002/0470011815.b2a15171/abstract

[6] Mahoney et al. Functional evaluation: The Barthel Index. Md State Med J. 1965 Feb;14:61-5. PMID: 14258950.

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