Statistical Range: Definition, Calculation, and Applications

The statistical range is the difference between the largest and smallest values in a dataset, calculated as maximum minus minimum. It is the simplest measure of variability in descriptive statistics, providing a quick sense of how spread out observations are. For students, researchers, and life-science professionals, the range serves as a first-pass check on data dispersion, though its sensitivity to outliers and dependence on only two data points limit its usefulness for making robust statistical inferences. This article explains how to calculate the range, demonstrates its application with worked examples, and compares it with other variability measures so you can choose the right tool for your data.

What the Range Measures in Descriptive Statistics

The range captures the total spread of a dataset by identifying the interval between its extreme values. In any collection of measurements, whether they are crop yields from experimental plots, blood pressure readings from clinical subjects, or sonar echo amplitudes from fish aggregations, the range tells you the full span of observed values. This single number answers a basic question about your data: how much do the smallest and largest observations differ?

The range belongs to a family of statistics known as measures of dispersion or variability. While measures of central tendency such as the mean and median describe the typical value in a dataset, measures of dispersion describe how much individual values differ from one another. The range is the most straightforward of these measures because it requires no calculations beyond identifying the extremes.

In research contexts, the range often appears in summary tables alongside the mean and standard deviation. For example, a meta-analysis of time in therapeutic range for patients with continuous-flow left ventricular assist devices reported weighted mean estimates with confidence intervals, and the underlying studies each reported the spread of their data using various dispersion measures including ranges where appropriate. The range provides immediate context for whether a reported average represents tightly clustered data or widely scattered observations.

How to Calculate the Range

The calculation of the range follows a simple two-step process. First, identify the maximum value in the dataset. Second, identify the minimum value. The range equals the maximum minus the minimum.

The formula is expressed as:

Range = Maximum value - Minimum value

For a dataset with values ordered from smallest to largest, the range is simply the difference between the last and first values. This calculation works for any level of measurement that has meaningful numeric spacing, including interval and ratio data.

Worked Example with a Small Dataset

Consider a researcher measuring the body weight of six pigs in kilograms: 62, 58, 71, 65, 59, and 68.

The maximum value is 71 kilograms. The minimum value is 58 kilograms. The range is 71 minus 58, which equals 13 kilograms.

This tells the researcher that the heaviest pig weighed 13 kilograms more than the lightest pig in this group. The range does not describe the distribution of weights between these extremes, only the total span.

Worked Example with a Larger Dataset

Suppose a study records the daily milk production in liters for ten dairy cows: 22, 25, 24, 28, 21, 26, 23, 27, 29, and 24.

The maximum value is 29 liters. The minimum value is 21 liters. The range is 29 minus 21, which equals 8 liters.

The range of 8 liters indicates that daily production varied by no more than 8 liters across these ten cows. However, this measure does not reveal whether most cows produced near 24 or 25 liters or whether production was evenly spread across the entire 8-liter span.

At a Glance: Range Compared with Other Variability Measures

The choice among these measures depends on your research question and the nature of your data. The range offers speed and simplicity, but the standard deviation and interquartile range provide more information about the distribution of values between the extremes.

The Range in Statistical Reporting

Researchers encounter the range in several reporting contexts. Descriptive summaries of study samples frequently include the range alongside measures of central tendency. Meta-analyses sometimes report the range of effect sizes across included studies to show the breadth of findings. Clinical studies may report the range of biomarker values to indicate the spectrum of disease severity in the study population.

In meta-analytic research, the range serves a practical purpose when studies report alternative summary statistics. Clinical studies frequently report the median, quartiles, and range instead of the mean and standard deviation. Methods have been developed to estimate the sample mean and standard deviation from these reported summaries, including approaches based on Bayesian order statistics that use the joint likelihood of observed order statistics together with weakly informative priors. These estimators perform competitively with existing approaches for estimating the mean and achieve superior performance for estimating the standard deviation, particularly in small-sample settings. The range is one of the inputs that makes such estimation possible.

When reporting the range in your own work, state both the minimum and maximum values instead of only the difference. Reporting the range as 13 kilograms is less informative than reporting the range as 58 to 71 kilograms. The latter form allows readers to understand the actual values observed, beyond the span between them.

Applications of the Range in Research and Practice

The range finds application across many fields because of its simplicity and ease of communication. In quality control, the range of measurements from a production batch provides a quick check on process consistency. In environmental monitoring, the range of pollutant concentrations across sampling sites indicates the spatial variability of contamination. In clinical research, the range of physiological parameters helps define normal and abnormal values.

Life Sciences and Clinical Research

In clinical and epidemiological studies, the range of a biomarker or risk factor across a study population helps researchers understand the spectrum of exposure or disease severity. For example, a study developing a pediatric comorbidity index used health-care utilization data to predict the 1-year risk of hospitalization based on predefined conditions. The resulting index provided a summary measure of disease burden for risk adjustment in epidemiologic studies. The range of index scores across the pediatric population would indicate how much disease burden varied among patients.

The range also appears in the assessment of measurement reliability. Interrater reliability represents the extent to which data collectors assign the same score to the same variable. While percent agreement was traditionally used to measure interrater reliability, the kappa statistic was developed to account for the possibility that raters guess on at least some variables due to uncertainty. Like most correlation statistics, kappa can range from -1 to +1. The range of possible kappa values provides context for interpreting the strength of agreement between raters.

Acoustics and Environmental Monitoring

In environmental acoustics, the range of echo magnitudes from sonar systems provides information about the statistical properties of scattering targets. A study of fish aggregations measured with long-range, mid-frequency sonar found that the fish aggregations were statistically characterized by highly non-Rayleigh distributions of echo magnitudes. The range of echo magnitudes in such data helps researchers distinguish fish aggregations from other sources of scattering in the water column.

Agricultural and Animal Science

For farmers and animal scientists, the range of production metrics such as weight gain, milk yield, or egg production across a herd or flock reveals the consistency of animal performance. A narrow range suggests uniform management conditions and genetic potential, while a wide range may indicate variability in health status, nutrition, or environmental conditions that warrants investigation.

Limitations of the Range

Despite its utility, the range has important limitations that researchers must understand before using it as a primary measure of variability.

Sensitivity to Outliers

The range depends entirely on the two most extreme values in a dataset. A single outlier can dramatically inflate the range, giving a misleading impression of the variability of the majority of observations. For example, if a herd of cattle has daily weight gains clustered between 1.0 and 1.2 kilograms but one animal gains 2.5 kilograms due to measurement error or an unusual health event, the range becomes 1.5 kilograms even though most animals vary by only 0.2 kilograms.

This sensitivity makes the range unreliable for datasets that may contain errors or extreme values. Researchers should examine their data for outliers before relying on the range as a summary measure.

Dependence on Sample Size

The range tends to increase with sample size because larger samples are more likely to include extreme values. A small sample of five observations will typically have a smaller range than a sample of fifty observations drawn from the same population, even though the underlying variability is identical. This property makes the range difficult to compare across studies with different sample sizes.

Ignoring Intermediate Values

The range uses only two data points from the entire dataset. All information about the distribution of values between the minimum and maximum is discarded. Two datasets can have identical ranges but very different distributions. One dataset might have values evenly spread across the range, while another might have most values clustered near the center with only a few extreme observations. The range cannot distinguish between these scenarios.

Lack of Mathematical Properties

The range does not have the useful mathematical properties of the variance and standard deviation. It cannot be easily combined across subgroups, and it does not lend itself to the calculation of confidence intervals or hypothesis tests. For inferential statistics, measures based on the sum of squared deviations are generally required.

When to Use the Range Versus Other Measures

The choice between the range and other variability measures depends on your specific goals and the characteristics of your data.

Use the Range When

The range is appropriate when you need a quick, easily communicated measure of spread. It works well for small datasets where all values can be inspected directly. It is also useful for quality control checks where you want to verify that all observations fall within acceptable bounds. In exploratory data analysis, the range provides an immediate sense of the data span before more sophisticated analyses are conducted.

Use the Interquartile Range When

The interquartile range is preferable when your data are skewed or contain outliers. Because it focuses on the middle 50 percent of observations, it resists the influence of extreme values. The interquartile range is often reported alongside the median for skewed distributions.

Use the Standard Deviation When

The standard deviation is the preferred measure when you plan to conduct inferential statistics or when you need a measure of variability that can be used in further calculations. The standard deviation is mathematically related to the variance and appears in formulas for confidence intervals, hypothesis tests, and effect sizes. It is also more stable across samples than the range.

Use the Variance When

The variance is useful in theoretical work and in analyses that require the additive properties of squared deviations. Analysis of variance and related techniques use the variance as their foundation. However, the variance is expressed in squared units, which makes it less interpretable than the standard deviation for descriptive purposes.

Practical Steps for Using the Range in Data Analysis

When incorporating the range into your data analysis workflow, follow these steps to ensure that you use it appropriately and interpret it correctly.

Step 1: Examine Your Data Before Calculating the Range

Before calculating the range, inspect your dataset for errors, missing values, and outliers. Plot the data if possible to visualize the distribution. The range is only meaningful if the minimum and maximum values are accurate measurements instead of data entry errors.

Step 2: Calculate the Range Alongside Other Measures

Do not rely on the range alone to describe variability. Calculate the standard deviation or interquartile range as well, and report all measures together. This provides a more complete picture of the data distribution.

Step 3: Report the Minimum and Maximum Values

When you report the range, include the actual minimum and maximum values instead of only the difference. This allows readers to understand the context of the spread and to assess whether the extremes are plausible.

Step 4: Consider the Sample Size

Remember that the range is sensitive to sample size. When comparing ranges across groups with different sample sizes, interpret differences cautiously. A larger range in a larger sample may reflect sample size instead of true differences in variability.

Step 5: Check for Outliers

If the range seems unexpectedly large, investigate the extreme values. Determine whether they represent genuine observations or errors. If outliers are present, consider reporting the interquartile range instead of or alongside the range.

Records and Measurements: Documenting the Range

In research and practice, documenting the range requires attention to the same standards that apply to other statistical measures. Record the following information when you use the range in your work:

The dataset from which the range was calculated, including the sample size and the source of the data. The minimum and maximum values, with their units of measurement. The date and conditions of data collection. Any data cleaning steps that affected the extreme values. The software or method used to calculate the range.

For example, a farm manager tracking pig weights should record also the range of weights for each batch but also the number of pigs weighed, the weighing equipment used, and any animals excluded from the analysis. This documentation allows others to assess the reliability of the reported range.

Common Failure Patterns in Using the Range

Several common errors occur when researchers and practitioners use the range in their work. Recognizing these patterns can help you avoid them.

Relying on the Range Alone

The most common failure is using the range as the sole measure of variability. This practice ignores the distribution of values between the extremes and can lead to incorrect conclusions about data consistency. Always supplement the range with other measures such as the standard deviation or interquartile range.

Failing to Check for Outliers

Another common failure is calculating the range without examining the data for outliers. A single erroneous value can distort the range and mislead interpretation. Always plot or inspect your data before relying on the range.

Comparing Ranges Across Different Sample Sizes

Comparing the ranges of datasets with different sample sizes can produce misleading conclusions. Because the range tends to increase with sample size, differences in range may reflect sample size instead of true differences in variability.

Misinterpreting the Range as a Measure of Typical Variability

The range describes the total span of the data, not the typical variability of individual observations. A dataset with a wide range may still have most observations tightly clustered. Do not use the range to describe how much individual values typically differ from the mean.

Using the Range for Inferential Statistics

The range does not support inferential statistics such as confidence intervals or hypothesis tests. Attempting to use the range for these purposes will produce invalid results. Use the standard deviation or variance for inferential analyses.

The Range in Meta-Analysis and Research Synthesis

In meta-analysis, the range plays a specific role in the estimation of study-level statistics. Clinical studies frequently report alternative summary statistics such as the median, quartiles, and range instead of the mean and standard deviation. To include such studies in meta-analyses, researchers need methods to estimate the sample mean and standard deviation from these reported summaries.

The Bayesian Order Statistics-based Estimator represents one approach to this problem. This method leverages the joint likelihood of observed order statistics together with weakly informative priors to obtain the full posterior distribution for the mean and standard deviation. It performs competitively with existing approaches for estimating the mean and achieves superior performance for estimating the standard deviation across evaluated scenarios, particularly in small-sample settings. The method remains robust and stable under non-normal distributions including skewed, heavy-tailed, and bimodal settings.

The range is one of the inputs to such estimation methods. When a study reports only the minimum, maximum, and sample size, the range provides information about the spread of the data that can be combined with distributional assumptions to estimate the standard deviation.

Statistical Heterogeneity and the Range

In meta-analysis, statistical heterogeneity refers to the variability in effect sizes across studies. Popular measures of statistical heterogeneity include the tau-squared and I-squared statistics. However, researchers often rely unduly on the I-squared statistic, using naive categorizations to gauge the extent of heterogeneity, leading to misuses of meta-analysis models and misleading conclusions.

A study of statistical heterogeneity in oral health meta-analyses found that almost all meta-analyses reported the I-squared value, while only about half reported the tau-squared value. Most meta-analyses based their heterogeneity interpretation on I-squared, and many chose their meta-analysis model based on the I-squared value. The median I-squared was 76 percent with a range of 0 to 100 percent, and the median tau-squared was 0.29 with a range of 0 to 2,632.

The range of I-squared values across meta-analyses illustrates the wide variability in heterogeneity reporting practices. This example shows how the range can be used to describe the spread of statistics across a collection of studies, even when the range itself is not the primary measure of interest.

The Range in Survival Analysis and Clinical Trials

In clinical trials and survival analysis, the range of event times provides context for understanding the duration of follow-up and the timing of outcomes. The log-rank test is commonly used to compare survival curves between treatment groups, but it may lose power in scenarios where survival curves remain similar for extended periods before diverging. Weighted versions of the log-rank test have been proposed to address this limitation, including the MaxCombo test and the modestly weighted log-rank test.

The range of event times in a clinical trial influences the choice of statistical methods. When survival curves exhibit non-proportional hazards, the standard log-rank test may not be the most appropriate primary analysis method. The range of follow-up times and the distribution of events across the follow-up period are important considerations in selecting the analysis approach.

The Range in Cost-Effectiveness Analysis

In health economics, the range of incremental cost-effectiveness ratios across different policies provides insight into the variability of value for money. A review of cancer prevention policies found that incremental cost-effectiveness ratios for clinical prevention policies ranged from under 2,000 to over 6,000,000 US dollars per life-year saved. The range of these ratios was enormous for environmental policies, with ratios ranging from 61,004 to over 24 billion US dollars per life-year saved.

The range of cost-effectiveness ratios across policies helps decision-makers understand the uncertainty in value for money and the potential for some policies to be far more cost-effective than others. However, the range alone does not indicate which policies are most likely to be cost-effective, and the wide ranges observed in this review highlight the importance of considering uncertainty in cost-effectiveness estimates.

The Range in Natural and Physical Sciences

The range appears throughout the natural and physical sciences as a descriptive statistic for observed phenomena. In solar physics, the variance of solar soft X-ray fluxes follows Taylor's law, which relates the variance to the mean flux through a power law relationship. The range of fluxes observed over an eight-year period provides context for understanding the variability of solar activity.

In neuroscience, the dynamic range of contrast encoding in the primary visual cortex has been investigated at both single-neuron and population levels. A study measuring neural responses to Gaussian stimuli over a wide range of Weber contrasts found that both average spiking responses and average population responses continued to increase robustly above 100 percent Weber contrast. The range of contrasts tested in this study was broader than in previous work, revealing that the dynamic range of contrast encoding in the primary visual cortex is broader than previously assumed.

In materials science, ion range statistics describe the distribution of penetration depths for ions implanted into materials. The range of ion penetration depths affects the properties of implanted materials and is an important consideration in semiconductor manufacturing.

The Range in Image and Signal Processing

In image processing, the range of pixel values in an image determines the contrast and dynamic range of the image. Statistics of range images have been studied in computer vision research to understand the properties of depth images and their use in object recognition and scene understanding.

In signal processing, the range of signal amplitudes affects the design of detection and classification algorithms. For example, the range of echo magnitudes from sonar systems influences the statistical models used to classify echoes as originating from fish aggregations or from other sources of scattering.

The Range in Polarization and Optical Studies

In optics, the degree of polarization is usually obtained from the coherency matrix or from the mean Stokes vector of a partially polarized optical field. A complementary geometric viewpoint represents normalized local Stokes vectors as random directions on the Poincare sphere. When these directions are described by an effective unimodal von Mises-Fisher distribution, the concentration parameter gives a direct one-to-one description of the degree of polarization through the mean resultant length.

The range of validity and limitations of the single von Mises-Fisher model are important considerations in this framework, particularly in relation to Gaussian-field statistics and to more general directional models needed for anisotropic or multimodal polarization fluctuations. The range of polarization states that can be described by a single von Mises-Fisher distribution limits the applicability of this model.

The Range in Temperature and Ecological Studies

In ecology, the range of temperatures that organisms can tolerate determines their geographic distribution and their response to climate change. A study of the maximum growth temperature for eukaryotes found that this temperature is thermodynamically driven but ecologically contingent. The range of maximum growth temperatures across eukaryotic species reflects both thermodynamic constraints and ecological adaptations.

For farmers and animal scientists, the range of temperatures that livestock can tolerate without stress is an important management consideration. Housing and ventilation systems are designed to maintain temperatures within an acceptable range for the species and production stage.

The Range in Genetic and Epidemiological Studies

In genetic epidemiology, the range of genetic variant effects across the genome provides context for understanding the genetic architecture of complex traits. Mendelian randomization studies use genetic variants as instrumental variables to assess causal relationships between exposures and outcomes.

A Mendelian randomization study of circulating vitamin D concentrations and risk of atrial fibrillation used summary statistics obtained for single-nucleotide polymorphisms identified from large genome-wide association meta-analyses. The study found no evidence to support a causal association between circulating vitamin D levels and risk of atrial fibrillation. The range of vitamin D concentrations in the non-deficient range was the focus of this analysis.

The Range in Educational and Behavioral Research

In educational research, the range of student performance scores provides context for understanding the variability of learning outcomes. A study of factors associated with students perceived competence using statistical software examined variables from expectancy-value theory and the technology acceptance model. The range of perceived competence scores across students reflected differences in attitudes, experience, and course-specific resources.

For educators, the range of student performance in a class can indicate whether instruction is effectively meeting the needs of all students or whether some students are struggling while others excel. A wide range may suggest the need for differentiated instruction or additional support for students at the lower end of the performance distribution.

The Range in Time Series and Memory Studies

In time series analysis, the range of observations over time provides a measure of the total variability of the series. The rescaled range statistic, known as the R/S statistic, is used to detect long-range memory in time series data. Strong laws of R/S statistics with long-range memory samples have been studied in the statistical literature.

For researchers analyzing time series data, the range of observations provides a quick check on the total variability of the series, while the R/S statistic provides information about the persistence or anti-persistence of the series.

Professional Escalation Criteria for Range-Related Issues

When working with the range in research or practice, certain situations warrant consultation with a statistician or other qualified professional. Seek professional assistance in the following circumstances:

When the range is unexpectedly large or small relative to your expectations, and you cannot identify the cause. When outliers are present and you are uncertain whether to exclude them or use alternative measures of variability. When you need to compare variability across groups with different sample sizes or different units of measurement. When you plan to use the range or related statistics in inferential analyses such as meta-analysis. When you are uncertain about the appropriate measure of variability for your data and research question.

A statistician can help you choose the appropriate measure of variability, assess the impact of outliers, and interpret your results correctly.

Safety and Welfare Context for the Range

In animal agriculture and life-science research, the range of physiological and production measurements has welfare implications. A wide range in body weights within a group of animals may indicate that some animals are not receiving adequate nutrition or that health problems are affecting a subset of the group. A wide range in temperature measurements in a livestock facility may indicate inadequate ventilation or heating system performance.

Monitoring the range of key welfare indicators can help identify problems early. For example, tracking the range of body condition scores across a herd can reveal whether all animals are maintaining adequate condition or whether some animals are losing condition. Tracking the range of water intake can reveal whether all animals have access to adequate water.

When the range of a welfare indicator exceeds acceptable bounds, investigate the cause and take corrective action. Consult with a veterinarian or animal welfare specialist if you are uncertain about the appropriate response.

The Range in Research Data Management

The National Institute of Standards and Technology supports the Research Data Framework, which provides guidance for managing research data throughout its lifecycle. Proper data management includes documenting the statistical measures used to summarize data, including the range. This documentation ensures that others can understand and reproduce your analyses.

The EQUATOR Network provides reporting guidelines for health research, including guidelines for reporting statistical methods and results. Following these guidelines ensures that your use of the range and other statistical measures is transparent and reproducible.

The NC3Rs Experimental Design Assistant provides guidance for designing experiments that use animals, including guidance on sample size calculation and statistical analysis. Proper experimental design ensures that the range and other variability measures are meaningful and interpretable.

Frequently Asked Questions

What is the formula for calculating the range in statistics?

The range is calculated by subtracting the minimum value from the maximum value in a dataset. The formula is Range equals Maximum minus Minimum. For example, if a dataset contains values from 10 to 45, the range is 35.

How is the range different from the standard deviation?

The range uses only the two extreme values in a dataset, while the standard deviation uses all values. The standard deviation measures the average distance of observations from the mean, while the range measures the total span between the smallest and largest values. The standard deviation is more stable and more useful for inferential statistics.

Why is the range sensitive to outliers?

The range depends entirely on the maximum and minimum values in a dataset. A single extreme value, whether genuine or erroneous, can dramatically change the range. For example, adding one very large value to a dataset can inflate the range even if all other values remain unchanged.

Can the range be used for inferential statistics?

The range is primarily a descriptive statistic. It does not have the mathematical properties needed for confidence intervals, hypothesis tests, or other inferential procedures. Use the standard deviation or variance for inferential statistics.

When should I use the interquartile range instead of the range?

Use the interquartile range when your data are skewed or contain outliers. The interquartile range focuses on the middle 50 percent of observations and resists the influence of extreme values. It is often reported alongside the median for skewed distributions.

How does sample size affect the range?

The range tends to increase with sample size because larger samples are more likely to include extreme values. This makes the range difficult to compare across datasets with different sample sizes. The standard deviation is more stable across samples.

What information should I report when using the range?

Report the minimum and maximum values along with the range itself. Include the sample size, the units of measurement, and any data cleaning steps that affected the extreme values. This documentation allows others to assess the reliability of the reported range.

How is the range used in meta-analysis?

In meta-analysis, the range is used to estimate the sample mean and standard deviation when studies report only the median, quartiles, and range. Methods such as the Bayesian Order Statistics-based Estimator use the range along with other summary statistics to recover the mean and standard deviation for inclusion in meta-analytic syntheses.

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References and Further Reading

• Research Data Framework. National Institute of Standards and Technology.
• EQUATOR Network. EQUATOR Network.
• Experimental Design Assistant. NC3Rs.
• NCBI Literature Resources. National Center for Biotechnology Information.
• PubMed. National Library of Medicine.
• Interrater reliability: the kappa statistic.. Biochemia medica, 2012.
• Development and Validation of a Pediatric Comorbidity Index.. American journal of epidemiology, 2021.
• Broadband classification and statistics of echoes from aggregations of fish measured by long-range, mid-frequency sonar.. The Journal of the Acoustical Society of America, 2017.
• Statistical Heterogeneity in Oral Health Meta-Analyses.. Journal of dental research, 2025.
• Robust Modestly Weighted Log-Rank Tests.. Pharmaceutical statistics, 2026.
• Meta-Analysis of Time in Therapeutic Range in Continuous-Flow Left Ventricular Assist Device Patients Receiving Warfarin.. Artificial organs, 2018.
• Willingness to pay for cancer prevention.. PharmacoEconomics, 2009.
• Self-designing clinical trials.. Statistics in medicine, 1998.
• Circulating Vitamin D Concentrations and Risk of Atrial Fibrillation: A Mendelian Randomization Study Using Non-deficient Range Summary Statistics.. 2022.
• Wide dynamic range of contrast-encoding in neural responses in macaque V1. 2026.
• The Maximum Growth Temperature for Eukaryotes Is Thermodynamically Driven but Ecologically Contingent. 2026.
• A Directional-Entropy Framework for Polarization Disorder: Information-Theoretic Insights from von Mises-Fisher Statistics. 2026.
• BOSE: A Bayesian Order Statistics-Based Estimator for Recovering the Sample Mean and Standard Deviation. 2026.
• Factors Associated with Students’ Perceived Competence Using Statistical Software. 2026.
• The variance of solar soft X-ray fluxes. 2026.
• Strong laws of R/S statistics with a long-range memory sample. Statistica Sinica, 2005.
• Statistics of range images. Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2000.
• Ion range statistics by a fourier series. Microelectronics Journal, 1983.

This article is educational and does not replace institutional policy, professional advice, or applicable safety and regulatory requirements.