Lane [1] presents the Rice University rule as a straightforward and practical method that appears to perform well across various sample sizes without assuming any specific distribution. Additionally, the author asserts that it enables the effective visualization of distribution shapes [2] . In practice, the rule is employed in applied research [3] . However, the rationale behind the rule is not provided, and it has received limited attention in the literature. One of the few studies investigating this rule is conducted by Sahann, Müller, and Schmidt [4] . In their research, they explore the correlation between the number of intervals required for a user to recognize a distribution and the number of intervals generated by four empirical rules. They conclude that the Scott, Rice University, and Freedman-Diaconis rules tend to overestimate the number of intervals. The Sturges rule emerges as the most suitable among the four, and they ultimately recommend using a fixed number of 20 class intervals. Their study incorporates four different distributions: uniform, normal, bimodal, and gamma, and four sample sizes: 100, 1000, 10,000, and 1,000,000. The user sample comprises 100 computer science students. Consequently, in this validation study, Rice’s rule does not stand out as a preferable choice.

The present study aims to provide a rationale for the Rice University rule based on the Scott and Freedman-Diaconis rules. Additionally, it seeks to assess the accuracy of the empirical histogram regarding six rules for determining the number of bins, including five classical rules: Pearson’s square root [5] , Sturges [6] , Doane [7] , Scott [8] , and Freedman-Diaconis [9] , along with the Rice University rule [1] . The study does not initially emphasize one rule over another. The assessment includes two additional factors: sample size and distribution type, considering both their principal effects and the interactions of second and third order among the three factors.

It begins with the presentation of a historical note on the histogram and six empirical rules to determine the number of class intervals. When the Rice University rule [1] is presented, its rationale is developed from the rules of Scott [8] and Freedman-Diaconis [9] .

The second objective of the study is pursued by randomly drawing five distinct samples from a standard continuous uniform distribution across seven different sizes: 20, 35, 50, 100, 200, 500, and 1000 data points. Employing the inverse transform sampling method, a total of 350 samples are generated, incorporating 10 distribution types. These distribution types encompass two mesokurtic symmetric distributions: normal N (μ = 5, σ² = 6.25) and beta (α = 30, β = 30); two leptokurtic symmetric distributions: Laplace (μ = 5, β = 2.5) and logistic (μ = 5, s = 2.5); two platykurtic symmetric distributions: arcsine (a = 1, b = 1) and semicircular (R = 2); two distributions exhibiting skewness and platykurtosis: triangular (a = b = 0, c = 1) and PERT (a = 1, b = 4, c = 5); and two distributions featuring skewness and leptokurtosis: exponential (λ = 1/2) and lognormal (μ = 0, σ² = 0.25). Subsequently, the six rules are applied to these 350 samples, resulting in the generation of 2100 empirical histograms (5 samples of 7 sizes of 10 distributions for 6 rules).

The probability of k class intervals in each of the 2100 empirical histograms was calculated using the cumulative distribution function of the corresponding distribution. This process generated the expected relative frequencies or theoretical probabilities. The distance between the relative frequencies and the expected probabilities of the class intervals was then measured using the average Euclidean distance, referred to as the average discrepancy.

On the other hand, a sample of 1000 uniformly distributed data with perfect symmetry was generated in the interval [0.001, 0.999], equispaced at a distance of 0.000998. Using the corresponding quantile function, 10 distributions were generated. A probability density function plot for each of the 10 distributions, bounded to the range of the random sample, served as a theoretical model to visually assess whether the empirical histogram accurately reproduces the distribution curve. The visual evaluation was conducted by an expert judge using a scale of four ordered categories, referred to as recognition level.

The means of the average discrepancy and the recognition level are compared, taking into account three factors: rule, distribution, and sample size, through analysis of variance with aligned rank transformation. Additional details are provided in the Method section. Following the explanation of the study’s methodology, the Result section is presented, and conclusions are subsequently drawn.

In 1833, the histogram was introduced by the French lawyer and statistician André Michel Guerry [10] as an approximation of a discrete empirical distribution to a continuous distribution function in the study of crimes and suicides in France. The English nurse Florence Nightingale [11] utilized histograms to compare the mortality of soldiers and civilians in her work on sanitation in the British army during the Crimean War against Russia, published in 1859. However, these authors simply named their graphic representations and did not use the neologism “histogram”, composed of the Greek words “histos” (ιστοσ) (which can be translated into English as “mast”) and “gramma” (γραµµα) (which can be translated as “graph” or “drawing”). Thus, this term, in its etymological sense, refers to a graph of masts or vertical bars [12] .

The statistical term “histogram” was coined by Karl Pearson (1857-1936) and first used in his lecture on maps and cartograms in 1891, during his tenure as a professor of geometry at Gresham College. In this lecture, Pearson explained that histograms could be used to represent historical information about reigns, sovereigns, or prime ministers of different periods [13] . However, it wasn’t until 1895 that the term “histogram” appeared in a written publication by Pearson when he presented his system of continuous distributions. In contrast to Guerry, Pearson discretizes a continuum of values into k class intervals to create the attached rectangles seen in the histogram, representing the areas under the continuous distribution [14] .

It is noteworthy that the histogram serves as a highly valuable graphical tool for comprehending the shape of a distribution and found extensive use in Karl Pearson’s development of his system of frequency curves [14] . The histogram allows for the graphical representation of the sampling distribution of a continuous random variable. This involves grouping the n sample values within a continuous range into k class intervals. Each interval is represented in the histogram by a rectangle, with its area corresponding to its relative frequency.

On the horizontal or abscissa axis, the k intervals are arranged and can be labeled with their two limits in brackets or parentheses [LI, LS) or with a singular value (the class mark). This axis can take values from 0 to +∞, from −∞ to +∞, or be bounded [−a, a], depending on the domain of the variable. Consequently, this horizontal axis may correspond to the upper two quadrants of a Cartesian coordinate axis when the bar chart uses only the first quadrant (of positive values). Contiguous intervals are placed next to each other, reflecting the continuous nature of the variable.

On the vertical or ordinate axis, the heights of the intervals are positioned. The height is determined by dividing the relative frequency by the width of the interval: h_i = f_i/a_i. These representations constitute the density histogram. While resembling a bar chart, the bars are connected, the X-axis values have a mathematical meaning, and their areas correspond to the relative frequency of the interval.

The most common practice is for the intervals to have the same width, a determination guided by empirical rules such as the square root, Sturges [6] , Doane [7] , Scott [8] , Freedman and Diaconis [9] , and the Rice University rule [1] . Another option is to maintain homogeneity in density or frequency within each interval while allowing variable width. This approach is recommended when utilizing Pearson’s chi-square test to assess the goodness-of-fit between the empirical distribution and a theoretical distribution [15] . Additionally, various algorithms exist for minimizing integrated mean square error [16] , hazard function [17] , or entropy [18] .

There exists a variant of the histogram that sees more general usage when dealing with datasets featuring numerous values that have been grouped into class intervals. This variant is known as a frequency histogram, wherein the intervals of values share the same width and are positioned adjacent to each other. For each interval, a rectangle is drawn to a height corresponding to its absolute or simple relative frequency. In this case, the area under the curve is not equal to the frequency of the interval, as in the density histogram, but is a value proportional to it. Consequently, it resembles a bar chart where the bars are connected to each other. This approach ensures the preservation of the data’s nature, avoiding the imposition of continuity on a discrete quantitative variable or the forced attribution of a quantitative and continuous character to an ordinal variable. The study uses this variant of the histogram for data analysis in order to facilitate its extrapolation to studies in the field of Psychology and related sciences.

When dealing with a continuous quantitative variable and aiming to construct a frequency table, it is necessary to establish class intervals for conducting the frequency count. Once the table is defined, the data can be visually represented through a histogram. Class intervals are contiguous and non-overlapping ranges of values for the variable that adhere to two key principles: ordering and completeness. The ordering principle ensures the intervals are continuous and consecutive, while the completeness principle ensures that all sample data falls within a single interval, maintaining exclusivity.

During the creation of the table, the challenge arises in determining the number of class intervals (k) and deciding whether the intervals will have the same width (w) or not (w_i; i = 1, 2, …, k). Here, the width of the class interval is defined as the difference between its upper limit (UL_i) and lower limit (LL_i): w_i = UL_i − LL_i.

It is recommended that the width of the intervals be constant (w), except for the two extreme intervals when long tails are generated by outliers. In this case, the width of these two extreme intervals can be much larger to encompass these low-frequency data points that are far away from the others.

There are several automatically applied or programmable rules for defining the number of class intervals, which can be classified into three groups [19] [20] . One group of rules starts by defining the constant amplitude, represented by a positive real number (w), and subsequently determines the number of intervals (k), which is a natural number. Within this group are rules designed to minimize an argument, typically either the integrated mean square error or the loss function [21] . Another set of rules begins by specifying the number of intervals (k), and from there determines the constant amplitude (w). In both groups of rules, the frequency or amount of data per interval (n_i) is variable. A third set of rules starts by setting the number of class intervals (k), and then establishes the homogeneous density or constant amount of data per interval (n), whereby the amplitude remains variable (w_i). This last strategy is employed to optimize the power of the goodness-of-fit chi-square test [15] , and is not developed in this paper.

On the one hand, 35 samples were randomly drawn from a standard continuous uniform distribution with seven different sizes (20, 35, 50, 50, 100, 100, 200, 500, and 1000), five samples for each size. The extraction was performed using the Excel random number generator.

u = { u i } i = 1 n ⊂ U [ 0 , 1 ]

Using the inverse transform sampling method, a total of 350 samples were generated from the initial set of 35 uniformly distributed samples. Seventy of these samples followed symmetric mesokurtic distributions. Specifically, 35 samples were drawn from a normal distribution with a location parameter μ = 5 and a squared scale σ² = 6.25 (Equation (15)), and the other 35 samples were generated from a beta distribution with shape parameters α = 30 and β = 30 (Equation (16)).

Φ − 1 ( x i ) = 5 + 2.5 Φ − 1 ( u i ) ;   X ~ N ( μ = 5 , σ 2 = 6.25 ) (15)

Q X ( u i ) = I u i − 1 ( α = 30 , β = 30 ) ;   X ~ Beta ( α = 30 , β = 30 ) (16)

Out of the total of 350 samples, 70 followed leptokurtic symmetric distributions, Specifically, 35 were drawn from the Laplace distribution with a location parameter μ = 5 and a scale parameter β = 2.5 (Equation (17)), and the remaining 35 were obtained from the logistic distribution with a location parameter x₀ = 5 and a scale parameter γ = 2.5 (Equation (18)).

Q X ( u i ) = 5 − 2.5 × signo ( u i − 0.5 ) × ln ( 1 − 2 | u i − 0.5 | ) ; X ~ Laplace ( μ = 5 , β = 2.5 ) (17)

Q X ( u i ) = 5 + 2.5 × ln ( u i 1 − u i ) ;   X ~ Logistica ( μ = 5 , s = 2.5 ) (18)

Out of the total of 350 samples, 70 followed symmetric platykurtic distributions. Specifically, 35 were drawn from the arcsine distribution with threshold parameters 0 and 1 (Equation (19)), and the remaining 35 from the semicircular distribution with a radius of 2 (Equation (20)). The standard arcsine distribution is equivalent to a beta distribution of shape parameters: α = 0.5 and β = 0.5, whose quantile function corresponds to an inverse regularized incomplete beta function: I u i − 1 (α = 0.5, β = 0.5), as seen in Equation (20) The semicircular distribution with parameter R (radius) corresponds to a beta distribution of shape parameters: α = 1.5 and β = 1.5, once transformed: X = 2 × R × (B − 1), where B ~ Beta (α = 0.5, β = 0.5) and X ~ Arcsine (R = 2), as shown in Equation (21).

Q X ( u i ) = I u i − 1 ( α = 0.5 , β = 0.5 ) ; X ~ Arcoseno ( a = 0 , b = 1 ) ≡ Beta ( α = 0.5 , β = 0.5 ) (19)

Q X ( u i ) = 4 I u i − 1 ( α = 1.5 , β = 1.5 ) − 2 ;   X ~ Semicircular ( R = 2 ) (20)

Out of the total of 350 samples, 70 followed distributions with skewness and platykurtosis. Specifically, 35 were drawn from the triangular distribution with parameters a = b = 0 and c = 1 (Equation (21)), and the remaining 35 from the PERT distribution with parameters a = 1, b = 4 and c = 5, where a is the minimum, b is the peak or mode and c is the maximum (Equation (22)).

X = 1 − 1 − u i ;   X ~ Triangular ( a = 0 , b = 0 , c = 1 ) (21)

α = 1 + 4 b − a c − a = 1 + 4 4 − 1 5 − 1 = 4 β = 1 + 4 c − b c − a = 1 + 4 5 − 4 5 − 1 = 2 Q X ( u i ) = 1 + 4 × I u i − 1 ( α = 4 , β = 2 ) ;   X ~ PERT ( a = 1 , b = 4 , c = 5 ) (22)

Finally, out of the total 350 samples, 70 followed distributions with skewness and leptokurtosis. Specifically, 35 were drawn from an exponential distribution with a rate parameter λ = 1/2 (Equation (23)), and the other 35 are obtained from a lognormal distribution with a location parameter μ_ln(X) = 0 and a scale parameter σ_ln(X) = 0.25 (Equation (24)).

Q X ( u i ) = − 2 ln ( 1 − u i ) ;   X ~ exp ( λ = 0.5 ) (23)

Q X ( u i ) = e 2 4 erf − 1 ( 2 u i − 1 ) = e 0.25 Φ − 1 ( u i ) ;   X ~ log N ( μ ln ( X ) = 0 , σ ln ( X ) = 0.25 ) (24)

The six empirical rules for determining the number (k) and width (w) of bins were applied to the 350 random samples, resulting in 2100 empirical histograms. The probability of the k class intervals (I) of each empirical histogram was calculated using the cumulative distribution function of the corresponding distribution to generate the theoretical histogram. The difference between the empirical and theoretical histograms was measured using the average Euclidean distance between the relative frequency f_n(I_i) and the expected probability p_X(I_i) for each bin ( i = 1 , 2 , ⋯ , k ). This difference is referred to as the Average Discrepancy (AD) and is shown in Equation (25).

AD = ∑ i = 1 k [ f n ( I i ) − p X ( I i ) ] 2 k p X ( I i ) = ∫ L L U L f X ( x ) d x ;   I i = ( L L i , U L i ] ;   i = 1 , 2 , ⋯ , k (25)

On the other hand, a sample of 1000 uniformly distributed data with perfect symmetry was generated in the interval (0, 1), starting at 0.001 and ending at 0.999, with a data spacing of 0.000998. Ten theoretical samples were created using the corresponding density functions to represent their density curves. Each density curve, confined within the sample range, was used to visually assess (by a single judge) whether the empirical histogram reproduces the curve corresponding to the distribution, using four ordered categories: 1 = not at all, 2 = a little, 3 = quite a lot, and 4 = completely, referred to as the Recognition Level (RL).

An index, termed the Accuracy Index (AI), was established using the Average Discrepancy (AD) and Recognition Level (RL). Initially, the difference between empirical and theoretical histograms (AD) is transformed into scores ranging from 1 to 4, where 4 signifies maximum accuracy and 1 denotes minimum inaccuracy among the 2100 samples (5 samples per triple condition: 10 distribution types × 7 sample sizes × 6 rules), as shown in Equation (26).

D T = 1 + 3 ( 1 − AD − min ( AD ) max ( AD ) − min ( AD ) ) = 1 + 3 ( 1 − AD − min ( { a d i } I = 1 2100 ) max ( { d i } I = 1 2100 ) − min ( { d i } I = 1 2100 ) ) = 1 + 3 ( 1 − AD − 0.00251545 0.21781641 − 0.00251545 ) (26)

Subsequently, AD and RL are summed, resulting in a random variable (D_T) with a potential range of 2 to 8 (Equation (27)). This combined variable is then transformed into the Accuracy Index (AI) with values ranging from 0 to 100, where 0 signifies total inaccuracy and 100 indicates total accuracy (Equation (28)).

S = D T + RL (27)

AI = 100 × S − min ( S ) max ( S ) − min ( S ) = 100 × S − 2 8 − 2 = 100 × D T + G R − 2 6 (28)

The normality of the distributions for AD, RL, and AI was assessed using Shapiro-Francia W’ test [36] and K² test [37] . Means for each of the three variables were compared across three factors: rule (six levels), sample size (seven levels), and distribution (ten levels). This was done through a three-factor aligned rank transformation analysis of variance [38] . Pairwise comparisons were conducted using Fisher’s least significant difference test [39] with Holm-Bonferroni correction applied to control the family rate error [40] . The effect size was estimated using the partial eta coefficient: η² = F × df₁/(F × fd₁ + df₂). Interpretation of η² values was as follows: less than 0.02 indicates a very small effect size, between 0.02 and 0.129 is considered small, between 0.13 and 0.259 is medium, and greater than or equal to 0.26 is considered large [41] [42] . Data analyses were performed using EXCEL 2023, IBM SPSS Statistics 29, and R 4.3. The significance level was set at 0.05.

The AD statistic is strongly influenced by sample size, exhibiting a linear relationship with a very high strength of association. To a much lesser extent, it is affected by distribution type and the rule used to determine the number and amplitude of class intervals, with these two factors having small effect sizes. The arcsine distribution, characterized as a bimodal platykurtic distribution, generates the maximum discrepancy, while the exponential distribution achieves the minimum discrepancy. Platykurtic distributions exhibit more discrepancy than leptokurtic distributions. The square root, Rice, and Doane rules, with means equivalent to each other, have significantly less discrepancy than the Scott, Friedman-Diaconis, and Sturges rules. Consequently, Rice’s rule, which can be based on the Scott and Friedman-Diaconis rules, generates less discrepancy between the empirical and theoretical histograms than these two rules. The interaction between rule and distribution type, as well as the triple interaction, are not significant. The second-order interactions of sample size with the rule and the distribution type are significant and small in size. The increase in sample size favors platykurtic distributions more, as well as the Scott and Friedman rules, that is, the conditions that generate the most discrepancy.

The Recognition Level is influenced by the sample size, with a large effect size. The relationship between sample size and Recognition Level is linear, demonstrating a strong association, although significantly lower than the Average Discrepancy. The type of distribution has a significant and large effect, with the least recognition given to the platykurtic semicircular distribution and the highest recognition to the leptokurtic exponential distribution. Rule type also has a significant, but small, effect size. Scott and Sturges rules have the highest recognition level, while the square root and Friedman-Diaconis rules have the least. Intermediate recognition levels are observed with Doane and Rice rules.

The increase in sample size primarily favors the recognition of the semicircular distribution. With small samples, the exponential and triangular distributions are the best recognized. The increase in size mainly favors the Doane and Rice rules and harms the Scott, Friedman-Diaconis, and square root rules. In the interaction between rule and distribution, it should be noted that the arcsine distribution, being a bimodal distribution, is poorly recognized by the Scott and Friedman rules, whereas the recognition of the triangular distribution is favored by these two rules. Recognition of Rice’s rule is independent of the distribution.

Consistent with the results for Average Discrepancy and Recognition Level, the sample size has a significant and large effect on Accuracy Index. The relationship between sample size and the Accuracy Index is a direct linear one with a very large strength of association. The distribution type also has a significant and large effect, with the lowest accuracy occurring with the symmetrical and platykurtic semicircular distribution, and the highest accuracy with the positive asymmetric and leptokurtic exponential distribution. However, as with the Recognition Level, no greater accuracy is observed with leptokurtic distributions than with platykurtic ones.

The effect of the Rule on Accuracy Index is significant, but small. The Accuracy Index achieves its highest accuracy with the Scott, Doane, and Sturges rules, and its lowest with the Friedman-Diaconis and square root rules, with Rice’s rule falling in between. Second-order interactions are significant. The increase in sample size favors the distribution where the Index has less accuracy, the semicircular one, and the Doane, Sturges, and Rice rules. The distribution where the Index shows more accuracy, which is the exponential, and the rules of Scott, Friedman-Diaconis, and square root, benefit less from the increase in population size. The rules of Scott and Friedman-Diaconis perform poorly with the arcsine distribution but show the greatest accuracy with the triangular one.

Rice’s rule improves with increasing sample size. It performs better than the Friedman-Diaconis rule, especially in relation to the arcsine distribution, but not as well as Scott’s rule, except with this same distribution. In terms of the Accuracy Index, its profile resembles that of the square root rule across the 10 distributions and is akin to Doane’s rule concerning sample size, differing from the profiles of the Scott and Friedman-Diaconis rules, which resemble each other. Among the seven rules, Scott’s stands out, except with the arcsine distribution. In this case, the square root and Rice’s rules are the better options. Consistent with other studies [19] , the square root rule exhibits the lowest accuracy, except with the arcsine distribution, and is the one that benefits the least from an increase in the sample.

As limitations of the study, it should be noted that rules based on the minimization of arguments, such as the rules of Rudemo [16] , Shimazaki and Shinomoto [17] , Liu, Hussain, Tan, and Dash [18] , or Knuth [21] , were not taken into consideration. This decision was made due to their complexity in programming and their absence in the statistical packages currently in use. Additionally, the rule based on a homogeneous density but a heterogeneous width of k bins [15] [45] was also excluded, since it is limited to goodness-of-fit tests. Very large sample sizes, such as 2000, 5000, 10,000, or more data points, were not included either. Nevertheless, the scope of the present study aligns with common data analyses in research in psychology and related fields. In these domains, rules determining the number of bins and sample sizes, such as the aforementioned, are not commonly encountered.

The accuracy of the empirical histogram in reproducing the shape of the distribution was assessed through average discrepancy (between the empirical and expected histogram), recognition level (of the theoretical histogram), and an accuracy index (a combination of the two previous variables). However, there are other measures, such as integrated mean square error or the Kullback-Leibler divergence [46] , which were more related to empirical rules that are not considered due to their computational complexity.

The author expresses gratitude the reviewers and editor for their helpful comments.

The author declares no conflicts of interest regarding the publication of this paper.

Rubia, J.M.D.L. (2024) Rice University Rule to Determine the Number of Bins. Open Journal of Statistics, 14, 119-149. https://doi.org/10.4236/ojs.2024.141006