From original Whipple to modified versions

Original Whipple’s index (WI) is calculated as a ratio of the people reporting their ages as a multiple of 5 (terminal digit 0 or 5) and the persons in the age range 23 and 62.

(1)

Where P_x is the population reporting age x in completed years. It is based on the rectangular distribution property and assumes a linear distribution of ages in a 5-year age range. In this original version, the Whipple index only checks the preferences (avoidances) at the ages with terminal digits of 0 and 5 without any distinctions. To cover this, the first change was suggested by Roger, observed in the book [14], which distinguishes the preferences for ages ending with terminal digits 0 and 5.

(2)

(3)

[For ease of presentation multiplier 100 is omitted from equation (2) to onward]. By taking the arithmetic mean of equations (2) and (3), we can get equation (1), i.e., original Whipple Index (WI)

(4)

This modification provides a meaning to distinguish between the ages ending with terminal digits 0 and 5, based on the property of rectangular distribution and linearity assumption over 10 years, like the original WI.

The second modification was suggested by Noumbissi [15], is based on the more sensible assumption of linearity across five years rather than a 10-year age range. Noumbissi feels that the 10-year assumption in the formulation of Roger’s and the original Whipple index is crude. Therefore, he suggested a modification in the Roger’s Whipple index, based on a more reasonable assumption of linearity, taking the age range of 5 years for terminal digits ‘0’ and ‘5’ [15]. He suggested the following formulas to calculate heaping at terminal digits ‘0’ and ‘5’:

(5)

(6)

Here, P_x is the number of individuals reporting their age as x years, and ₅P_x is the number of individuals reporting their age in the age range (x, x + 4). Similarly, age heaping can be calculated for other terminal digits (1,2, …, 9) as:

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

When there is no digit preference/avoidance, these digit-specific modified Whipple indices are equal to 1 [15].

The third modification is suggested by Spoorenberg [16]. He felt the extensions proposed by Noumbissi to all ten digits are not practical for spatial, temporal, and spatiotemporal comparisons. He introduced a summary index based on the modified Whipple index proposed by Noumbissi and named it the Total Modified Whipple Index (W_tot).

(15)

where W_i is the digit specified modified Whipple index for each of the ten digits (0–9) developed by Noumbissi.

If no preference is observed, then

and

If all reported ages end in 0 or 5, then and all other . Hence, W_tot reaches the maximum value of 16. This index can be used as a general measure of the quality of age reporting in complement to Noumbissi’s previous development [16].

Mathematical inconsistencies in Noumbissi’s Indexes

There are a few shortcomings in the calculation of the equation for individual digits given in equations (5–14) by Noumbissi. If we expand the numerator of the above equations (5–14), the equations will be as follows;

(5e)

(6e)

(7e)

(8e)

(9e)

(10e)

(11e)

(12e)

(13e)

(14e)

It can be seen that all expanded equations (5e–14e) are not based on a complete age range (23–62) years. Moreover, for equations (7e, 8e, 9e, and10e), few values of the denominator are beyond the age range (23–62). These equations give unrealistic and poor predictions of the indexes for digits (1, 2, 3, and 4). Therefore, there is an obvious shortcoming in the indexes proposed by Noumbissi. The approach to calculating individual digit (0–9) preferences in Noumbissi’s version is unrealistic and not based on mathematical formulation. Spoorenberg’s total modified Whipple index (W_tot) is based on Noumbissi’s modification. Therefore, it does not give an appropriate summary value of all terminal digits robustly.

Theoretical foundation

The evolution of the Whipple index from its original form to the modified versions reflects a continuous effort in demography to refine statistical tools to measure age misreporting. However, the existing methods for quantifying age misreporting, such as the modified Whipple’s Index proposed by Noumbissi and subsequently adopted by Spoorenberg, exhibit limitations in their mathematical formulation, particularly when calculating indices for individual terminal digits. An examination of the expanded equations for individual digit indices (equations 5e-14e from Noumbissi’s work) reveals that the denominators do not consistently encompass the entire age range of interest (typically 23–62 years). Furthermore, for several digits (specifically 1, 2, 3, and 4, corresponding to equations 7e, 8e, 9e, and 10e), the denominator includes age values falling outside this defined range. This inconsistency in the reference age range leads to potentially unrealistic and inaccurate predictions of the individual digit indices. Consequently, the approach used by Noumbissi for calculating individual digit preferences appears mathematically flawed and lacks a robust theoretical basis.

Since Spoorenberg’s total modified Whipple index (W_tot) is derived from Noumbissi’s modification, it inherits these shortcomings and may not provide a reliable summary measure of age misreporting across all terminal digits. A need exists for a summary index that accurately captures age misreporting or heaping across the entire specified data range (23–62 years) without the aforementioned mathematical inconsistencies. To address these limitations of existing indices, this research proposes a re-modification of the Total Modified Whipple Index. The new index is constructed in the logic of the original Whipple index, applying its structure systematically to each terminal digit using the mathematically consistent framework of the Rogers Whipple Indices (WI_i). In doing so, this research aims to contribute to the ongoing development of demographic quality measures by presenting the Robust Modified Whipple Index (RMWI) as a more reliable and comprehensive tool for assessing age misreporting.

Robust Modified Whipple Index (RMWI) [Proposed Index]

Following the Roger’s Whipple index, the indexes for each terminal digit (0–9) can be calculated as;

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

Based on indexes (WI₀ to WI₉) in the equations (15–24), a modified version of Spoorenberg’s total modified Whipple index (W_tot) can be calculated as;

(25)

If there is no heaping at any digits, then

and

If all reported ages are heaped at a multiple of 10, i.e., end at digit 0, then WI₀ = 10 and all other WI_i = 0, and hence, RMWI = 18.

The analysis in this study focuses on the age range of 23–62 years, a commonly used span in demographic research that helps reduce distortions from age misreporting, especially at younger and older ages. This range effectively captures the core adult population, where age heaping is typically most evident, making it a suitable base for evaluating digit preference or avoidance. However, the proposed Robust Modified Whipple Index (RMWI) is flexible and can be applied to any rectangular age distribution where each terminal digit (0–9) has an equal probability. For example, alternative ranges such as 15–64 or 20–79 can be used depending on the research objectives and the quality of the available data.

A proposed criterion for the quality of age reporting for RMWI

To smooth the interpretation of the RMWI value and provide a standardized assessment of age reporting quality, a criterion is proposed that parallels the widely used United Nations (UN) standards for the original Whipple Index. The United Nations proposed the following (Table 1) criteria to assess the quality of age reporting based on the Whipple Index [17].

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Table 1. UN criteria for the quality of reported ages.

https://doi.org/10.1371/journal.pone.0330519.t001

The original Whipple’s index is a summary measure of the quality of age data, which emphasizes the heaping at digits 0 and 5. It ignores the digit preferences (avoidance) on all other 8 digits (1, 2, 3, 4, 6, 7, 8, and 9). The RMWI, by incorporating the deviations from expectation for all ten terminal digits, offers a more comprehensive measure of age misreporting. A set of ranges for the RMWI value is established to categorise the quality of age reporting (Table 2). To calculate the range for RMWI, the following cases are considered

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Table 2. Criteria for quality of reported ages using RMWI.

https://doi.org/10.1371/journal.pone.0330519.t002

1. If all ages are correctly reported, i.e., WI₀ = WI₁ = WI₂ = … = WI₉ = 1, RMWI = 0.
2. If all reported ages are heaped at a multiple of 10, i.e., end at digit 0, WI₀ = 10 and all other WI_i = 0, RMWI = 18. Similarly, if all reported ages ended in a single digit, i.e., any one WI_i = 10 and all other WI_i = 0, RMWI = 18.
3. If any two digits are reported only (say 0 and 5), then WI₀ = WI₁ = 5 and all other WI_i = 0, RMWI = 16.

Hence, the value of RMWI ranges between 0 and 18. A value of “0” represents the perfection of data. As the value of RMWI deviates from 0, it approaches the imperfection of the reported ages, implying that the ages are misreported. These theoretical bounds of the RMWI (ranging from 0 to 18) represent benchmarks for perfect accuracy and extreme misreporting, respectively. Based on the index’s structure and its sensitivity to varying patterns of digit preference, the following ranges are proposed to classify the quality of age data based on RMWI (Table 2).

The proposed classification in Table 2 offers a practical scale for assessing the quality of age reporting using the RMWI index. A RMWI value between 0.00 and 0.19 indicates perfect accuracy of age reporting, reflecting negligible digit preference, with less than 1% of individuals misreporting their age. Values from 0.20 to 0.99 suggest fairly accurate data, indicating 1% to 4.99% age misreporting likely due to weak digit preference. A range of 1.00 to 1.99 signals moderate age misreporting (5% to 9.99%), indicating clear evidence of age misreporting. Values between 2.00 and 2.99 represent poor or rough data quality, pointing to widespread age misreporting. Finally, a RMWI value exceeding 3.00 suggests very poor data quality, where at least 15% of individuals are likely misreporting their ages. The percentage ranges of RMWI are derived from the mathematical structure and its sensitivity to departures from uniformity of terminal digits. A higher RMWI reflects a greater deviation from expected digit uniformity, suggesting more significant age misreporting. This interpretive framework adds practical value to the RMWI by enabling comparisons across datasets and populations. By incorporating all ten terminal digits with equal weight and offering a robust summary measure, the RMWI and its associated criterion provide a more comprehensive and statistically consistent approach than earlier modifications.

The following sections illustrate the practical application of the RMWI and this criterion using real-world and simulated datasets, demonstrating its effectiveness in identifying and quantifying age misreporting.

Application to data

To test the proposed index, RMWI, we applied it to the Standard Demographic and Health Survey (DHS) datasets from three countries: India, Turkey, and Pakistan. A comparison with existing indies was carried out to enhance the analysis. Standard DHS surveys collect primary data using four types of model questionnaires. Among these questionnaires, a household questionnaire is used to collect data on the characteristics of the household for all the usual residents of the household. This is the main questionnaire that includes all respondents who are used to get information for other types of questionnaires. In this household schedule, information is available on age, sex, education, relationship to household head, residence, and other details. Reported age data from individuals from this household questionnaire is used for this research. The sample size used for this study is given in Table 3 [18]. More details about the DHS surveys can be found at the DHS website (www.dhsprogram.com).

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Table 3. Sample size of the individual respondent in the DHS series of three selected countries.

https://doi.org/10.1371/journal.pone.0330519.t003

Among the three selected countries for analysis, the quality of age reporting is considered poor in India and Pakistan [2], and average in Turkey. Appendix Tables A1, and A2 give the digit-specific Whipple indexes (WI_i) and digit-specific modified Whipple indexes (W_i) for India, Turkey, and Pakistan respectively.

Fig 1 shows the digit preference (avoidance) using the digit-specific Whipple indexes WI_i (Roger’s indexes) and modified Whipple indexes W_i (Noumbissi’s indexes). Technically, if all digits are correctly reported, then the value of all digit-specific indexes must be 1. If the value of an index for any digit is greater than 1, it is the preferred digit by people telling their age at the time of data collection. This may cause heaping at those digits. If the value of the index is less than 1, this shows the avoidance of that digit.

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Fig 1. Quality of age reporting: Digit-specific Whipple’s index (WI_i) and digit-specific Modified Whipple’s index (W_i) for the series of DHS data of India, Turkey, and Pakistan.

Data Source: Standard DHS of India (2005–2021), Turkey (1993–2018) and Pakistan (1990–2018) [18], Figures based on the Author’s calculations of digit-specific Whipple indexes.

https://doi.org/10.1371/journal.pone.0330519.g001

Age reporting in India and Pakistan follows a classical pattern of age misreporting. A strong age heaping at the digits 0 and 5 is reflected in both digits’ specific Whipple index as well as in the original Whipple index. Digits 1 and 9 are the most avoided in age reporting in both countries. Preferences for 0 and 5 have declined a little bit over time but the situation of the accurate reporting of age is still looking like a nightmare in both countries. The situation in Turkey is less extreme. Preferences for 0 and 5 declined over time and were very near to 1 in the year 2008 to onwards. In Turkish datasets, digit 3 is as likely to be preferred as 0 or 5. The most observed pattern of digit preference in India is 5, 0, and 8 for digit-specific Whipple’s index (WI_i) and 5, 0, and 2 for digit-specific Modified Whipple’s index (W_i). The digit “2” replaces its position from digit “8” due to the overestimation of digit “2” in the digit-specific Modified Whipple’s index in all three-survey datasets of India. Similarly, digit “3” also changes its position over digit “4” due to the overestimation of the index for digit 3. Although the digits “1 and 4” have not changed their position these two digits also overestimated the digit in the digit-specific Modified Whipple’s index. In the case of Pakistan, digit “5” is ranked as the highest preferred digit in digit-specific Whipple indexes; however, for the PDHS 2005–06 and PDHS 2012–13, the digit “0” ranked as the highest preferred digit in the digit-specific modified Whipple index. Similarly, digits 2 and 3 ranked at earlier positions due to overestimation in the digit-specific modified Whipple index. In Turkish datasets, all overestimated digits “1, 2, 3, and 4” in digit-specific modified Whipple indexes moved to earlier ranks as compared to digit-specific Whipple indexes. In the digit-specific modified Whipple index (Noumbissi’s indexes), digit 3 is preferred over digit 5 due to the unstable calculating equation for digit 3. Digits 1 and 9 are the most avoided in all populations (see Fig 1, Appendix A1 and A2).

Comparison of digit specific indexes: W_i vs WI_i

The accuracy of the total index (calculated for all ages) depends on the accuracy of digit-specific indexes. In this section, we compare two digit-specific indexes used to calculate the total modified Whipple index (W_tot) and the proposed RMWI index.

If no preference (avoidance) of any specific digit is present, then the digit specified index for any single terminal digit must be equal to 1, corollary, sum of all ten indexes must be equal to 10. Statistically, for any population where age is reported correctly, then the probability of each terminal digit of age must be equal to 0.1, and for all terminal digits i = 0, 1, 2, …, 9. Practically, some digits are preferred (P_i > 0.1), and some are avoided (P_i < 0.1), however, the sum of all digit probabilities must be 1, and corollary, the sum of all digit indexes (DIs) must be 10. Therefore, in any population following a rectangular distribution and assuming the linearity assumption, the following results can be obtained for individual terminal digit probability (P_i) and digit index (DI_i).

Avoided digit (under-reporting of age)

Ideal digit (correct reporting of age)

Preferred digit (over-reporting of age)

or

Digit-specified indexes calculated in tables Appendix A1 and A2 showed that in all thirteen datasets of three countries, few digits are preferred (DI_i > 1) and others are avoided digits (DI_i < 1). This is expected in all countries with age misreporting. As a result, the probability of individual terminal digits is greater than 0.1 (preferred) or less than 0.1 (avoided). To compare the age misreporting indices, look deeply at Appendix A1 and A2. In the calculation of digit specified indexes, two methods, W (modified Whipple i.e. Noumbissi’s indexes), and WI (original Whipple), are used. Apparently, all digit-specified indexes calculated by the two methods look similar. However, only the WI method has an equal probability of selection of individual terminal digit i = 0, 1, 2, …, 9. In the W method, terminal digits 1, 2, 3, and 4 have some added probability than the remaining six digits. These four terminal digits are overestimated as the probability of selection of each of these terminal digits exceeds 0.1 (see equations 7e–10e). As a result, for the W method, the sum of probabilities exceeds 1 and the sum of digit indexes exceeds 10 (see the last column of tables Appendix A1 and A2). Statistically, the sum of all ten terminal digit indexes must be ‘10’ in any method. This is not plausible in the W method due to over selection of terminal digits 1, 2, 3, and 4. Therefore, this method fails to give a robust index value for individual terminal digits. Only the WI method is based on the mathematical formulation of the selection of equal probability for all possible terminal digits of human age (0, 1, 2, …, 9). Hence, in the digit-specified indexes, only WI methods give a robust index for all terminal digits. Summary index W_tot based on the individual digit specified indexes with an unequal probability of selection of each terminal digit, is non-robust. In the WI method, all terminal digits have an equal probability of being included, and hence all digit-specified indexes have equal weightage for any terminal digits. Therefore, the newly proposed summary index RMWI is based on the WI method, is robust and based on a mathematical formulation. Bar charts presented in Fig 2 give the side by side comparison of digit-specific Whipple indices from Noumbissi’s method (W_i) and the Whipple method (WI_i) using Turkey (2013–2018) DHS data. Clearly, terminal digit 1, 2, 3 and 4 gives overestimation in both datasets.

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Fig 2. Side-by-side comparison of digit-specific Whipple indices from Noumbissi’s method (W_i) and the Whipple method (WI_i) using Turkey (2013 and 2018) DHS data.

Data Source: Standard DHS of Turkey (2013 and 2018) (ICF, 1985–2023), Figures based on the Author’s calculations of digit-specific indexes.

https://doi.org/10.1371/journal.pone.0330519.g002

An example: Turkey demographic and health survey datasets

To strengthen and support our claim, we include the raw datasets from the Tukey demographic and health surveys (TDHS) in Appendix Table A4 for ages 21–64. We calculate new digit indexes using equations 5e to 14e with the expanded age range of 21–64 years. We named these indexes as W_i[21–64] for i = 0, 1, 2, …, 9. Denominators of the digit indexes ‘1, 2, 3, and 4’ ranges beyond the range 23–62 [equations: 7e, 8e, 9e, and 10e] change their digit index values. Table 4 gives three different types of digits specified indexes. The first part of Table 4 shows the digit-specific Modified Whipple index W_i’s (Noumbissi’s method) based on the age range of 23–62 years (calculated in the previous section). The second part gives the new type of digit-specified index based on the age range of 21–64 calculated using the digit-specific Modified Whipple index W_i[21–64] (Noumbissi’s method). The third part gives the digit-specific Whipple index WI_i’s (calculated in the previous section). The last column of the table shows the sum of all digit indexes for all ten digits. In all six survey datasets, results for digit-specific Modified Whipple indexes showed that digit ‘3’ is a more preferred digit than digit ‘5’ (see columns W₃ and W₅). However, when we use a complete range of denominators to calculate the digit-specific Modified Whipple index W_3[61–64] for terminal digit ‘3’, the results are reversed. Digit 5 is more preferred than digit ‘3’ [see column W_3[61–64] and W_5[61–64]]. This is all due to the overestimation of digit 3 in Noumbissi’s method. Additionally, digit ‘2’ is pretended as a preferred digit in five datasets, while originally the digit ‘2’ was avoided in a few survey years. Digits 1, 2, 3, and 4 are overestimated in the digit-specific Modified Whipple index method. The pattern and rank of terminal digits calculated using the age range (21–64) are quite similar to the digit-specific Whipple index. Therefore, the digit-specific Whipple indexes give robust results as compared to the digit-specific Modified Whipple indexes W_i’s. The sum of all ten digits for any method must be equal to 10. The sum of digit indexes W_i’s are greater than 10 due to the overestimation of digits 1, 2, 3, and 4. The sum digit indexes W_i[21–64] are very near to 10, while the sum of digit indexes WI_i’s are exactly equal to 10. Additionally, in raw datasets of Turkey, the sum of all ages with the terminal digit ‘5’ is more than the terminal digit ‘3’ (see Appendix Table A4).

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Table 4. Comparison of digit-specific Whipple’s indexes (WI_i) Digit-specific modified Whipple’s indexes (W_i), and digit-specific modified Whipple’s indexes (W_i[21–64]) for Turkey data.

https://doi.org/10.1371/journal.pone.0330519.t004

Comparison between Whipple index (WI), Total modified Whipple’s Index (W_tot), and Robust Modified Whipple’s Index (RMWI)

Comparably, all three indices give the same patterns over the years and can be used for spatial, temporal, and demographic comparisons within nations and globally. The original Whipple index is a good choice to check the quality of age data when we are interested in checking the heaping at digit 0 or 5 only. The other two indices will be used to check the overall quality of reported ages. However, digit-specific modified indexes (Noumbissi’s indexes) give overestimated values for digits 1, 2, 3, and 4. This indicated a misestimation of preference for the digits 1, 2, 3, and 4. For example, Noumbissi’s index for digit 3 in the Turkish dataset showed that it is more reported than digit 5. Which is not the actual case. Noumbissi’s indexes calculated for digits 1, 2, 3, and 4 give the wrong representation of the original reported digit due to the inaccuracy of equations (see equations 8–10 or 8e–10e). The W_tot based on these inconsistent equations did not provide reliable and accurate information about the quality of age data for individual digits. The new RMWI index is based on mathematical equations (equations 16–26) with the same methodology proposed in the original Whipple index and by Roger et al. This modification gives more reliable and robust results based on mathematical models as compared to the previous proposal by Spoorenberg.

Although the comparison figures of W_tot and RMWI both give the same pattern (Fig 3). However, W_tot, based on Noumbissi’s version, is impractical for comparison and doesn’t give a summary value for any dataset. It is also based on unrealistic and non-mathematical formulations and, therefore, not robust. RMWI is based on the Whipple index, and the whole data range is thus preferred over W_tot. RMWI gives more precise, robust, and accurate results as compared to W_tot.

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Fig 3. Total modified Whipple’s Index (W_tot), and Robust modified Whipple’s Index (RMWI).

https://doi.org/10.1371/journal.pone.0330519.g003

Simulation study

To evaluate the performance of the proposed modification, we applied it to several simulated datasets of different sizes. All series are randomly generated, and there is no observed heaping at the digits 0 and 5. Fig 4 gives the digit-specified Whipple and modified Whipple indexes for all ten digits for different series, while Fig 5 presents a side-by-side comparison of digit-specific Whipple indices from Noumbissi’s method (W_i) and the Whipple method (WI_i), clearly illustrating the overestimation issue in Wi and the balanced distribution in WI_i. For ease of presentation, we compared RMWI with W_tot and WI, respectively (Table 6). In 1^st comparison of RMWI and W_tot, consider series S1, where WI₀ = 1.229 is most preferred followed by WI₉, WI₈, WI₃, WI₂, WI₁, WI₇, WI₅, WI₆, and WI₄ = 0.723 (most avoided digit) while W₃ = 1.264 is most preferred followed by W₀, W₂, W₇, W₈, W₉, W₁, W₅, W₆ and W₄ = 0.843 (most avoided). Digit 3 ranked up from place 4^th to 1^st, and digit 2 ranked up from place 5^th to 2^nd due to over-estimation of digits 2 and 3 in W_tot or simply an over-estimation in Noumbissi’s indexes. This type of over-estimation in Noumbissi’s indexes makes W_tot penniless, and W_tot remains incapable of accessing the quality of age data. Hence, RMWI gives a more accurate and robust estimate for the quality of age distribution.

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Table 5. Comparison between Whipple index (WI), Total modified Whipple’s Index (W_tot), and Robust modified Whipple’s Index (RMWI).

https://doi.org/10.1371/journal.pone.0330519.t005

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Table 6. Comparison between Whipple index (WI), Total modified Whipple’s Index (W_tot), and Robust modified Whipple’s Index (RMWI) in simulated data series.

https://doi.org/10.1371/journal.pone.0330519.t006

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Fig 4. Quality of age reporting: Digit-specific original Whipple’s index (WI_i) and digit-specific Modified Whipple’s index (W_i) for the simulated data series.

https://doi.org/10.1371/journal.pone.0330519.g004

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Fig 5. Side-by-side comparison of digit-specific Whipple indices from Noumbissi’s method (W_i) and the Whipple method (WI_i) using Simulated datasets.

https://doi.org/10.1371/journal.pone.0330519.g005

In all simulated data series, the value of the Whipple index (WI) is at its perfection level (Table 5). However, RMWI says that data in Series S1 and S2 are moderate, in Series S6, fairly accurate, and in Series S7, S9, and S10 are perfectly accurate. RMWI presents the actual picture of all data series, taking all terminal digits. We can observe that in series S1, there is a visible avoidance at digit 4 (WI₄ = 0.723), but WI ignores it at all. Similarly, in series S2, digit avoidance at digit 3 (WI₃ = 0.741). These preferences and avoidance are accurately covered by RMWI and ignored by WI. The WI emphasizes only heaping at digits 0 and 5 and ignores all other digits. Therefore, if heaping or preferences (avoidance) is present at any(some) other digit(s) (1, 2, 3, 4, 6, 7, 8, 9), WI fails to assess the overall quality of age distribution. Fig 5, based on simulated datasets, presents a side-by-side comparison of digit-specific Whipple indexes from Noumbissi’s method (W_i) and the Whipple method (WI_i). These bar charts clearly illustrate the overestimation present in Noumbissi’s method, particularly for digits 1–4, whereas the WI_i used in the proposed index remain more balanced across all digits.