Rounding a series of decimal numbers to whole numbers can create situations where the sum of the rounded series no longer matches the original sum (control total). The Greatest Mantissa rounding algorithm is one possible solution.

In the realm of demography and population studies, precise and accurate estimates are of paramount importance. These estimates serve as the foundation for a multitude of applications, from urban planning and resource allocation to policy-making and social research.

However, the process of producing these estimates often creates fractional numbers, which, while accurate, can be challenging to work with in practical scenarios.

That’s an issue that the U.S. Census Bureau encounters with their subcounty population estimates. The sum of the population of the cities, towns and unincorporated parts of a county must align with the county’s published totals. However, the Bureau’s process of producing the estimates necessitates rounding so that the number of people in each community is expressed as whole integers.

Unfortunately, the traditional “round to nearest” approach can result in subcounty populations that no longer align with the originally published county totals. Over large datasets, this lack of consistency can severely impact the reliability of the data. Achieving this alignment while maintaining the relative proportions of the subcounty estimates can be a complex task.

This is where the concept of “Greatest Mantissa Rounding” comes into play. This rounding methodology allows us to adjust the fractional estimates to integers while ensuring that the sum of these rounded estimates still matches the given total. This method is particularly useful in situations where we need to maintain the integrity of total values while dealing with individual estimates that are expressed as decimals.

In the following sections, we will delve deeper into this methodology, exploring its application in the context of producing controlled subcounty population estimates. Follow along and you will have a solid grasp of the Greatest Mantissa Rounding method and its significance, as well as a better idea about how to apply it in your applications.

### Problem

Consider a county with a total population of 1,000 people. The population of each town in the county is estimated as follows:

- Town A: 550.7 people
- Town B: 249.5 people
- Town C: 199.8 people

The sum of these estimates is 1000, which matches the county’s total population.

Because the numbers should represent whole persons, we want to round the estimates to an integer with the rounded totals still summing to the county’s total population (1,000). The traditional “round to nearest” approach yields the following integer set:

- Town A: 551 people

- Town B: 250 people
- Town C: 200 people

By rounding to the nearest whole number, the sum is now 1,001. This is a common occurrence in numerical computations and over large datasets, the difference between the original sum and the sum of the rounded numbers can be significant. This discrepancy is known as a round-off error.

### Solution

The original total can be preserved using the Greatest Mantissa Rounding method. Here’s how:

**Step 1:** Round down the estimates to the nearest integer:

- Town A: 550.7 people -> Rounded down to 550 people
- Town B: 249.5 people -> Rounded down to 249 people
- Town C: 199.8 people -> Rounded down to 199 people

The sum of these rounded estimates is 998, which is two less than the county’s total population of 1000.

**Step 2: **The mantissa is the portion of the original town population figures to the right of the decimal point. The largest mantissas will be selected to round up. These should be ordered from greatest to least.

- Town C: 0.8
- Town A: 0.7
- Town B: 0.5

Town C has the largest mantissa (0.8) and is shown first in the list, followed by Town A (0.7).

**Step 3:** Add “ones” to the towns one by one until the control total of 1,000 is reached. Since the rounded sum, 998, is two below the county’s 1,000-person population, we know that two towns need to be rounded up by one to reach the county’s total. The first “one” is added to Town C because its mantissa is the largest.

- Town A: 550 people
- Town B: 249 people
- Town C: 199 people -> 199 + 1 = 200 people

Now, the sum of the rounded estimates is 999, which is still less than the county’s total population of 1000. We need to round one more town up to reach the county’s total. Town A had the second largest mantissa so “one” is added to its rounded value.

- Town A: 550 people -> 550 + 1 = 551 people
- Town B: 249 people
- Town C: 200 people

Now, the sum of the rounded estimates is 1000, which matches the county’s total population.

This is how Greatest Mantissa Rounding works when distributing the “ones” across multiple towns.

## Conclusion

The Greatest Mantissa Rounding method allows us to round decimal estimates to integers while ensuring that the sum of the rounded estimates matches a given total. This method can be particularly useful in situations where we need to maintain the integrity of total values while dealing with individual estimates that are expressed as decimals.

### References

Microsoft’s Bing Assistant on ‘Greatest Mantissa Rounding and Population Estimates’, Knoxville, Tennessee, United States, November 30, 2023.

Coleman, C. D. 2006b. “The Greatest Mantissa Algorithm.” Manuscript, Washington, DC: U.S. Census Bureau.

Coleman, C. D. 2015. “SAS R Macros for Constraining Arrays of Numbers.” Manuscript, Washington, DC: U.S. Census Bureau