# The Numbers Game: Significant Digits, Rounding and Truncating

It is important for those who work with data to know and understand some of the technical aspects of numbers, both when precise measurements and rounded or estimated figures are involved. In this article, we will look at three related aspects of numbers: significant digits, rounding and truncating.

## Significant Digits

Not every digit in a number is what is called “significant” in mathematical terms. Significant digits (also called significant figures) speak to the precision of the measurement with the aim of distinguishing between a very precise measurement and a rounded or estimated figure. There are rather specific rules for when digits are and are not significant:

All non-zero digits and zero digits between two non-zero digits are significant.

Examples:

493 has three significant digits

23,407 has five significant digits

90,000,401 has eight significant digits

## Trailing Zeros: Not Significant

Trailing zeros in numbers without decimal points are not significant. (This is because with trailing zeros, we do not know if the figure is rounded to the rightmost significant digit or not. If the zeros are significant, additional information about the figure must be known.)

Examples:

4,000 has one significant digit (just the 4)

98,310 has four significant digits (9, 8, 1, 3)

320 has two significant digits (3, 2)

## Trailing Zeros: Significant

Trailing zeros in numbers with decimals points are significant. (The presence of the decimal point indicates the precision of the measurement.)

Examples:

319.000 has six significant digits (It is assumed that the measurement was to the third decimal place.)

250.0 has four significant digits

Leading zeros are not significant, whether they are in numbers with decimal points or not.

Examples:

005 has one significant digit (just the 5)

0.0567 has three significant digits (5, 6, 7)

004.008 has two significant digits (4, 8)

0.0000700 has three significant digits (7 and the two 0s following 7)

## Rounding and Truncating

When calculations are carried out on data, such as in finding the mean and standard deviation of a data set, we often have to round the results. The question arises as to what place value we should round. The generally-accepted answer is to round to the same number of significant digits as the value with few number of significant digit in the raw data.

Because of the need to round, it is important to know the precision of the measuring instrument, which is indicated by the significant digits in the data. For example, if we were calculating the average (mean) attendance at seven football games and the attendance figures were rounded to the thousands place (with two significant figures), then our result should be rounded to the thousands. For example, if the attendance figures are 78,000; 82,000; 75,000; 76,000; 79,000; 80,000; and 81,000, then we get:

Because the raw data values have only two significant digits, this number can only be accurate to two significant digits, so we must round to the thousands place. Using standard rounding rules (round up if the digit to the right of the place value to which the number is being rounded is 5 or greater), this figure should be rounded to 79,000 (because the digit to the right of the thousands place is a 7).

Given this rule for rounding, however, it is important to consider the context of the data. For example, if the values are ages, then it would be appropriate to round to two significant digits even if some of the ages were single digits. For example, if the ages of a group of students are 9, 10, 10, 11, 11, 11, 12, 13, and 14, and we calculate the mean, we get 11.222…. It would make sense to round this to 11 (two significant digits) instead of 10 (one significant digit) even though one of the raw data values (9) has only one significant digit.

Truncating is different than rounding. In truncating, we just drop any decimal places and use the whole-number value. As an example, in some testing situations, such as on a test of competency, there is a “cut score,” the minimum score a test-taker must receive to pass the exam. The scores are truncated, not rounded, because reaching the cut score is an indication of being competent in that subject area or skill, and not reaching the cut score is an indication of not being competent. So if a passing score is 140, and a candidate receives a score of 139.8, it is reported as 139, and the candidate does not pass the exam.

## Round-off Error

There are a variety of errors that can occur when using rounded and truncated values. One of the most common and important errors to avoid is rounding intermediate values when doing calculations with numbers instead of rounding only the final result. For example, if we were to calculate the standard deviation of the values given in the example above (attendance at a football game), we have to use the mean of the data to do so. For accuracy, we would want to use the value 78,714.29… instead of the rounded value of 79,000 in those calculations. After doing the calculations, we would round the calculated value to two significant digits (2,600; the more precise value is 2563.48).