Thoughts and Ideas on Significant Digits in Excel

Steve Aprahamian contributed some great observations and thoughts on the topic of significant digits and working in Excel. Steve works with numbers and their statistical variability as a large part of his profession. (He is a Ph.D. chemist.) His thoughts are presented in this tip and should be helpful to many readers.

It does not matter how many significant figures you use in displaying values in Excel, as the program will always use 15 significant digits in its calculation. You can ROUND the values (changing the displayed decimal places does not change the value used in the calculation, ROUND does), but the accepted practice has always been to include all the figures you have for all your calculations and ROUND at the end to the appropriate number of significant digits. This actually improves the precision of your calculations.

The easiest thing to do is to ignore significant digits and just display all final calculations with two or three decimal places. Fewer does not provide enough information, while more gives people the impression that you are anal-retentive, pedantic, or ignorant in the meaning of significant figures! (I once had a college instructor who deducted points if you did not use significant digits. He hated "calculator diarrhea." If the problem, as stated, resulted in an implied 1.23 result, answering 1.2345678901—the result displayed by the calculator—was considered incorrect and points were deducted.)

If you want to keep track of significant digits, the easiest way is to keep a value of significant digits for each number you have. With a value and the number of significant digits, Excel can calculate the number of decimal places and you can display your final value with this information. Significant digits and the number of decimal places are not independent; the decimal places come from the significant digits and the value.

Given a value (X) and the number of significant digits (SD), the number of decimal places is given by the following formula:

=SD - INT(LOG10(X)) - 1

You can then convert the appropriate value to text by using:

=FIXED(X, SD - INT(LOG10(X)) - 1)

ROUND will not work, since you must also change the number of fixed decimal places in the display. The VALUE function can be used with the "FIXED text" to convert the text to a numeric value for calculations.

Keeping track of significant digits is probably going to be a real pain, however. It is important to be aware of the following of a couple of things. First, when you multiply or divide you must pay attention to significant digits. The number of significant digits in the result is the same as the minimum number of significant digits for each of the operands. For instance, consider the following equation:

3.1 * 3.45678

The answer to this equation would only have two significant digits, because one operand has two and the other has six, and the result will have the minimum of those two. Thus, the answer to the equation would be 11, which has two significant digits. Note, as well, that even though both operands had at least one decimal place, the result does not, as that would give the result too many significant digits. Decimal places are not considered in the result.

Second, when you add or subtract, you must pay attention to decimal places, not to significant digits. The number of decimal places in the result is the same as the minimum number of decimal places for each of the operands. For instance, consider the following equation:

23.1 + 103.789120

The answer to this equation would have one decimal place, because one operand has one decimal place and the other has six, and the result will have the minimum of those two. Thus, the answer to the equation would be 127.9, which has a single decimal place. The number of significant digits in the operands is not considered in the case of addition and subtraction.

It can be noted that an even better way to accomplish the control of significance is to use propagation of error techniques. You can keep track of the variability (variance, standard deviation, or some other determination of "error") and actually determine the variability of the final result. You would report a value like 1.23 +/- 0.05 or 1,234 +/- 45. This method gives a better determination of what the actual significant digits are. This approach, of course, is best left to a separate discussion.

Without an explicit variability, significant digits, for the most part, only imply the variability in the value. The implication is that the number is +/- one-half of the last significant digit. For instance, a result of 1.23 suggests (with no other information) an actual value between 1.225 and 1.235, while a result of 12 suggests an actual between 11.5 and 12.5.

Some additional comments on significant digits:

• All digits 1-9 are significant. (12 has two significant digits, 1.234 has four significant digits, and 1,234.5 has five significant digits.)
• The digit 0 is significant when it is between two significant digits. (102 has three significant digits and 1.0204 has five significant digits.)
• The digit 0 appearing on the left of a number is only a placeholder and is not significant. (0.0123 has three significant digits and 0.0000000000000000005 has only one significant digit.)
• The digit 0 appearing on the right of a number may or may not be a significant digit and must be defined by the user. (1.2300 has five significant digits; 1.20 has three significant digits; and 100 could have one, two, or three significant digits—the number alone is not enough to make a determination.)
• Constants have an infinite number of significant digits and should never determine the significance of the final result. For instance, if you double something you are multiplying it by 2.00000000000...(infinite); you are not multiplying it by 2, which would limit the result to one significant digit. Also, when you use pi (a constant) in a calculation, you use more significant digits for pi than the significant digits used in other operands in the equation.

ExcelTips is your source for cost-effective Microsoft Excel training. This tip (1983) applies to Microsoft Excel 97, 2000, 2002, and 2003. You can find a version of this tip for the ribbon interface of Excel (Excel 2007 and later) here: Thoughts and Ideas on Significant Digits in Excel.