Significant Figures and Standard Deviations

Standard Deviation. A large number of trials in the measurement of a particular quantity will result in a "normal" distribution. 68 % of the values will be within one "standard deviation" of the average. 95 % will be within two "standard deviations" of the average. For the relatively small number of trials normally done in the laboratory, the "estimated standard deviation" can be calculated as shown on page 7 of your lab manual.

SIGNIFICANCE: If a measurement is more than 3 standard deviations from another measurement, the difference is significant. If the values are within 2 standard deviations, the difference is not significant.

To calculate averages and standard deviations using the spreadsheet EXCEL the formulae are =AVERAGE(A1:A3) and =STDEV(A1:A3) where A1 and A3 are the first and last cell in the column to be averaged.

Standard deviations and arithmetic calculations: For two numbers, A and B, with standard deviations, a and b, you can calculate the standard deviation of the SUM or DIFFERENCE as follows:

(A ± a) + (B ± b) = (C ± c)

A + B = C and c = sqrt( a2 + b2 ) where sqrt is the square root in EXCEL the formula is =SQRT()

You can calculate the standard deviation of the PRODUCT or QUOTIENT as follows:

(A ± a) * (B ± b) / (C ± c) = (D ± d) where * indicates multiplication (as in EXCEL)

A * B / C = D and d = D * sqrt ((a/A)2 + (b/B)2 + (c/C)2}

Standard deviations and significant figures:

The standard deviation provides an estimate of how well you know a particular measurement. It is, however, an estimate. The standard deviation should be expressed to ONE significant figure (unless the number is between 11 and 19 (times some power of ten, in which case you can use two significant figures. The number modified by the significant figure should be expressed to AGREE IN PLACE with the standard deviation. For example, a value resulting from a spreadsheet calculation of an average and standard deviation might be 10.1298 ± 0.2595. This should be expressed as

10.1 ± 0.3 or 10.1 (0.3) where the number in parenthesis is taken to be the estimated standard deviation. The estimate indicates that the value is only known to within three tenths of a unit of measurement. The figures beyond the tenths place, are NOT SIGNIFICANT.

VERY LOW standard deviations: Any measured number is assumed to have some uncertainty in the measurement. If your calculated standard deviation is LESS THAN 1 in the last significant figure as determined by using the rules on page A15-17 of the text by Zumdahl, ROUND-UP to 1. For example, 2.34 ± 0.0002 would be expressed as 2.34 ± 0.01.

More examples

 555.15983 ± 0.0289435 555.15983 ± 0.0012329 555.159 ± 0.000234 555.15983 ± 28.9435 wrong 555.16 ± 0.03 555.1598 ± 0.0012 555.159 ± 0.001 560 ± 30 correct

Margaret Kastner, Bucknell University