Significant digit in calculations

There are three rules for reporting significant digits in calculated answers. These rules are summarized in Table 1.5. Reflect on how they apply to your previous experiences. Then examine the Sample Problems that follow. 

Table 1.5 Rules for Reporting Significant Digits in Calculations...

Notice that a result of this type, in order to be interpretable, must comprise three numbers the mean, the (relative) standard deviation, and the number of measurements that went into the calculation. All calculations are done using the full precision available, and only the final result is rounded to an appropriate precision. The calculator must be able to handle >4 significant digits in the standard deviation. (See file SYS SUITAB.xls.)... 

Sres is easy to calculate and, since the relevant information resides in the first significant digits, its calculation places no particular demands on the soft- or hardware (cf. Section 3.3) if the definition of r,- in Table 2.1 and Eqs. (1.3a)-(1.3d) is used. 

When you take measurements and use them to calculate other quantities, you must be careful to keep track of which digits in your calculations and results are significant. Why Your results should not imply more certainty than your measured quantities justify. This is especially important when you use a calculator. Calculators usually report results with far more digits than your data warrant. Always remember that calculators do not make decisions about certainty. You do. Follow the rules given below to report significant digits in a calculated answer. 

Examine each group s data and calculated value for density. Note how the number of significant digits in each value for density compares with the number of significant digits in the measured quantities. 

If we just leave the answer the way our electronic calculator gives it to us, anyone could assume that the measurement had been carried out with a precision of 1 part in 86,387, which is not true. We must reduce the number of significant digits in the answer to two because the factor with fewer significant digits has two. Thus, we change the answer to 86 cm. ... 

The answer is 3.00 g/cm. It must have three significant digits because both the dividend (4.92) and the divisor (1.64) have three significant digits. In this case, add two zeros to the answer given by the electronic calculator (3) to get the correct number of significant digits. 

Step 4 If necessary, multiply all the numbers of moles by the same small integer to clear fractions. Round off the result to an integer only when the number of moles is within 1% of the integer. Always use at least three significant digits in empirical formula calculations otherwise, rounding errors may produce an incorrect empirical formula. 

Calculate the formula mass of each of the following to two decimal places twice, first by rounding each atomic mass to two decimal places, and second by using the entire number of significant digits in the atomic masses and rounding the formula mass ... 

LeChateher s principle (Section 18.3) allows us to make qualitative predictions about the effects of changes of conditions on an equihbrium system but does not ahow quantitative calculations. However, at equihbrium at a given temperature, a certain ratio of concentration terms is very nearly constant for all solutes and gases involved in any given reaction. (Solids and pure liquids are not included in the ratio.) Because it is not exactly constant, we will use two or at most three significant digits in equihbrium constant calculahons. For the general reaction... 

We must apply the rules wheu reportiug the answer. For example, the reciprocal of 9.00 is really 0.111, but the calculator displays something like 0.111111111. Similarly, dividing 5.34 by 1.78 should yield 3.00, but the calculator displays 3. We must report only the three significant digits in the first example and must add the two significant zeros in the second example. 

Use at least three significant digits in formula mass calculations. 

Note that the calculated limits do not appear to be equidistant from the point estimate, as we might have expected. This is the result of rounding to two significant digits in the calculations. The calculated 95% confidence interval about the difference in proportion of participants reporting headache as an AE is written as follows ... 

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