How do you relate the significant figures used on a measured value to the standard deviation of
several measurements taken? For example, where do you round the following measurement
and its standard deviation: 782.5729 ±0.38914 g?
To relate significant figures to the standard deviation of several measurements, you need to consider the uncertainty or precision of the measurements. Significant figures represent the number of reliable digits in a measurement and indicate the precision of the measurement.
In the example you have provided, the measurement is given as 782.5729 g with a standard deviation of ±0.38914 g. Here's how you can round the measurement and its uncertainty:
1. Start by identifying the least precise measurement. In this case, it is the standard deviation of ±0.38914 g, which has five digits.
2. Count the number of significant figures in the least precise measurement. Since the standard deviation represents uncertainty, it is usual to consider only one significant figure. So, in this case, the standard deviation has one significant figure.
3. Round the measurement based on the number of significant figures in the standard deviation. In this case, since the standard deviation has one significant figure, round the measurement to one decimal place. Therefore, the measurement becomes 782.6 g.
4. Express the uncertainty using the same number of decimal places. In this case, since the measurement was rounded to one decimal place, the uncertainty should also be presented to one decimal place. Therefore, the uncertainty becomes ±0.4 g.
So, the rounded measurement would be 782.6 ±0.4 g, with one significant figure to match the precision of the standard deviation.