The mass of a gold coin was measured three times and each measurement was made to five digits. The mass values were 23.319g, 23.341 g, and 23.296g. the average mass was reported as 23.32 g. The actual mass of the coin is 25.5631g.
Why is the average mass of the gold coin reported to only four significant figures?
Looking at 23.319, the 233 and 1 are known but the last number is uncertain. The average is 23.3198 which rounds to 23.320 but the 0 is in the uncertain column so it is reported as 23.32.
Well, it seems that the average mass of the gold coin thought, "Hey, four significant figures should be plenty! Why bother with more?" It's like going to a party and deciding to bring just enough snacks to keep everyone happy, without going overboard. The average mass didn't want to be too showy with its measurement accuracy, so it settled for four significant figures, leaving the remaining digits for moments when they're truly needed. It's modestly efficient and avoids unnecessary decimal traffic jams.
The average mass of the gold coin is reported to only four significant figures because the least precise measurement out of the three is rounded to four decimal places.
In this case, the least precise measurement is 23.341 g, which has four visible decimal places. Rounding this value to four significant figures results in 23.34 g.
Since the average mass is determined by calculating the sum of the three measurements and dividing it by three, the rounded value of 23.34 g is used in the calculation.
As a result, the average mass of the gold coin is reported as 23.32 g, which is consistent with the significant figures of the least precise measurement.
The average mass of the gold coin is reported to only four significant figures because that is the lowest number of significant figures common to all three measurements.
To determine the number of significant figures in a measurement, we consider all the digits that are known with certainty, plus the first uncertain digit. In this case, the mass values 23.319 g, 23.341 g, and 23.296 g each have five digits, so they are measured to five significant figures.
When calculating the average, we add together the three measurements and divide by the number of measurements (in this case, three). Doing so gives us a sum of 69.956 g. Dividing this sum by three yields an average mass of approximately 23.3186667 g.
However, when we report the average mass, we need to consider the number of significant figures present in the original measurements. Since the lowest number of significant figures among the three measurements is four (due to the 23.296 g measurement), we round the average to four significant figures, yielding 23.32 g.
Therefore, the average mass of the gold coin is reported to only four significant figures to reflect the precision of the original measurements.