# 1.6 Significant figures  (Page 3/12)

 Page 3 / 12

Some factors that contribute to uncertainty in a measurement include the following:

• Limitations of the measuring device
• The skill of the person taking the measurement
• Irregularities in the object being measured
• Any other factors that affect the outcome (highly dependent on the situation)

In our example, such factors contributing to the uncertainty could be the smallest division on the ruler is 1/16 in., the person using the ruler has bad eyesight, the ruler is worn down on one end, or one side of the paper is slightly longer than the other. At any rate, the uncertainty in a measurement must be calculated to quantify its precision. If a reference value is known, it makes sense to calculate the discrepancy as well to quantify its accuracy.

## Percent uncertainty

Another method of expressing uncertainty is as a percent of the measured value. If a measurement A is expressed with uncertainty δA , the percent uncertainty    is defined as

$\text{Percent uncertainty}=\frac{\delta A}{A}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}100%.$

## Calculating percent uncertainty: a bag of apples

A grocery store sells 5-lb bags of apples. Let’s say we purchase four bags during the course of a month and weigh the bags each time. We obtain the following measurements:

• Week 1 weight: 4.8 lb
• Week 2 weight: 5.3 lb
• Week 3 weight: 4.9 lb
• Week 4 weight: 5.4 lb

We then determine the average weight of the 5-lb bag of apples is 5.1 ± 0.2 lb. What is the percent uncertainty of the bag’s weight?

## Strategy

First, observe that the average value of the bag’s weight, A , is 5.1 lb. The uncertainty in this value, $\delta A,$ is 0.2 lb. We can use the following equation to determine the percent uncertainty of the weight:

$\text{Percent uncertainty}=\frac{\delta A}{A}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}100%.$

## Solution

Substitute the values into the equation:

$\text{Percent uncertainty}=\frac{\delta A}{A}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}100%=\frac{0.2\phantom{\rule{0.2em}{0ex}}\text{lb}}{5.1\phantom{\rule{0.2em}{0ex}}\text{lb}}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}100%=3.9%\approx 4%.$

## Significance

We can conclude the average weight of a bag of apples from this store is 5.1 lb ± 4%. Notice the percent uncertainty is dimensionless because the units of weight in $\delta A=0.2$ lb canceled those inn A = 5.1 lb when we took the ratio.

Check Your Understanding A high school track coach has just purchased a new stopwatch. The stopwatch manual states the stopwatch has an uncertainty of ±0.05 s. Runners on the track coach’s team regularly clock 100-m sprints of 11.49 s to 15.01 s. At the school’s last track meet, the first-place sprinter came in at 12.04 s and the second-place sprinter came in at 12.07 s. Will the coach’s new stopwatch be helpful in timing the sprint team? Why or why not?

No, the coach’s new stopwatch will not be helpful. The uncertainty in the stopwatch is too great to differentiate between the sprint times effectively.

## Uncertainties in calculations

Uncertainty exists in anything calculated from measured quantities. For example, the area of a floor calculated from measurements of its length and width has an uncertainty because the length and width have uncertainties. How big is the uncertainty in something you calculate by multiplication or division? If the measurements going into the calculation have small uncertainties (a few percent or less), then the method of adding percents    can be used for multiplication or division. This method states the percent uncertainty in a quantity calculated by multiplication or division is the sum of the percent uncertainties in the items used to make the calculation . For example, if a floor has a length of 4.00 m and a width of 3.00 m, with uncertainties of 2% and 1%, respectively, then the area of the floor is 12.0 m 2 and has an uncertainty of 3%. (Expressed as an area, this is 0.36 m 2 [ $12.0{\phantom{\rule{0.2em}{0ex}}\text{m}}^{2}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}0.03$ ], which we round to 0.4 m 2 since the area of the floor is given to a tenth of a square meter.)

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