Uncertainties and significant figures

Physics is a quantitative science about measurements and calculations. These estimations include measurements with uncertainties or errors in analysis. In physics, uncertainty of measured value is an interval around that value in a manner that the repetition of the measurement will generate a new result which lies within that particular interval. 

Uncertainty is mainly related to accuracy or precision in theory and practice as well. 

So we have to calculate the uncertainties to avoid the errors. In this article, we will discuss the uncertainties and significant figures in physics. Along with the rules stating the uncertainties in physics.

Significant figures 

Significant figures are the number of digits that give the specificity of the value. We start figuring significant figures at the first non-zero digit. It can help to calculate the number of significant figures for an assortment of numbers. Uncertainty illustrates the dependability of the assertion that the stated measurement result represents the value of the measuring. 

The difference between the result of a measurement and the mean that would result from an infinite number of measurements, taken under the same conditions. It is often equated to the standard deviation, which is a measure of the expected random uncertainty.

There are three major types of uncertainties. They are called random uncertainties, reading uncertainties and systematic effects.

Absolute uncertainty 

The absolute uncertainty is the number that gives a range of true values when 

combined with a reported value. Absolute uncertainties always have the same units as the reported value with which they are associated. Usually, the final results that include uncertainty should have absolute uncertainty. 

Relative uncertainty 

The relative uncertainty is the ratio of the absolute uncertainty to the reported value. Relative uncertainties are always unitless. Multiplying the relative uncertainty by the reported value yields the absolute uncertainty. 

The following rules are for deciding the number of significant figures in a measured quantity:

• All nonzero digits are significant.

1.234 g has four significant figures

• Zeros between non-zero digits are significant.

1002 kg has four significant figures,

3.07 ml has three significant figures.

• Zeroes to the left of the first non-zero digits are not significant. Such zeroes completely indicate the degree of the decimal point.

0.001 C0 has only one significant figure

0.012 g has two significant figures.

• Zeroes to the right of a decimal point in a number are significant.

0.023 ml has two significant figures

• When a number ends in zeros that are not to the right of a decimal point, the zeroes are not necessarily significant

190 miles may have two or three significant figures; 50,600 calories may have three, four, or five significant figures.

Systematic error 

Systematic errors mainly cause a distribution’s exactness. Normal reasons for systematic error embrace data-based error, impaired tool exercise, and environmental interference. 

For example, forgetting to tear or zero a balance produces errors in measurements. Typical causes of systematic errors include observational error, imperfect instrument calibration, and environmental interference.

Classifications of systematic errors

1. Instrumental errors: These arise when an instrument is not calibrated properly at the time of manufacturing. It can be adjusted by selecting precise equipment. 

2. Flaws in the experimental method: It is due to the limitation in the experimental setup. 

A systematic error may occur because the tool has been wrongly calibrated or due to a weakness in the physical structure of the tool. 

Random error 

Random errors in the experiment occur due to unknown and unpredictable changes in the experiment. 

Random error is a chance difference between the observed and true values in a measurement. Random errors can happen because of the format of the device.

Random measurement errors lead to inconsistent measurable values when the same quantity is measured multiple times. Random errors often have a Gaussian normal distribution. Hence, statistical methods can help analyse the data. Random errors are tiny errors that move in each direction and have an inclination to cancel one another. 

The mean m of several measurements of the same quantity is the best estimate of that quantity, and the standard deviations show the accuracy. The standard error of the estimate m is s/sqrt(n), where n is the number of measurements. 

Rules for stating uncertainty 

Experimental uncertainties should be stated to a 1-significant figure. 

The uncertainty is simply an associated estimate and so it can’t be additional precise than the simplest estimate of the measured price.

Ex. v = 31.25 ± 0.034953 m/s 

v = 31.25 ± 0.03 m/s (correct) 

Conclusion

From the above, we can conclude that the uncertainties are important to avoid errors in calculation. To get more accurate values we use the uncertainty of significant numbers. It can help to find the significant digits. Experimental errors can be reduced by using the uncertainty of significant numbers. Scientific uncertainty normally implies that there is a range of possible values within which the valid value of the measurement lies.

 Uncertainty is important for science because it stimulates scientists to engage in further analysis and research. It is important to understand that scientific uncertainty does not mean science is flawed. Reasonably it means an absence of certainty and in science, it’s okay to have uncertainty.