Error in the Measurement and Significant Figures

This article introduces two key concepts:

1. Error in the measurement 
2. Experimental uncertainty

The article explains each concept and provides examples for each one.

One can become aware of how measurements are performed and how we communicate them to others.

Everyday examples of measurement errors exist. For example, when someone buys clothes at a store, they will take different sizes and compare the fit, length, and width. The person judges whether the clothes fit correctly by observing them with the naked eye. This can be different from the way a measuring tape guides the person.

Measurement:

In science and everyday life, measurement is used to describe physical phenomena. We measure length, mass, volume, and many other quantities to indicate how large or small we consider these things to be. 

But measuring something doesn’t necessarily give us a good idea of what the thing is or how it behaves. For example, the formula for calculating the volume of a cube a³ tells us that each side of the cube is one unit long. But this formula doesn’t say anything about whether there are three sides or four sides or its shape. 

Similarly, let’s give someone a large number of measurements of an irregular-shaped object, and they plot the numbers on a graph. They will be able to draw a straight line through the points, but that doesn’t tell us anything about what the object is beyond the fact that it is approximately straight. We know nothing about its shape or what it might look like.

To describe general aspects of a physical phenomenon, we must use more than just the number of measurements. We need to make certain assumptions about what the numbers mean and how they relate to other numbers in the system. This is where experimental uncertainty comes into play.

Error in the measurement and significant figures

Error in the measurement: 

Due to instrument limitations or human error, the error can be introduced in the measurement process. For example, a thermometer may have a large error if it is not calibrated accurately, or a metre stick may introduce large errors if it is only accurate to one millimetre.

Significant figures: 

Significance is a property of measurements used in statistics and can be demonstrated in four ways. They can be “correctly” significant, “unreliably” significant, “falsely” significant, or “not at all” significant. 

The correct significance of measurement will be displayed to the appropriate degree of accuracy on most modern computers. The significant, incorrect, false, and not at all significant are called significant figures. Several different expressions for the significance of a measurement can be used for both scientific and technical purposes.

The significant figures of a number are those digits that can be determined with the precision of the measurement instrument. The values obtained using all the possible digits, also called “all significant figures”, are often preferred to those obtained using only two or four digits. One decimal place is called one significant figure. 

For example, 23.55 is the correct way to write the result of subtracting 18 from 30. However, 5.55 is the correct way to write the result of subtracting 05 from 10.

Meaningful digits:

In a result written in scientific notation, the number of digits to the right of the decimal point is called meaningful digits. The total number of digits, including any leading and trailing zeros, are all significant figures.

Experimental uncertainty:

It is the error associated with measuring a physical quantity or its value or magnitude over a range of values. It symbolises the precision to which values can be measured within an experiment.

The uncertainty in measuring something means that the value obtained from a particular measurement is not necessarily accurate or precise; there may be some error. The greater the uncertainty, the more error present in our measurement. Consequently, we have less confidence in extrapolating our results to larger or smaller scales.

Measurement involves deciding on which measurement to perform next, based on what is known and what can be observed. For example, suppose that we want to find the mass of an object. We could make sure that we have measured the object’s length using a ruler (length measurement uncertainty) and then make sure that we have weighed this object’s length (weight measurement uncertainty). 

We could then use the relationship between mass and weight to find the object’s mass. But to do this, we would have to make certain assumptions about how long the object is and what weight it should have.

Conclusion: 

In summary, measuring errors is not just reporting individual errors in a single measurement. The effect of errors can be cumulative, and the uncertainty associated with experimental results will be much greater than just the sum of individual errors. 

Error in the measurement of the experiment is an analytical concept related to the risk of getting it wrong and is calculated as part of statistical analysis. The first step in reducing errors in the experimental measurements is to be aware of their potential to affect the data’s accuracy and interpretation.