What is Error Analysis and Significant Figures?

Error analysis is a method of discussing the possible errors that have been encountered in physical measurements. It consists of tabulating the source of error to the observed magnitude. It allows us to determine whether our analysis is valid.

An estimated error, in mathematics, has precision attached to it since a measurement unit used helps determine its uncertainty.

Error analysis and significant figures are the two terms you might encounter when you read about measurement uncertainty. Error analysis considers the sources of error, and significant figures give a measure of how much error is to be considered in certain calculations.

Error Analysis

Error analysis allows us to know the sources of error by observing the magnitude of the error. Errors in physical measurements are classified as: “Random errors” and “Systematic errors”.

Random Errors

Random errors could be mechanical vibrations or irregularities in the instruments, incorrect readings or human error. The magnitude of such errors increases with time, and they are not likely to be constant over a period of time. They can be further classified as “A-B” type, where A is a very large number and B is a small number.

Systematic Errors

Systematic errors in the measurement processes are more likely to be constant over a period of time. They could be due to an instrument or an operator error.

Importance of Error analysis 

In analysing how much uncertainty is associated with a particular measurement, it is important to know where the error occurred. The amount of uncertainty can be quantified by the associated error “A”, which multiplies a random error and is the product of systematic error and random error. The resulting quantity is called “Percentage Error Extension”.

Estimated error

Estimated error is a part of the uncertainty analysis, and it shows the percentage of the uncertainty added to the value of an asset. The analysis calculates the root mean square error (RMSE) of an asset under different scenarios.

A solution in which all unknowns are known and all other variables are constant is called a linear model. Linear models can be used when there is no need for estimating to show how much a variable will change with changes in unobservable variables. However, in real-life scenarios, data are usually non-linear. Non-linear solutions are used to calculate the uncertainty of a variable rather than the estimated error.

Accuracy of measurement

The key factor considered while analysing a system is the reliability of the measurements, such as accuracy and repeatability.

Accuracy of measurement is defined as the closeness to the actual value of a variable.

Repeatability of measurements is defined as the stability and accuracy over time in which values remain constant or unaffected. 

The estimated error is a calculation that shows how close an observed value is to the true value. It is a measure of uncertainty that shows what percentage of the true value would be calculated.

There are several ways to estimate the errors. Some of them are:

Instrument sensitivity: Using a mathematical model, the values of input variables are adjusted so that observed values match the true value. This helps in estimating precision and sensitivity simultaneously. 

Waveform comparison:  It compares the measured variable to the reference waveform and corrections to the input variable. 

Frequency domain version: frequency-domain version is used to calculate precise errors by calculating precise components of error in the time domain.

Statistical technique: Another way to calculate the estimated error is to use a linear regression technique, which shows a relationship between X and Y variables. 

It can also be used for more complex situations involving multiple lines of causality and interactions. The linear regression technique gives the estimated error in a single parameter range.

Accepted convention

If the error analysis is conducted, it must be accepted by the analysts. The known and accepted convention is used for calculating estimated errors and significant figures.

The estimated error gives a more detailed view of a model rather than estimated uncertainty. It can be calculated by measuring an object over and over again, which would determine the variability of measurements.

Significant figure

Significant Figures define how precise the number that is being analysed is. Significant figures avoid logarithmic misreading and enhance clarity when writing numbers in scientific applications.

Significant figures are different from significant magnitude, the scale of measurement. It is also known as AS or ASN (absolute accuracy) in scientific applications.

The measured quantity is the uncertainty of the measurement. It is measured in significant figures and can be measured in either type of significant figures (significant magnitude or significant figures). Since quantities are usually denoted in BCD (Binary Coded Decimal), the number 1.0001 is four significant figures, and 1.0003 is five significant figures. When there are more significant figures than the precision of measurements, it leads to under-precision, but if there are less than three significant figures, it leads to over-precision.

Conclusion

The estimated error is the most important aspect of the estimation. It is used in financial modelling and also in statistical analysis. In some cases, it can also calculate expected profit or expected loss. The estimation error is calculated by measuring objects repeatedly and calculating the difference between them, which shows the variability of measurements. Significant figures are used to avoid logarithmic misreading. They are also used in financial applications and statistical analysis, where the measured quantity is the uncertainty of the measurement. It is measured in significant figures and can be measured in either in significant magnitude or significant figures.