1.12. Various Errors in Measurements

All measurements are subject to errors, irrespective of the instrument and method used. The 'true value' of a measured quantity is thus never known. The angular, linear and elevation measurements might have three basic errors present (Punmia et al., 206); (i) natural errors, (ii) instrument errors, and (iii) personal errors.

This chunk introduces the concept that every measurement contains some level of error. No matter how precise we believe a measurement to be, its 'true value' is always uncertain. Errors can arise from various factors, and they are classified into three main types: natural errors (which could arise from environmental conditions like temperature or humidity), instrument errors (caused by imperfections or calibration issues in the measuring tools), and personal errors (which are related to human mistakes in reading or recording measurements). Understanding these errors is crucial for improving measurement accuracy.

Think of measuring the height of a plant with a ruler. The ruler might not be perfectly calibrated (instrument error), the weather could be very windy causing the plant to sway (natural error), or you might misread the ruler (personal error). These examples help illustrate that even simple tasks can involve various forms of errors.

Broadly, these errors will fall under two categories: (a) Systematic or cumulative, and (b) Accidental, random or compensating. Systematic errors can always be identified and corrected because their magnitude and nature (sign) can both be determined. For example, a measuring tape is designed for its standard length under a particular pull and temperature, but if the pull or temperature changes during the field work, its effect on the length (increase or decrease in length) of the tape can be computed.

Errors in measurements can be classified into two main categories. Systematic errors are consistent, predictable errors that can occur due to known factors, such as temperature or pressure, and they can usually be corrected. On the other hand, accidental errors are more unpredictable. They occur without a fixed pattern and can vary, often due to factors such as random environmental influences. This distinction is important because it helps in developing strategies for reducing different types of errors.

Imagine a kitchen scale that always reads 200 grams more than the actual weight because of a calibration issue. This is a systematic error that can be fixed. Alternatively, if you weigh the same ingredient multiple times and get slightly different results each time due to slight hand movements or distractions, those variations represent random errors. Understanding these types of errors helps ensure more accurate recipes.

Accidental, random or compensating errors are subject to chance, and hence follow the laws of probability. The magnitude and sign of errors are not known, as they are sometimes positive and sometimes negative, sometimes of small magnitude, and sometimes of large magnitude, and hence can’t be determined or eliminated. To minimise it, we take a large number of observations to make an estimate of magnitude of such error that is likely to occur.

Accidental errors are unpredictable and can vary widely. Because their effects can cancel each other out (one measurement might be higher and another lower), they are often addressed by taking multiple measurements and averaging the results. This reduces the impact of random variations and gives a more reliable estimate of the true value.

Imagine you're a student measuring your height using a tape measure. If you measure on three different days and get heights of 160 cm, 161 cm, and 159 cm, the differences may confuse you. However, if you average those measurements, you get a better estimate of your actual height. This averaging helps smooth out the random errors inherent in single measurements.

In fact, there is one more error, i.e., mistake or blunder, but that cannot be classified under any category of error as these are mainly due to the carelessness of the observer. Mistakes can be corrected only if discovered, and comparison of several measurements of the same quantity could be used in isolating the mistakes.

It's essential to differentiate between errors and mistakes. While errors can be systematic or random, mistakes are more about human carelessness or oversight. For example, if someone miswrites a measurement in their notes, this is a mistake. Discovering such mistakes is crucial for ensuring the integrity of measurements, and doing multiple measurements can help catch these sloppy errors.

Consider a teacher grading papers and accidentally skipping a student's work due to oversight. This mistake could lead to an unfair assessment. If the teacher double-checks each grade against a list, they can catch these mistakes and ensure every student is evaluated fairly. This highlights the importance of careful checking in all fields, including measurement.

There are two more terms involved when we deal with the errors in measurements; (i) accuracy, and (ii) precision. Accuracy is the closeness or nearness of the measurements to the 'true' or 'actual' value of the quantity being measured. The term precision (or repeatability) refers to the closeness with which the measurements agree with each other.

Accuracy and precision are two key concepts in measurement. Accuracy refers to how close a measurement is to the true or actual value, whereas precision refers to how consistently you can reproduce the same measurement. A set of measurements can be accurate but not precise if they are spread out far from the true value. Conversely, they can be precise (close together) but not accurate if they are consistently wrong. Understanding this distinction is essential for evaluating the quality of measurements.

Imagine playing darts. If you throw all your darts close to the bullseye (the center of the target), your throws are accurate. If you throw them all together around the outer edge, your throws are precise but not accurate. Ideally, you want to be both: hitting the bullseye consistently with all your throws, showcasing both accuracy and precision.

Statistically, precision can be measured by means of a quantity σ, known as standard deviation or standard error, and is given by- (1.2) Where v2 is the sum of the squares of the residuals, n is the number of measurements. The smaller the value of σ becomes, the greater is the precision.

Precision can be statistically quantified using standard deviation (σ), a measure that tells us how spread out the measurements are around their average value. If measurements are very close to one another, the standard deviation is small, indicating high precision. Conversely, a larger standard deviation signifies that the measurements vary widely, indicating lower precision. This statistical approach helps in assessing the reliability of the measurements.

Think of a company's sales figures. If the monthly sales numbers fluctuate slightly around an average, the standard deviation is low, indicating that the company maintains stable performance. However, if the sales vary drastically from month to month, the standard deviation is high, signaling potential instability in sales performance. This helps the business understand its operational consistency and make necessary adjustments.

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