Errors in Measurement: Types, Calculation and Combination of Errors

Errors in measurement are the amount of inaccuracy. The difference between the true or actual value and the measured value is characterized as measurement error. 

• The measured value is the precise value, whereas the true value is the average of an unlimited number of measurements. 
• Measurements are an essential element of life; we measure time, steps taken to calculate calories burned, ingredients used in cooking, and clothing sizes to ensure a great fit. 

Errors in measurement can be classified as

• Gross errors
• Random errors
• Systematic errors

The least count error is the smallest value of the measurement that can be directly taken from a measuring instrument.

Backlash error is a motion error that occurs when the tip of the screw remains stationary for a portion of the rotation after reversing the thimble's rotational direction.

Key Terms: Percentage error formula, Measurement of physical quantities, Errors, Average or mean value, Absolute and relative error

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Errors

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An error is defined in science and engineering as a discrepancy between the desired and actual performance or behavior of a system or object. 

• The above definition acts as the foundation for many types of control systems, which define an error as the difference between a fixed point and the process value. 
• A thermostat in a home heating system is an example of this; the functioning of the heating equipment is regulated by the difference (the error) between the thermostat setting and the detected air temperature. 
• Engineers attempt to create devices in such a way that the effects of error, whether intentional or unintentional, can be reduced. 
• These errors in a system might be hidden design errors that go undiscovered for years until the perfect mix of conditions develops to activate them. 
• Other errors in constructed systems can occur as a result of human error, such as cognitive bias. 
• Human factors engineering is frequently used in design to reduce this type of error by making systems more forgiving or error-tolerant.

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Measurement

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The measurement is a numerical representation of an object's properties for comparison with other objects. 

• Measurement, in other words, is the act of defining how large or little a physical amount is in comparison to a fundamental reference quantity of the same kind. 
• All experimental study depends on measurement. All of the major technical advancements would not have been possible without ever-increasing levels of measuring precision. 
• Amounts are measured using international standards that are entirely accurate when compared to others. 
• To understand the concept of measurement errors, we need to understand the two words that determine the error. 
• There are two types of value: true value and measured value. 
• Through experimental means, it is impossible to find the true value. 
• It is defined as the mean of an unlimited number of measured values. 
• The measured value is a single, exact measurement of the object.

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Types of Errors in Measurement

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Based on the source the error arises, it is classified as

• Gross errors
• Random errors
• Systematic errors

Gross Errors

This type of error mostly considers human error and other errors made during reading, recording, and reading. The most common type of error, human error in measuring, falls within this category of measurement errors. For example, the individual obtaining the reading from the instrument's meter may read 23 as 28. 

Gross errors can be avoided by using the following two measures

• Reading and recording data should be done with caution. Also, the calculation of errors must be accurate.
• We may reduce gross errors by increasing the number of experimenters. If each researcher takes various readings at different points, we can reduce gross errors by taking the average of many measurements.

Random Errors

The errors which cannot be associated with any systematic or constant cause are called random errors. 

• These errors occur irregularly and can randomly have any sign i.e. positive or negative. 
• The magnitude of the size of errors can also vary randomly.
• These errors can arise due to unpredictable fluctuations in the experimental conditions. 
• For example, random changes in pressure, temperature, voltage supply, etc.
• If we make a large number of observations of a value and take their arithmetic mean, the effect of positive errors will get reduced due to the effect of negative errors and vice-versa.
• Hence, the arithmetic mean of these observations is much closer to the true value, than any of the individual observations.

Systematic Errors

The errors which occur in one direction only, i.e. either positive or negative are called systematic errors. It is also known as constant errors.

• If the measured value is greater than the true value, the error is said to be positive.
• If the measured value is less than the true value, the error is said to be negative.

Some of the sources of the systematic errors are as follows

• Environmental Errors
• Observational Errors
• Instrumental Errors

Environmental Errors

This type of error occurs in measurements as a result of the influence of external factors on the measurement. 

• Temperature, pressure, and humidity are all examples of external conditions. 
• If we are measuring our body temperature under the armpits and the electricity goes out and the room gets heated, our body temperature will rise, altering the reading.

Observational Errors

These are errors caused by a person's bias, a lack of correct instrument setup, or a person's carelessness in taking observations. Measurement errors could include incorrect readings caused by parallax problems.

Instrumental Errors

These errors arise when the measuring instrument itself has some defect in it, such as

• Improper designing and calibration: It means the instrument is not graduated properly. For example, if an ammeter reads a current of 1.5 A when a 2 A current is actually flowing through the circuit, it has an imperfect calibration.
• Zero error: If the zero mark of the vernier scale does not coincide with the zero mark of the main scale, the instrument is said to have zero error. A meter scale having worn off zero mark also has zero error.

Different types of errors in measurement

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Least Count Errors

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The smallest value of the measurement that can be directly taken from a measuring instrument is called its least count. It is also called the resolution of the instrument.

• Thus the least count error is related to the precision provided by the measuring instrument.
• These errors can be reduced by using high-precision instruments and improved experimental techniques.
• The least count error belongs to the random error but is within a limited size.
• It occurs with both systematic and random errors.

Also Read:

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Errors Calculation

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Though measured with the utmost care, all measurements are liable to have errors as discussed above. While dealing with a large number of careful observations, we calculate the following errors

Absolute Error

The magnitude of the difference between the individual measured value and the true value of the quality is called the absolute error of the measurement.

Let a₁, a₂, a₃,.....,a_n be the n number of measurements, then the mean of the measured value is given by

a_mean = (a₁ + a₂ + a₃ +......+ a_n) / n

Absolute errors are given as

 Δa₁ = | a₁ – a_mean |

Δa₂ = | a₂ – a_mean |

Δa₃ = | a₃ – a_mean |

Δa_n = | a_n – a_mean |

Mean Absolute Error

The arithmetic mean of the individual absolute errors is called the mean absolute error in the measurement.

Δa_mean = (|Δa₁| + |Δa₂| +......+ |Δa_n|) / n

Thus the value of the physical quantity obtained in an experiment is written as

a = a_mean – Δa_mean

Relative Error

The relative error is the ratio of the mean absolute error Δamean to the mean value amean of the measured quantity. It is also called fractional error or proportional error sometimes.

Relative error = Δa_mean / a_mean

Percentage Error

The relative error expressed in percentage gives the percentage error. It is denoted by δa.

δa = (Δa_mean / a_mean) x 100

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Combination of Errors

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In general, a physics experiment consists of a series of measurements taken with various equipment. The final result is then determined using various mathematical techniques. The final outcome error is determined by the errors in the individual measurements as well as the type of essential mathematical operations.

The error of a sum or a difference

When two quantities are added or subtracted, the maximum permissible error in the final result is the sum of the absolute errors in the individual quantities.

Let two quantities a and b, then the error of the sum or difference of these two quantities is given by

Δz = Δa + Δb

The Error of a Product or a Quotient

When two quantities are multiplied or divided, the relative error in the result is the sum of the relative errors in the individual quantities.

Δz/z = Δa/a + Δb/b

Errors in case of a measurement of a quantity raised to a power

When a quantity is raised to a power n, the relative error in the final result is n times the relative error in that quantity.

If z = xⁿ then

Δz/z = n(Δx/x)

If z = a^mbⁿ / c^q then

Δz/z = m(Δa/a) + n(Δb/b) + q(Δc/c)

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How To Reduce Errors In Measurement

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Keeping an eye on the method and performing the steps mentioned below can help to reduce errors.

• Check that the measurement formulae are correct.
• To increase accuracy, double-check the measured value of a quantity. 
• Use the most precise tool available.
• Pilot testing measuring instruments is recommended for improved accuracy.
• For the same construct, use numerous measurements. 
• Take note of the measurements taken under controlled conditions.

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Things to Remember

• Measurement error is defined as the difference between the true or actual value and the measured value.
• Errors can be classified as Gross errors, Random errors, and Systematic errors.
• The least count is the smallest value of the measurement that may be obtained directly from a measuring device.
• The absolute error is the magnitude of the difference between each individual measured value and the true value of the quality.
• The mean absolute error in the measurement refers to the arithmetic average of the individual absolute errors.
• The ratio of the mean absolute error to the mean value of the measured quantity is known as the relative error.

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