The performance characteristics of the measuring instrument are classified as static and dynamic characteristics. Error is one of the stable characteristics. The difference between the true value and the measured value does not change over time. The true value is the average of an infinite number of measurements, and the measured value is the exact value.
The errors that arise during any measurement are called measurement errors. The error can arise due to many reasons.
Types of Measurement Errors
Measurement errors can be classified into the following five types.
- Constant Errors
- Systematic Errors
- Random Errors
- Absolute Errors
- Relative Errors
Constant Errors
Constant errors are those that can affect the result by the same amount.
For example: If a thermometer placed in melting ice at normal pressure has a reading of 1°C, the instrument has an error of 1°C.
Systematic Errors
If the equipment continuously produces an error of uniform deviation during its operation, it is known as a systematic error. Systematic errors occur due to the characteristics of the material used in the instrument.
Errors are arranged according to a certain law for a known reason and occur in one direction, either positive or negative. Systematic errors can be reduced by selecting better equipment, improving experimental techniques or procedures, and removing as many individual errors as possible. For a given experimental set-up, these systematic errors can be calculated to a certain extent and the necessary corrections can be applied to the observed readings.
Types of Systematic Errors
- Instrumental Errors: Instrumental errors arise due to the use of instruments or measuring instruments. Errors may arise due to incomplete design or calibration of the measuring instrument, zero error in the instrument, deficiencies and loading effects, etc. It depends on the range or resolution of the measuring instrument.
- Gross error: A gross error is a systematic error caused by the observer’s lack of experience when taking a measurement, called a gross error. The values of macroscopic errors will vary from observer to observer. Sometimes, even gross errors can occur due to improper selection of equipment. Gross errors can be reduced by following these two steps.
- Choose the most appropriate instrument based on the range of values to be measured.
- Note down the readings carefully.
- Environmental Errors: These errors arise due to changes in the environment and external conditions like temperature, wind speed, pressure, humidity, electric field or magnetic field, etc.
- Observational Errors: Some errors are due to the meter reading being taken by the observer. Parallax errors are included in this type of error. These errors may also arise due to imperfections in the experimental setup.
Random Errors
Errors caused by unknown sources during the measurement time are called random errors. It is also called chance error. Therefore, it is not possible to eliminate these errors. However, random errors can be minimized by following these two steps.
- Step 1 – Take more readings by different observers.
- Step 2 – Perform statistical analysis on the readings obtained in Step 1.
If the number of observations is ‘n’ times, the random error is reduced to 1/n times.
So when a physical quantity is measured, a_{1}, a_{2}, a_{3} ….., there are n different readings. Then their most accurate value is the arithmetic mean value. It is shown as,
Absolute Errors
The magnitude of the difference between the actual value of the measurement quantity and the measured value is the absolute error. When the exact value of the quantity measured is not known, the arithmetic mean of the measured values can be taken as the actual value.
If the measured values of a certain quantity are a_{1}, a_{2}, ……, then a_{1}, a_{2}, ……… are the errors in the measurement. It is shown as,
The arithmetic mean of the measurement of all absolute errors is the final absolute error.
The value obtained in a single measurement may be in the range.
Relative Errors
Relative error or fractional error is the ratio of the actual value of the quantity measured and the absolute error. Since the arithmetic mean value is assumed to be the actual value, the relative error is represented as,
The relative error is expressed in percentage,
Measurement Statistical Analysis
Measurement statistical analysis (MSA) is the practice of using statistical tools such as a gauge R&R (repeatability and reproducibility) to determine whether a measurement system is capable of accurate measurement.
It allows analytical determination of the uncertainty of the final test result. A large number of measurements are usually needed for statistical analysis to be meaningful. Systematic errors should be smaller than random errors because statistical analysis of measurement data cannot remove a certain bias inherent in all measurements.
The following are the parameters used in statistical analysis.
- Mean
- Median
- Variance
- Deviation
- Standard Deviation
Mean
The arithmetic mean of all the readings measured in a variable is called the mean. The best approximation is possible when the number of readings of the same quantity is very large.
If x_{1}, x_{2}, x_{3},….,x_{n} be the n readings of a particular measure. So the mean or mean value of these readings is calculated as:
Where,
- m = Arithmetic Mean or Average Mean.
- x_{n} = Number of Reading taken.
- n = Total Number of Readings.
Median
If the number of readings for a suitable particular measure is large, it may be difficult to calculate the mean or average value. Calculating the median value will be approximately equal to the mean value.
To calculate the median value, the readings of a particular measure have to be arranged in ascending order. If the number of readings is an odd number, calculate the median value using the following formula:
When the number of readings is an even number, the median value is calculated by the following formula:
Deviation from Mean
The difference between the reading and the mean value of a particular measurement is called the deviation from the mean. This is also called deviation. it can be represented as,
Where,
- d_{i} = the deviation of i^{th} reading from the mean.
- x_{i} = the value of i^{th} reading.
- m = the mean or average value.
Average Deviations
The mean deviation represents the accuracy of the instrument used in the measurement. The mean deviation is the sum of the absolute values of the deviation divided by the number of readings. It can be represented as,
Where,
- D_{av} = average deviation.
- |d_{1}|, |d_{2}|, |d_{3}|, ……., |d_{n}| = absolute value of deviations.
- n = Total Number of Readings.
Standard Deviation
The root mean square of the deviation is called the standard deviation. It can be represented as,
The above formula is the number of readings n is valid for greater than or equal to 20.
If the number of readings, n, is less than 20, the standard deviation is represented by the following formula:
Where,
- σ = the standard deviation.
- d_{1}, d_{2}, d_{3}, ..…, d_{n} = the deviation of the first, second, third,…, n^{th} readings from the mean respectively.
If the standard deviation value is smaller, the measurement reading values will have greater accuracy.
Variance
The square of the standard deviation is called the variance. it can be represented as,
Where,
- V = the variance.
- σ = the standard deviation.
The mean square of the deviation is also called the variance. it can be represented as,
The above formula is the number of readings n is valid for greater than or equal to 20.
If the number of readings, n, is less than 20, the variance is represented by the following formula:
- V = the variance.
- d_{1}, d_{2}, d_{3}, ..…, d_{n} = the deviations of first, second, third, …, n^{th} readings from mean respectively.