In the process of measurement, due to the detection method used, the instrumentation involved, and the influence of environmental and operational conditions, even when measuring the same object under identical conditions, slight differences in the obtained data are common. These variations are typically attributed to measurement errors.
First, let's explore the measurement method and classification of measurement errors. One of the most commonly used representations is the **absolute error**, which is defined as the difference between the measured value (x) and the true value (xo). Mathematically, it can be expressed as:
$$ \Delta x = x - x_o $$
However, since the true value is often unknown, a standard reading is usually taken as an approximation for xo, referred to as the "actual value." For example, if a calibrated circuit has an actual current of 4.63A, and the ammeter reads 4.65A, the absolute error would be +0.02A. The unit of the absolute error matches that of the quantity being measured.
To improve accuracy, the concept of a **correction value** is introduced. The correction value (c) is defined as the difference between the actual value and the measured value:
$$ c = x_o - x = -\Delta x $$
The correction value has the same magnitude as the absolute error but with the opposite sign. When added to the measured value, it yields the actual value.
For instance, consider a pressure gauge G that measures a pressure of 1000.2 N/m², while the actual pressure measured by the system is 1000.5 N/m². In this case, the correction value for the gauge would be:
$$ c = 1000.5 - 1000.2 = 0.3 \, \text{N/m}^2 $$
It’s important to distinguish between **error** and **deviation**. Error refers to the difference between the measured value and the true value, while deviation is the difference between a single measurement and the average of all measurements. Although these terms are sometimes used interchangeably, they have distinct meanings.
Another important measure is the **relative error**, which expresses the absolute error as a percentage of the actual value:
$$ \text{Relative Error} = \frac{\Delta x}{x} \times 100\% $$
This helps compare the precision of different measurements. For example, if two instruments both have an absolute error of 1mA, but one measures 100A and the other 100mA, their relative errors will differ significantly, showing that the former is much more accurate.
The **reference error** is another key metric, calculated as the ratio of the absolute error to the full-scale range of the instrument:
$$ \text{Reference Error} = \frac{\Delta x}{L} \times 100\% $$
Unlike relative error, reference error uses the instrument’s range instead of the actual value. This makes it easier to evaluate performance across different ranges, though it may not always reflect the true accuracy at every point within the scale.
The **maximum reference error** is the largest reference error observed over the entire measurement range. It is considered the primary indicator of a system's accuracy and is often referred to as the **basic error** of the system.
Now, let’s classify the causes of errors:
1. **Systematic Errors**: These occur consistently and follow a predictable pattern. They are caused by instrument inaccuracies, incorrect methods, or environmental influences. Systematic errors can often be corrected through calibration or adjustments.
2. **Random Errors**: These occur unpredictably and vary in magnitude and sign. They result from numerous small, uncontrollable factors and are best analyzed using statistical methods. Precision is used to describe the magnitude of random errors.
3. **Gross Errors**: Also known as blunders, these are large deviations that occur due to human mistakes, equipment failure, or improper procedures. Such errors should be identified and excluded from the data set.
It’s worth noting that these types of errors can sometimes overlap or transform into each other depending on the context. For example, complex systematic errors may appear as random errors in some cases, while previously classified random errors may later be understood as systematic after further investigation.
In modern measurement systems, correction and compensation techniques can significantly reduce or eliminate systematic errors. After proper calibration and verification, only random errors remain, which can then be analyzed statistically to estimate the overall uncertainty of the measurement. This approach ensures more reliable and consistent results.
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