What is the meaning of accuracy and precision? How to express it?

Accuracy and precision: In any analysis, when the same sample is tested using the same method, the results obtained after multiple measurements are never exactly identical, which indicates the presence of errors in the measurements. To this end, we must understand the causes of errors and their representation methods, and minimize errors as much as possible to improve the accuracy of the analysis results. I. Accuracy and Error Accuracy refers to the degree of agreement between the measured value and the true value. The level of accuracy is often measured by the magnitude of the error. In other words, the smaller the error, the higher the accuracy ； The larger the error, the lower the accuracy. There are two ways to express error: absolute error and relative error. Absolute error (E) = Measured value (x) – True value (T). Relative error (E%) = |Measured value (x) – True value (T)| / True value (T) × 100. To determine the accuracy of a measured value, it is necessary to know its error or relative error. To determine the error, the true value must be known. But the true value is usually unknown. In practical work, standard methods are commonly used, with the arithmetic mean obtained from multiple repeated measurements taken as the true value. Since the measured value (x) may be greater than or less than the true value (T), both the absolute error and the relative error can be positive or negative. Example: If the measured value is 57.30 and the true value is 57.34, then: Absolute error (E) = x – T = 57.30 – 57.34 = –0.04. Relative error (E%) = E/T × 100 = (–0.04/57.34) × 100 = –0.07. Example: If the measured value is 80.35 and the true value is 80.39, then Absolute error (E) = x – T = 80.35 – 80.39 = –0.04. Relative error (E%) = E/T × 100 = –0.04/80.39 × 100 = –0.05. In these two examples, the absolute error is the same for both measurements, but the relative errors differ significantly; this shows that the two concepts have different meanings. The absolute error represents the difference between the measured value and the true value, while the relative error represents the percentage of that error relative to the true value. For values obtained from multiple measurements, their accuracy can be calculated using the following formula: Absolute error (E) = ∑Xi/n – T. Where: Xi ---- the result of the i-th measurement ； n----- Number of measurements ； T----- True value. Relative error (E﹪) = E/T × 100 = (–T) × 100/T. For example, if the results of 3 measurements are 0.1201 g/L, 0.1185 g/L, and 0.1193 g/L, and the content of the standard sample is 0.1234 g/L, then the absolute error and relative error can be calculated. Solution: The average value = (0.1201 + 0.1193 + 0.1185) / 3 = 0.1193 (g/L). The absolute error (E) = x – T = 0.1193 – 0.1234 = –0.0041 (g/L). The relative error (E%) = E / T × 100 = –0.0041 / 0.1234 × 100 = –3.3. It should be noted that sometimes, in order to indicate the measurement accuracy of certain instruments, the absolute error is a clearer indicator. For example, the error of an analytical balance is ±0.0002 g, and the reading error of a constant-volume burette is ±0.01 ml; these are all expressed in terms of absolute error. II. Precision and Bias Precision refers to the degree to which the results of n repeated measurements under the same conditions are consistent with each other. The degree of precision is expressed by the deviation; the smaller the deviation, the higher the precision. 1. Deviation: There are absolute deviation and relative deviation. Absolute deviation (d) = x – Relative deviation (d%) = d / ×100 = (x – ) / ×100. Where: --- the average value of the n measurements ； x---- Result of a single measurement ； d---- Absolute deviation of the measurement result ； d﹪----Relative deviation of the measurement result. From the above formula, it can be seen that the absolute deviation refers to the difference between an individual measurement and the average value. Relative deviation refers to the percentage that the absolute deviation represents of the average value. It can be seen that absolute deviation and relative deviation can only be used to measure the degree of deviation of an individual measurement result from the average value. To better illustrate precision, the mean deviation (d_mean) is commonly used in routine analytical work. 2. Mean deviation: The mean deviation refers to the sum of the absolute values of the deviations of each individual measurement value from the average value, divided by the number of measurements. That is, the mean deviation (d_mean) = (|d1| + |d2| + … + |dn|)/n = ∑|di|/n. The relative mean deviation (d_mean%) = d_mean × 100/% = ∑|di|/n × 100. Where: d_mean – mean deviation; n – number of measurements; the average value is obtained from n measurements; x1 – result of a single measurement; d1 – absolute deviation of a single measurement from the average value, where di = ||*|| ； ∑︱di ︱----the sum of the absolute differences of the absolute deviations from n measurements ； The mean deviation represents the deviation of any value within a set of measured values. Therefore, the average deviation does not consider positive or negative values. Example: Calculate the mean value ( ), average deviation (d_mean), and relative deviation (d_mean %) for the following set of measurement values. Solution: 55.51, 55.50, 55.46, 55.49, 55.51. Mean value = ∑x/n = (55.51 + 55.50 + 55.46 + 55.49 + 55.51)/5 = 55.49. Average deviation = ∑|x_i – μ|/n = ∑|x_i – 55.49|/5 = (0.02 + 0.01 + 0.03 + 0.00 + 0.02)/5 = 0.016. Average relative deviation = |∑|x_i – μ|/n| × 100% = 0.016/55.49 × 100% = 0.028%. III. The relationship between accuracy and precision. After understanding the definitions and methods for determining accuracy and precision, we should be aware that they are two distinct concepts, but there is a certain relationship between them. It should be noted that the higher the precision of the measurement, the closer the results will be to the true value. However, it cannot be assumed absolutely that high precision implies high accuracy, as the presence of systematic errors does not affect the precision of the measurements; on the contrary, without good precision, it is highly unlikely to achieve high accuracy. It can be said that precision is a prerequisite for ensuring accuracy.

According to the definition in JJF1001-2011 \"General Terms and Definitions in Metrology\": Accuracy: The degree of agreement between the measured value of a quantity and its true value. Precision: Under specified conditions, the degree of consistency among the readings or values obtained from repeated measurements of the same or similar test objects.