Two types of infrared tympanic thermometers were adopted in this study. One is the OMRON MC-510 Gentle Temp model (OMRON Co., Japan). The other is the BRAUN IRT-3020 Themoscan model (Braun Co., Germany). The operating procedures for ear temperature measurement were described in detail in their manual. The specifications of the two thermometers are listed in Table 1.

The standard temperature was prepared by a temperature calibrator and a blackbody cavity. This standard temperature was simulated as the tympanic temperature to evaluate the performance of two types of infrared thermometers. The standard temperature served as the standard for performance testing.

The model of temperature calibrator is TC-2000 Scan Sense (Instrutek Co., Norway). The operating temperature was ranged from −40 °C to 150 °C. The inner temperature of oil bath for calibration was detected by a four-wire Pt-100 thermometer. The uncertainty of this calibrator is 0.03 °C according to its specifications.

An aluminum cylinder was inserted into the oil bath of the calibrator. The length of the cylinder was 14.5 cm and the diameter was 0.485 cm. The emissivity of this cylinder was assembled to be unity calculated from the calculation equation of blackbody cavity [20].

The target temperature for calibration of infrared thermometer was maintained at 34.5, 36.0, 37.5, 39.0 and 40.5 °C. The performance test experiment was executed in an environmental chamber at three levels of ambient temperature: 25 °C, 30 °C and 35 °C. The variation of the setting temperature was controlled within ±1 °C.

Each infrared thermometer was randomly measured at five sets of standard temperatures at the same ambient temperature. Five replicates were made for each standard temperature. As one measurement was made, the infrared thermometer was taken out of the blackbody cavity for five minutes and then put into the cavity again for further measurement.

The performance of infrared thermometers was assessed by their accuracy and repeatability. The definition of accuracy is the closeness with which a measurement approaches true value [11]. The error is defined as: (1) Ei = Tr − Tswhere Tr is reading value of the infrared thermometer in °C; and Ts is standard value of the infrared thermometer in °C.

The standard deviation of data sets was calculated from replicates. These values were then further analyzed by one and two way ANOVAs (Analysis of Variance) and other statistical procedure. Significance was taken as P < 0.05. The P value is the smallest level of significance that would be to reject the hypothesis.

The regression analysis technique was applied to establish the calibration equation from the relationship between the standard values and the reading values. The criteria for selecting of the best equation are the coefficient of determination R², t-tests of each parameters and residual plots [21,22].

To establish the calibration equation, the standard values of the standard temperature made by a temperature calibrator were selected as the dependent value y. The reading values of the infrared thermometer were viewed as an independent variable. The ambient temperature of the experiment testing is assumed as the other independent variable. This calibration equation f(x) is called as the inverse equation [23–25]: (2) y = f ( x i )

The form of y(x) can be expressed as follows: (3) y = b 0 + b 1 X 1 + b 2 X 1 2where X₁ is the reading values of infrared thermometer, b₀, b₁, b₂ are parameters.

Many texts dealing with the calibration equation reported that the reading values of the thermometer were viewed as the independent variable and the standard values of temperature were viewed as the dependent variable. This equation is called as the classical equation. However, the inverse equation has shown the better predictive ability [23–25].

The standard uncertainty due to the estimate values of standard deviation of replicates u_x is a Type A uncertainty. It was easy to calculate from the replicates of the measurement made using the IR thermometer at different conditions.

The standard uncertainty due to the calibration of infrared thermometer is a Type A uncertainty. The uncertainty in a predicted value u_y is calculated by follows: (4) u y = s 1 p + 1 n + ( y − y ) 2 ¯ ∑ ( y i 2 ) − ( ∑ y i ) 2 / nwhere s is standard deviations of calibration equation, p is the number of replicates in the calibration; and n is the number of the sample data.

The uncertainty source of reference standard was provided by the manufacturer’s specifications. The distribution of uncertainty was assumed as normal distribution. The estimate of uncertainty for the standard temperature is: (5) u ref = U ref 2where U_ref is the uncertainty source of TC-2000 temperature calibrator, and u_ref is the uncertainty due to the reference temperature.

The source U_non due to nonlinearity and repeatability is specified by manufacturers. The variation response for this error source is assumed as a rectangular distribution. The uncertainty due to nonlinear and repeatability u_non is calculated as: (6) u non = ± U non 2 3where U_non is the uncertainty source due to the nonlinear and repeatability that represents a specification interval provided by the manufacturer; and u_non is the uncertainty due to the nonlinear and repeatability made using the IR thermometer.

The uncertainty measurement due to resolution is assumed to be a rectangular distribution. It is considered as ±½ of the display scale value. The uncertainty value due to resolution u_res is estimated as the follows: (7) u res = ± U res 2 3where U_res is the uncertainty source due to the resolution effect, and u_res is the uncertainty due to the resolution effect.

The uncertainty source of ambient temperature was not mentioned by manufacturers. In this study, the effect of the ambient temperature on the variation of measurements made using the IR thermometer was evaluated by two-way ANOVA test. If the ambient temperature has significant effects on the variation of measurement, the uncertainty due to the ambient temperature can then be further estimated. The variation response from the temperature variation is assumed a rectangular distribution. The uncertainty due to temperature variation is: (8) u tem = ± U tem 3where U_tem is the uncertainty source due to the ambient temperature; and u_tem is the uncertainty due to the ambient temperature.

The uncertainty due to the reference temperature, nonlinearity and repeatability, resolution, and ambient temperature are classified as Type B uncertainty.

These uncertainty effects did not have significant correlations, and the combined standard uncertainty u_c then be calculated as follows:

Uncertainty estimation of a fixed measuring point. (9) u c 1 = u x 2 + u ref 2 + u non 2 + u res 2 + u tem 2

The relationship between reading values of MC-510 thermometer versus standard values is shown in Figure 1. The reading values of this tympanic thermometer are close to each other. The effect of ambient temperature on the reading values was inconstant. The error distribution of OMRON MC-510 thermometer at three ambient temperatures is shown in Figure 2. The error band is ±0.3 °C and the error values moves from positive to negative with the increase of standard temperature. According to the standard of the ASTM standard for allowable errors of clinical thermometers is 0.1 °C in the temperature range 37–39 °C, 0.2 °C in the temperature range 36–37 °C and 39–41 °C. The allowable error of clinical thermometers was within a band of ±0.3 °C in the range below 36 °C. From the data distribution of Figure 2, the performance of reading values was corresponding with the requirement of the ASTM standard in the temperature range below 37 °C. However, the reading values were diversity in the temperature of 37.5 °C. Only the average values of several measurements could reduce the measured errors. As the higher temperature ranged higher than 38 °C, the reading values were underestimated.

The relationship between standard temperature values and reading values of MC-510 thermometer was established by regression analysis. The best equation is: (11) T sta = 25.1634 − 0.4739   T red + 2.1563   T rea 2 R 2 = 0.992 ,             s = 0.1965where T_sta is the standard temperature and T_rea is the reading temperature of IR thermometer.

The predicted errors of calibration Equation (11) are presented in Figure 3. All the predicted errors in the range of standard temperature were below 37 °C and higher than 39 °C all within the ASTM standard. However, the predicted errors at the standard temperature of 37.5 °C revealed high spreading of the measurement. Above the standard temperature of 37 °C, the accuracy of the MC-510 infrared tympanic thermometer had improved significantly.

The relationship between reading values of BRAUN IRT-3020 thermometer versus standard values is shown in Figure 4. The closeness of the data distribution at each standard temperature indicated a good replication of the performance. The error distribution of this infrared thermometer at three ambient temperatures is presented in Figure 5. The error bands were within 0.2 °C of the whole measuring range except at 37 °C. The reading values of IRT-3020 thermometer could meet the requirements of the ASTM standard in the range below 37 °C. Above the standard temperature of 37 °C, the errors values of this thermometer were higher than the ASTM standard.

The relationship between standard values and reading values of IRT-3020 thermometer was established by regression analysis. The adequate equation is: (12) T sta = 0.2637 + 0.9871   T red       R 2 = 0.999 ,             s = 0.0696

The predicted errors of Equation (12) are shown in Figure 6. All the predicted errors were located within the requirement range of the ASTM standards except for the standard temperature of 37.5 °C. However, the mean value of several replicates at this temperature could reduce the predictive errors.

The standard deviation of five measurements of MC-510 thermometer at different standard temperatures and three levels of ambient temperatures are shown in Figure 7.

At the standard temperature of 36 °C, the standard deviations for three ambient temperatures were higher than that of other standard temperatures. To evaluate the factors affecting the standard deviations of MC-510 thermometer, two-way ANOVA table was established and the significant tests were executed by F test with a 95% confidence level. The result is listed in Table 2.

In comparison with the variations of measurements, the standard temperature has significantly affected the variations of standard deviations. However, the ambient temperature showed no significant effects on the variation of measurements (P = 0.1305>0.05). The ambient temperature did not influence the variation of standard deviation of each measurement. The uncertainty source of ambient temperature was not considered for the evaluation of uncertainty of the MC-510 thermometer. The distribution of standard deviation of the Braun IRT-3020 measurement is presented in Figure 8. The ambient temperature did not have the significant effect on the standard deviations. At the standard temperature of 39 °C, the standard deviations were closed at 39 °C.

The two-way ANOVA table for the evaluation of the factors affecting the standard deviations of the IRT-3020 thermometer is listed in Table 3. The results indicated that the standard temperatures (P = 0.9355 > 0.05) and the ambient temperatures (P = 0.2274 > 0.05) did not have the significant effects on the standard deviation of IRT-3020 measurements. The effect of ambient temperature on the uncertainty of IRT-3020 was not considered in further analysis.

The adequate calibration equation for OMRON MC-510 thermometer is the polynomial equation and the best calibration equation for BRAUN IRT-3020 is a linear form. The uncertainty due to the replications of measurement is the standard deviation of each measurement. However, the uncertainty derived from the calibration equation is calculated by Equation (4). The type A uncertainty of several observations for two-types of infrared tympanic thermometers are listed in Table 4. For the OMRON MC-510 thermometer, the uncertainty of calibration equation did not reduce as compared to the uncertainty calculated from standard deviations of each observation. The same results could be found for the BRAUN IRT-3020 thermometers.

The type B uncertainty analysis for the two infrared tympanic thermometers was calculated by Equation (5), (6) and (7). This result is listed in Tables 5–6. Comparing these sources of uncertainty for two-type infrared tympanic thermometers, the main source of the uncertainty included the specification of nonlinear and repeatability (type B) and the standard deviations (type A).

The u_c values for two infrared tympanic thermometers at five observations are listed in Table 7.

According to Equation (9), the values of u_c1 are calculated at 34.5, 36.0, 37.5, 39.0 and 40.5 °C of the standard temperature for MC-510 thermometer. The numeric values are 0.2519, 0.2499, 0.2316, 0.1851 and 0.2318 °C, respectively. For the polynomial form of calibration equation, the combining standard uncertainty calculated by Equation (10) to evaluate at these standard temperatures were 0.2685, 0.2324, 0.2321, 0.2246 and 0.2741, respectively. The result indicated that the polynomial calibration did not decrease the uncertainty for the MC-510 thermometer.

The values of u_c are calculated at 34.5, 36.0, 37.5, 39.0 and 40.5 °C of the standard temperature for IRT-3020 thermometer. They are found to be 0.1491, 0.0873, 0.0872, 0.089 and 0.1324 by Equation (9), respectively. For the linear calibration equation, the combined standard uncertainty evaluated at these standard temperatures was 0.1395, 0.0965, 0.0965, 0.0967 and 0.1394 by Equation (10), respectively. This result indicated that the calibration equation did not improve the uncertainty of the IRT-3020 thermometer.

This study evaluated the accuracy and uncertainty for two types of infrared tympanic thermometers. If the reading values of both thermometers were recognized as the true values of tympanic temperature, the accuracy of these thermometers could meet the requirement of the ASTM standards in the temperature range below 37 °C. At the standard temperature of 37.5 °C, there was wide variation in measurement in both thermometers. The average value of several measurements could be used to improve the accuracy. That is, more data needs to be taken at this temperature point to ensure the accuracy. At the temperature range above 37.5 °C, the reading values of both types of infrared tympanic thermometer need to be transformed by calibration equation to improve their accuracy.

The errors of measurement could be classified as systematic and random errors. The systematic errors could be expressed as errors and improved by its calibration equation. These procedures have been studied [18,26]. The random errors can only be improved by selecting of sensing elements or increasing sample numbers. The quantitative expression of this random error has been established in the international standards [14].

In the previous study, the uncertainty of measurement for thermometer, hygrometer and rough rice moisture meter could be improved by establishing the adequate calibration Equations [18, 19, 26]. In this study, the measuring errors could be reduced by the calibration equations. That is the systematic errors could be decreased. However, the uncertainty of measurement for two infrared tympanic thermometers could not be improved by the calibration equations. The reason could be explained by the form of predictive uncertainty [Equation (4)]. The estimated values of standard deviations for calibration, Equations (11) and (12), were 0.1965 and 0.0696, respectively. These numeric values are all almost higher than the standard deviation of replicates for different standard temperature and ambient temperature. That is the reason that the predictive uncertainty calculated by Equation (4) is almost higher than that of standard deviations.

Many studies have been reported that compare the performance of the rectal thermometer and infrared tympanic thermometer [4,5,7]. However, the performance of accuracy and uncertainty of the traditional mercury glass thermometer and the infrared tympanic thermometer were not mentioned by these researchers. That may be the reason for the inconsistency of their results.