To calculate the mean and standard deviation of the error probability density function for your temperature measurement system, we need to consider the contributions from each component: the thermistor, the bridge, and the recorder. Each of these components has its own mean and standard deviation, which we will combine to find the overall error in the system.
Understanding the Components
In a typical temperature measurement system, the thermistor measures the temperature, the bridge converts this measurement into a usable signal, and the recorder logs the data. Each of these components introduces some level of error, which we can model statistically.
Model Equations
Assuming that the errors from each component are independent and normally distributed, we can use the following relationships:
- Mean of the total error: This is simply the sum of the means of the individual components.
- Standard deviation of the total error: This is calculated using the square root of the sum of the squares of the individual standard deviations.
Calculating the Mean Error
Let’s denote the mean values of the thermistor, bridge, and recorder as μ₁, μ₂, and μ₃, respectively. The mean error (μ_total) can be calculated as:
μ_total = μ₁ + μ₂ + μ₃
Calculating the Standard Deviation of the Error
For the standard deviation, let’s denote the standard deviations of the thermistor, bridge, and recorder as σ₁, σ₂, and σ₃. The total standard deviation (σ_total) is given by:
σ_total = √(σ₁² + σ₂² + σ₃²)
Example Calculation
Assuming you have the following values from Table Prob. 7:
- Thermistor: μ₁ = 320 K, σ₁ = 1 K
- Bridge: μ₂ = 0 K, σ₂ = 0.5 K
- Recorder: μ₃ = 0 K, σ₃ = 0.2 K
Now, let’s calculate the mean and standard deviation of the error:
Mean Error Calculation
Using the means:
μ_total = 320 K + 0 K + 0 K = 320 K
Standard Deviation Calculation
Now for the standard deviation:
σ_total = √(1² + 0.5² + 0.2²) = √(1 + 0.25 + 0.04) = √(1.29) ≈ 1.14 K
Final Results
For a true input temperature of 320 K, the mean of the error probability density function is:
Mean Error: 320 K
And the standard deviation of the error probability density function is approximately:
Standard Deviation: 1.14 K
This means that while the system is designed to measure a temperature of 320 K, the actual readings can vary, with a standard deviation of about 1.14 K due to the combined errors from the thermistor, bridge, and recorder. This understanding is crucial for assessing the reliability of your temperature measurements.
