dA = Am− At
Where:
The absolute value of the error cannot be determined exactly because the true value is unknown.
(b) Product of Two Quantities:
Consider two quantities a1 and a2. The product is expressed as A = a1 × a2. By taking the logarithm of both sides and differentiating with respect to A, the resultant limiting error can be calculated. The resultant error is the sum of the relative errors in the measurement of terms. For power factors, the relative error would be n times in this case.
Gross Errors
Gross errors result from human mistakes during reading, recording, or calculating. Examples include reading a value incorrectly, such as mistaking 21 for 31. Gross errors can be minimized by:
Systematic Errors
Systematic errors are consistent and repeatable errors that arise due to the following:
Random Errors
Even after accounting for systematic errors, some errors remain, which are known as random errors. These errors may arise from unknown or unpredictable factors, and although they cannot be fully eliminated, their effects can be reduced through careful experimentation.
When making any measurement, some degree of uncertainty is inevitable, regardless of the precision of the measuring tool. Estimating and reporting this uncertainty is crucial for providing a complete and accurate description of the measurement.
Precision and Accuracy of the Instrument: The inherent precision and accuracy of the measuring instrument directly influence the uncertainty of the measurement. Precision refers to the smallest unit that the instrument can reliably measure, while accuracy refers to how close the measurement is to the true value.
Experimental Conditions: Various factors, such as environmental conditions, the skill of the experimenter, and the specific characteristics of the object being measured, can affect the uncertainty. For example:
In both examples, the uncertainty is larger than the smallest divisions marked on the measuring tool (e.g., 1 mm for the meter stick and 0.05 mm for the Vernier caliper), showing that other factors besides the instrument's resolution contribute to the overall uncertainty.
Reporting Uncertainty
When reporting a measurement with its uncertainty, it’s important to clearly communicate what the uncertainty represents. The format is typically:
Measurement = (measured value ± standard uncertainty) unit of measurement
The "± standard uncertainty" indicates an approximate 68% confidence interval, meaning that there is a 68% chance that the true value lies within this range, based on the normal distribution of possible errors.
Example:
Diameter of Tennis Ball:
Diameter = 6.7 ± 0.2cm
This notation tells us that the measured diameter is 6.7 cm, with an uncertainty of 0.2 cm, indicating that the true diameter is likely between 6.5 cm and 6.9 cm, with about 68% confidence.
In conclusion, the experimenter must evaluate and quantify the uncertainty based on all the influencing factors and report it in a way that accurately reflects the measurement's reliability.
Calculate the Mean:
Determine Deviations:
Square the Deviations:
Sum the Squared Deviations:
Calculate the Variance:
Find the Standard Deviation:
Let the N measurements be x1, x2,…,xN, and let the mean of these values be . The deviation for each measurement is given by . The standard deviation s is calculated as follows:
Where:
Let's consider a previous example where the average width is 31.19 cm. Suppose the deviations of five measurements are:
The standard deviation s can be calculated as:
For a large number of measurements, the distribution of the data tends to follow a bell-shaped curve known as the Gaussian or Normal Distribution, where the standard deviation provides a measure of the spread of the data around the mean.
Where:
The standard error is smaller than the standard deviation by a factor of √N, reflecting that the uncertainty of the mean decreases with the number of measurements.
Example Calculation
In a previous example, the standard deviation s was 0.12 cm, and the number of measurements N was 5.
To find the standard error:
Thus, the average value should be reported as:
Average paper width = 31.19 ± 0.05cm
Handling Anomalous Data:
Formula for Fractional Uncertainty
In the previous example:
Interpreting Fractional Uncertainty:
Defining Significant Figures
Non-Zero Digits: Always significant.
Zeroes Between Non-Zero Digits: Always significant.
Leading Zeroes: Not significant. They only place the decimal point.
Trailing Zeroes in Decimal Numbers: Significant.
Trailing Zeroes in Whole Numbers: Ambiguous without additional context. They may or may not be significant.
Example: If you measure the radius of a circular field as 9 meters (1 significant figure) and calculate the area as A=πr2, the result from a calculator might be 254.4690049 m². Since the radius is only known to 1 significant figure, the area should be reported as 3×102 m² (1 significant figure).
1 Significant Figure: Implies a relative uncertainty of about 10% to 100%.
2 Significant Figures: Implies a relative uncertainty of about 1% to 10%.
3 Significant Figures: Implies a relative uncertainty of about 0.1% to 1%.
Note: For practical purposes, uncertainties in the last digit of significant figures can be approximated. The example with 99, implying ±1, highlights that the exact uncertainty can vary based on rounding conventions but gives a general sense of precision.
In summary, significant figures help in conveying the precision of measurements and ensure that calculations reflect the limitations of the data accurately.
In scientific and experimental contexts, determining whether measurements agree with each other or with theoretical predictions involves evaluating their uncertainty ranges. Here’s a concise overview:
Agreement with Theoretical Predictions:
Consistency Between Measurements:
Discrepancy:
Avoiding Unethical Practices:
In summary, agreement is determined by overlapping uncertainty ranges, while discrepancies arise when these ranges do not overlap. The interpretation should be grounded in the context of the uncertainties involved, and any significant discrepancies should be investigated to identify potential sources of error.
3 videos|39 docs|22 tests
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1. What is the difference between limiting errors and guarantee errors in error analysis in electrical engineering? |
2. How is the standard deviation used in error analysis for electrical engineering measurements? |
3. What is the significance of the standard deviation in error analysis for electrical engineering experiments? |
4. How can the standard error of the mean be calculated in error analysis for electrical engineering experiments? |
5. How can anomalous data be identified and addressed in error analysis for electrical engineering experiments? |
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