Basic Concepts of Experimental Error

Basic Concepts

• Experimental error is the difference between a measurement and the true value or between two measured values.
• Experimental error is demonstrated by accuracy and precision.
• Accuracy of a measurement refers to how close the measured value is to the true or accepted value.
• Precision refers to how close together a group of measurements are to each other.
• Precision has nothing to do with the true value of a measurement
• Precision is sometimes referred to as repeatability or reproducibility.
• A highly reproducible measurement gives values close to each other.
• Accuracy can be determined by a single measurement.
• Precision can only be determined with multiple measurements.

Types of Experimental Errors

• Experimental errors are not mistakes in calculations or miscalculations.
• Systematic errors affect the accuracy of a measurement.
• Systematic errors are one-sided.
• Repeated measurements yield results that differ from the true or accepted value by the same amount
• Systematic errors cannot be improved by repeating measurements
• Systematic errors are difficult to detect but can be reduced by refining measurement method or technique.
• Random errors affect the precision of a measurement.
• Random errors are two-sided errors.
• Repeated measurements fluctuate above and below the true value.
• Random errors are easily analyzed by statistical analysis.
• Random errors can be reduced by refining the measurement method or technique.

Causes of Systematic Error

• Faulty calibration of measuring instruments or poorly maintained instruments can cause systematic error.
• Faulty reading of instruments by the user, also called parallax error. This consistently gives high or low readings
• "Zero error" occurs when an instrument gives a reading when the true reading is zero.
• Systematic errors can be reduced by recalibrating the instrument or subtracting/adding an error value to data
• Taking more measurements does not reduce systematic error

Causes of Random Error

• Unpredictable fluctuations in the readings of measurement apparatus.
• The experimenter's interpretation of the instrumental reading.
• Random errors can be reduced by repeating measurements several times and taking the average.
• Statistical analysis can be used to analyze random errors.

Calculating Experimental Error

• Reporting experimental results should describe accuracy and precision of the experimental measurements.
• Significant figures estimate the precision of a measurement. The number of significant figures is the number of figures that are known with some degree of reliability.
• The number 13.2 has 3 significant figures, 13.20 has 4.

Rules for Significant Digits

• All non-zero digits are significant.
• Zeros: leading zeros are not significant, captive zeros are significant, trailing zeros are significant only if the number contains a decimal point
• When multiplying or dividing, the answer should have as many significant digits as the measurement with the fewest significant digits.
• When adding or subtracting, the answer should have as many decimal places as the measurement with the fewest decimal places.

Rules for Rounding Off Numbers

• If the digit to be dropped is greater than 5, the last retained digit is increased by 1.
• If the digit to be dropped is less than 5, the last retained digit remains the same.
• If the digit to be dropped is 5, and there are other non-zero digits following it, the last retained digit is increased by 1.
• If the digit to be dropped is 5, and only zeroes follow it, the last retained digit is increased by 1 if it is odd, but remains the same if even.

Percent Error

• Percent error measures the accuracy of a measurement.
• % Error = [(Measured value - True value)/ True value] x 100

Mean and Standard Deviation

• Mean is the average of a set of data.
• Mean = sum of data points / number of data points
• Standard deviation tells how spread out measurements are from the average.
• A low standard deviation means measurements are close to the average, while a high standard deviation means measurements are spread out from the average

Reporting the Results of an Experimental Measurement

• The result of an experimental measurement should be reported with two parts:
  • Best estimate of measurement (e.g. mean)
  • Variation of the measurements (e.g. standard deviation of the measurements)
  • Measured quantity = x ± standard deviation

Straight Line Graphs

• Straight line graphs display data variables and trends clearly, aiding in predictions.
• They can display multiple dependent variables against one independent variable.
• Equation of a straight line: A = mB + c, where m is slope, and c is y-intercept

SI Units

• SI units has seven base quantities and base units
• Quantities such as length, mass, time, temperature, amount of substance, electric current, luminous intensity can be derived.
• There are derived SI units whose expressions can be calculated using seven base units. For example, Area = Length x Length, Velocity = Distance/time.

SI Prefixes

• SI prefixes are used to represent multiples or submultiples of a unit.