Error Analysis in Physics

Every measurement has some error in it. Error analysis is a method to calculate this error and to determine if it is significant enough to nullify any conclusions made from the data collected during the experiment. An error analysis consists of four steps: 1) estimating absolute error for each measuring instrument used during an experiment; 2) Calculating the percentage and combination error for each data point; 3) Using the percentage or combination error to construct error bars for each data point; 4) Determining how much confidence should be given to the conclusions drawn from the experiment. Each of these steps is described below.

Step 1: Estimating Absolute Error

Absolute error is a pessimistic estimate of the error associated with each measuring device. At its smallest it is ½ of the smallest division used on the measuring device. However larger values are common. Below is a table which list some common measuring devices used in physics laboratories and their most optimistic absolute error.

Instrument / Absolute Error
Metric Ruler or Meter Stick / + or - .0005 meters or .05 cm
Triple beam balance / + or - .005 grams
Electric scale / + or - .0005 grams
Stopwatch / + or - .1 second
CBL and Photogate / + or - .0005 seconds
Spring Scale / + or - .1 N
CBL and Force Probe / + or - .001 N

Step 2: Calculating Percentage and Combination Error

Percentage error converts the error inherent in a measurement into a percentage. If a data point requires several measurements to becalculated then the percentage errors for each measurement are added to get the combination error of the data point. Percentage error is calculated using the equation: percentage error = (absolute error / measurement) x 100%. Combination error is simply the sum of all the percentage errors of the quantities involved in the calculation of the data point. To be clear, each data point should have its own percentage error. A data point that requires a calculation such as velocity or acceleration require adding thepercentage errors to get a combination error.

The example below shows the calculation of the combination error for velocity data points in which the distance measurement had an absolute error of .1 m and the time measurement had an absolute error of .05 seconds.

Distance Measurement (m) / Time Measurement (seconds) / % Error Distance / % Error Time / Combination error
1.0 m / 1.05 s / (.1/1) x 100 =10% / (.05/1.05) x 100 =4.7 % / 10% + 4.7% = 14.7 %
2.2 m / 3.10 s / (.1/2.2) x 100 =4.5% / (.05/ 3.10) x 100 = 1.6 % / 4.5% + 1.6% =6.1 %
3.2 m / 5.22 s / (.1/3.2) x 100 =3.1 % / (.05/ 5.22) x 100 = .95% / 3.1% + .95% =4.05%
4.4 m / 7.23 s / (.1/4.4 )x 100 = 2.2 % / (.05 / 7.23) x 100 = .69% / 2.2% + .69% = 2.89%
5.0 m / 9.15 s / (.1/5.0) x100 = 2% / (.05/9.15) x 100
= .54 % / 2% + .54% = 2.54%

Step 3: Using Percentage or Combination Error to Construct Error Bars:

Error bars are placed on the graph and used to represent the actual range of values for a data point. They are calculated by multiplying the data point value by the percentage or combination error to get a delta value which tells how much higher or lower the actual measurement could be. The delta value is then added and subtracted to the data point to get the maximum and minimum values for the point. These are then placed on the graph and error bars are drawn between the data point and these values. Below the delta, maximum and minimum values are calculated for the first three data points from the above data. The velocities have also been calculated.

Velocity / Delta Calculation / Maximum Value / Minimum Value
1/ 1.05 = .95 m/s / 14.7% x .95 m/s = .147 x .95 = .14 m/s / .95 m/s + .14 m/s = 1.09 m/s / .95 m/s - .14 m/s = 0.80 m/s
2.2 m / 3.10 s = 0.71 m/s / 6.1 % x 0.71 m/s = .061 x .71 m/s = .043 m/s / 0.71 m/s +.043 m/s = 1.1 m/s / 0.71 m/s - .043 m/s= .68 m/s
3.2 m/ 5.5 s= .61 m/s / 4.05% x .61 m/s = .0405 x .61 m/s = .025 m/s / .61 m/a +.025 m/s = 0.63 m/s / .61 m/s - .025 m/s = 0.58 m/s

The above maximum and minimum points would be added to the graph and error bars would be drawn. These bars represent the range of possible values for the data point.

Step 4: Determining Confidence in Conclusions:

The confidence in the data is determined by the amount of overlap in the error bars, the more overlap the less confidence. If the error bars overlap one another so that direct and inverse relationship trend lines fit the data, report a low confidence in the conclusions of the experiment. If the error bars allow only a direct or inverse relationship trend line to fit the data then moderate confidence in the conclusions is warranted. If the error bars allow only a direct or inverse relationship and only a linearor curved trend lineto fit the data then there is high confidence in the conclusions. An exception to these rules is if a horizontal trend line is produced. Here the confidence in the conclusions depends on the percentage or combination error with less than 10% being high confidence, 11-20% being moderate confidence and more than 20% being low confidence.