Observational error

They therefore alter the value observed and repeated identical measurements do not reduce such errors.

For example, length measurements with a ruler accurately calibrated in whole centimeters will be subject to random error with each use on the same distance giving a slightly different value resulting limited precision; a metallic ruler the temperature of which is not controled will be affected by thermal expansion causing an additional systematic error resulting in limited accuracy.

Random errors show up as different results for ostensibly the same repeated measurement.

Random error can be caused by unpredictable fluctuations in the readings of a measurement apparatus, or in the experimenter's interpretation of the instrumental reading; these fluctuations may be in part due to interference of the environment with the measurement process.

The higher the precision of a measurement instrument, the smaller the variability (standard deviation) of the fluctuations in its readings.

If you consider an experimenter taking a reading of the time period of a pendulum swinging past a fiducial marker: If their stop-watch or timer starts with 1 second on the clock then all of their results will be off by 1 second (zero error).

Distance measured by radar will be systematically overestimated if the slight slowing down of the waves in air is not accounted for.

Systematic errors may also be present in the result of an estimate based upon a mathematical model or physical law.

For instance, the estimated oscillation frequency of a pendulum will be systematically in error if slight movement of the support is not accounted for.

If this cannot be eliminated, potentially by resetting the instrument immediately before the experiment then it needs to be allowed by subtracting its (possibly time-varying) value from the readings, and by taking it into account while assessing the accuracy of the measurement.

Hopings systematic error is present if the stopwatch is checked against the 'speaking clock' of the telephone system and found to be running slow or fast.

Clearly, the pendulum timings need to be corrected according to how fast or slow the stopwatch was found to be running.

Such errors cannot be removed by repeating measurements or averaging large numbers of results.

Stochastic errors added to a regression equation account for the variation in Y that cannot be explained by the included Xs.

Thus, the exact formulation of a survey question is crucial, since it affects the level of measurement error.

However, if one or more independent variables is measured with error, then the regression coefficients and standard hypothesis tests are invalid.