defination of systematic error

Systematic errors are biases in measurement which lead to measured values being systematically too high or too low. See also biased sample and errors and residuals in statistics. All measurements are prone to systematic error. A systematic error is any biasing effect, in the environment, methods of observation or instruments used, which introduces error into an experiment and is such that it always affects the results of an experiment in the same direction. Distance measured by radar will be in error if the slight slowing down of the waves in air is not accounted for. The oscillation frequency of a pendulum will be in error if slight movement of the support is not accounted for. Incorrect zeroing of an instrument leading to a zero error is an example of systematic error in instrumentation. So is a clock running fast or slow. See also observational error and errors and residuals in statistics.
Constant systematic errors are very difficult to deal with, because their effects are only observable if they can be removed. Such errors cannot be removed by repeating measurements or averaging large numbers of results. A common means to remove systematic error is the observation of a known process, i.e. through calibration. Another means to remove systematic error is by a subsequent measurement with a more sophisticated experiment equipment.
In statistics and optimization, the concepts of error and residual are easily confused with each other.
Random Error is a misnomer; an error is the amount by which an observation differs from its expected value; the latter being based on the whole population from which the statistical unit was chosen randomly. The expected value, being the average of the entire population, is typically unobservable. If the average height of 21-year-old men is 5 feet 9 inches, and one randomly chosen man is 5 feet 11 inches tall, then the "error" is 2 inches; if the randomly chosen man is 5 feet 7 inches tall, then the "error" is −2 inches. The nomenclature arose from random measurement errors in astronomy. It is as if the measurement of the man's height were an attempt to measure the population average, so that any difference between the man's height and the average would be a measurement error.
A residual, on the other hand, is an observable estimate of the unobservable error. The simplest case involves a random sample of n men whose heights are measured. The sample average is used as an estimate of the population average. Then we have:

The difference between the height of each man in the sample and the unobservable population average is an error, and

The difference between the height of each man in the sample and the observable sample average is a residual.

Residuals are observable; errors are not.
Note that the sum of the residuals within a random sample is necessarily zero, and thus the residuals are necessarily not independent. The sum of the errors need not be zero; the errors are independent random variables if the individuals are chosen from the population independently.

Errors are often independent of each other; residuals are not independent of each other (at least in the simple situation described above, and in many others).

Definitions
All experimental uncertainty is due to either random errors or systematic errors. Random errors are statistical fluctuations (in either direction) in the measured data due to the precision limitations of the measurement device. Random errors usually result from the experimenter's inability to take the same measurement in exactly the same way to get exact the same number. Systematic errors, by contrast, are reproducible inaccuracies that are consistently in the same direction. Systematic errors are often due to a problem which persists throughout the entire experiment.

Note that systematic and random errors refer to problems associated with making measurements. Mistakes made in the calculations or in reading the instrument are not considered in error analysis. It is assumed that the experimenters are careful and competent!