Bias (statistics)
Statistical bias is a systematic tendency which causes differences between results and facts. The bias exists in numbers of the process of data analysis, including the source of the data, the estimator chosen, and the ways the data was analyzed. Bias may have a serious impact on results, for example, to investigate people's buying habits. If the sample size is not large enough, the results may not be representative of the buying habits of all the people. That is, there may be discrepancies between the survey results and the actual results. Therefore, understanding the source of statistical  bias can help to assess whether the observed results are close to the real results. Bias can be differentiated from other mistakes such as accuracy (instrument failure/inadequacy), lack of data, or mistakes in transcription (typos). Bias implies that the data selection may have been skewed by the collection criteria. Bias does not preclude the existence of any other mistakes. One may have a poorly designed sample, an inaccurate measurement device, and typos in recording data simultaneously. Also it is useful to recognize that the term “error” specifically refers to the outcome rather than the process (errors of rejection or  acceptance of the hypothesis being tested). Use of "flaw" or "mistake" to differentiate procedural errors from these specifically defined outcome-based terms is recommended. Mathematical Form. Statistical bias is a feature of a statistical technique or of its results whereby the expected value of the results differs from the true underlying quantitative parameter being estimated. The bias of an estimator of a parameter should not be confused with its degree of precision, as the degree of precision is a measure of the sampling error. The bias is defined as follows: let formula_1 be a statistic used to estimate a parameter  formula_2, and let formula_3 denote the expected value of formula_1. Then, is called the bias of the statistic formula_1 (with respect to formula_2). If formula_8, then formula_1 is said to be an unbiased estimator of formula_2; otherwise, it is said to be a biased estimator of formula_2. The bias of a statistic formula_1 is always relative to the parameter formula_2 it is used to estimate, but the parameter formula_2 is often omitted when it is clear from the context what is being estimated. Types. Statistical bias comes from all stages of data analysis. The following sources of bias will be  listed in each stage separately. Data selection. Selection bias involves individuals being more likely to be selected for study than others, biasing the sample. This can also be termed selection effect, sampling bias and "Berksonian bias". Hypothesis testing. Type I and type II errors in statistical hypothesis testing leads to wrong results. Type I error happens when the null hypothesis is correct but is rejected. For instance, suppose that the null hypothesis is that if the average driving speed limit ranges from 75 to 85 km/h, it is not considered as speeding. On the other hand, if the average  speed is not in that range, it is considered speeding. If someone receives a ticket with an average driving speed of 7 km/h, the decision maker has committed a Type I error. In other words, the average driving speed meets the null hypothesis but is rejected. On the contrary, Type II error happens when the null hypothesis is not correct but is accepted. Estimator selection. The bias of an estimator is the difference between an estimator's expected value and the true value of the parameter being estimated. Although an unbiased estimator is theoretically preferable to a biased estimator , in practice, biased estimators with small biases are frequently used. A biased estimator may be more useful for several reasons. First, an unbiased estimator may not exist without further assumptions. Second, sometimes an unbiased estimator is hard to compute. Third, a biased estimator may have a lower value of mean squared error. Interpretation. Reporting bias involves a skew in the availability of data, such that observations of a certain kind are more likely to be reported.