Question

give a example 0f a measurment error .describe a situation in which it might occur and why is poses a problem for statistic if the sample size increases and everything els remains the same ,in what will the confidence change?

Answer 1

Measurement Error (also called Observational Error) is the difference between a measured quantity and its true value. It includes random error (naturally occurring errors that are to be expected with any experiment) and systematic error

For example, let’s say you were measuring the weights of 100 marathon athletes. The scale you use is one pound off: this is a systematic errorthat will result in all athletes body weight calculations to be off by a pound. On the other hand, let’s say your scale was accurate. Some athletes might be more dehydrated than others. Some might have wetter (and therefore heavier) clothing or a 2 oz. candy bar in a pocket. These are random errors and are to be expected. In fact, all collected samples will have random errors — they are, for the most part, unavoidable.

When discussing the statistical properties of an exam, one may hear the term “error” or measurement error used by psychometricians. Error can be considered information contributing to a person’s exam score beyond the person’s true or actual ability. So from computational and classical test theory perspective, error is

E = O – T               or            Error = Observed – True

The Observed score is the actual score on the exam and True score is the person’s actual ability. Error is the difference between observed and true scores.

Error can be random or systematic. According to Crocker and Algina (1986), Introduction to Classical and Modern Test Theory, random error typically occurs during one administration. This may include guessing, misreading a question or a candidate not feeling well. In this instance of random error, the error would not likely occur during a subsequent administration. Systematic errors are typical attributes of the person or the exam that would occur across administrations. These errors typically do not have much to do with the content being measured. An example could be an exhaustive item with excess verbiage that asks a simple math problem and the simple math problem is what is intended to be measured, not the candidate’s ability to sort through the verbiage.

Why is measuring error important?

Reliability, theoretically speaking, is the relationship (correlation) between a person’s score on parallel (equivalent) forms. As more error is introduced into the observed score, the lower the reliability will be. As measurement error is decreased, reliability is increased. With that said, administering two forms of an exam to one candidate to calculate reliability is not practical.

Because creating perfectly parallel exam forms and administering two forms for a given candidate is not practical, we estimate reliability using a single form methodology. Coefficient Alpha, and its variations, has been a popular reliability estimate used for single exam form administrations.

Why is all of this important? Validity.

If systematic errors occur, there is a threat to the validity of the exam program. Having an exam that measures something other than what it is intended to will result in inaccurate exam results and inappropriate score interpretations. This would be considered a major flaw in the program.

Developing certification test requires many steps. It is these steps that reduces the probability of systematic error. This includes defining the content of the exam (i.e., job analysis), writing and reviewing items with multiple and separate SMEs, conducting standard setting and equating studies, reviewing item and exam form statistics, and conducting proper scoring procedures. It is very important that organizations develop certification exams using best practices to avoid threats from error.

I think by the increase of sample size the mean value would be more and more accurate and also precise hence measurement errors are going to be reduced gradually thus increasing the confidence level

The size of our sample dictates the amount of information we have and therefore, in part, determines our precision or level of confidence that we have in our sample estimates. An estimate always has an associated level of uncertainty, which depends upon the underlying variability of the data as well as the sample size. The more variable the population, the greater the uncertainty in our estimate. Similarly, the larger the sample size the more information we have and so our uncertainty reduces.

Suppose that we want to estimate the proportion of adults who own a smartphone in the UK. We could take a sample of 100 people and ask them. Note: it’s important to consider how the sample is selected to make sure that it is unbiased and representative of the population

The larger the sample size the more information we have and so our uncertainty reduces.

If 59 out of the 100 people own a smartphone, we estimate that the proportion in the UK is 59/100=59%. We can also construct an interval around this point estimate to express our uncertainty in it, i.e., our margin of error. For example, a 95% confidence interval for our estimate based on our sample of size 100 ranges from 49.36% to 68.64% (which can be calculated using our free online calculator). Alternatively, we can express this interval by saying that our estimate is 59% with a margin of error of ±9.64%. This is a 95% confidence interval, which means that there is 95% probability that this interval contains the true proportion. In other words, if we were to collect 100 different samples from the population the true proportion would fall within this interval approximately 95 out of 100 times.

What would happen if we were to increase our sample size by going out and asking more people?

Suppose we ask another 900 people and find that, overall, 590 out of the 1000 people own a smartphone. Our estimate of the prevalence in the whole population is again 590/1000=59%. However, our confidence interval for the estimate has now narrowed considerably to 55.95% to 62.05%, a margin of error of ±3.05% – see Figure 1 below. Because we have more data and therefore more information, our estimate is more precise.