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- The goal in a statistical research
- Layout and interpretation of the survey's results
- What the error margin is
- Functionality of the error margin
- Variability in a population
- Calculation of the error margin
- Final remarks
We start a statistical research to find out the representative value of some characteristic in a population. In other words, by collecting data in a small section of a huge population we will be able to project the characteristics of the whole population (also called 'universe'). Check this example: in the year 2010 in a country called Omega two parties contend for the presidency, Cyan versus Amber. Population size is 10 million voters, and each person chooses either Omega or Amber. BetaStats, a survey firm, has surveyed a sample of 600 voters and has found that 360 people are is voting for Cyan and 240 for Amber.
Layout and interpretation of the survey's results
Let's check now the layout of the information collected in the survey. First of all, we note that 60% of the sample chooses Cyan and 40% chooses Amber. Here's the report:
Betastats – Presidential elections survey Omega 2010
Report
- Cyan: 60%
- Ambar: 40%
- Number of persons surveyed: 600
- Error margin: ±4%
The error margin
A solid theoretical model supports the concept of the error margin, including several algebariv operations which give structure to the analysis. I already wrote a paper referring to this, and the same paper shows every step necessary to build a formula for the sample size. The error margin is built initially as an algebraic tool.
The error margin originates in the concept of the the confidence interval. In few words, the CI is formally built to contain the actual value of some statistical characteristic with a defined likelihood, which is generally 95% of confidence (we can say also a 5% risk).
Let's see a very simple example: if it is attended that men in Omega have an average height of 6', a CI at a level of 5% may take the form of the range 5'9" to 6'1", so the amplitude will be 1". If this CI or any other confidence interval having the value 6' could be prepared with the sample data, then there will be no basis to refuse the hypothesis of the average height being 6'.
Let's come back to the easy part of this matter. Transform the CI in such a way to turn the center of the interval in the unit, i.e., number 1 or 100%. The distance from the center of the interval to any of its boundaries is what we call the error margin. For example, once 6' has been turned into 100%, we see an interval from 97.37% to 102.63%, so that the amplitude is 2.63%, and we say that the error margin is ±2.63%. Thus, the error margin is just a way to show the amplitude of our transformed interval.
Functionality of the error margin
The error margin allows us to gauge the precision of our results. The most popular way to use the conept is to add / substract the value of the amplitude to the value measured in the research. Coming again to the Omega situation, Cyan will likely gain some figure between 64% and 56% of the votes, so that Amber will accordingly reach a figure between 36% and 44%. Actually, this sort of interpretation of the theoretical approach skips some important issues, so that I prefer to consider that there exist better interpretations and applications of this amplitude. Anyway, I will not cover those interpretations here, just mention them.
I rather prefer to follow the general interpretation of the error margin concept in this discussion. Let's say then that our research states that Cyan's final performance will take a value between 56% and 64%, and Amber's will reach the 36% to 44% range.
Variability
An underlying concept in the study of confidence intervals is the variability. We note this as the ratio σ/μ (square root of the variance divided by the average value). This variability is a measure of the average separation of a value regarding the population average value. Let's suppose that the variability of the height in Omega men was 7%. In this case, the average separation from the middle (or average) value will be that of 5", so that tall men will reach an average height of 6'5" and short men in average will be 5'5" tall.
Error margin calculation
In building our model, the following expression came out:
in which error represents the error margin, n is the number of individuals surveyed, Z1-α is a constant which depends on the confdence we choose for the interval, and σ/μ is variability.
Thde expression says that there stands a trade-off between the error margin and the number of individuals surveyed. This originates in Z1-α and σ/μ being constants while the error margin and the size of the sample are engaged in an infinite number of combinations.
Therefore, we may choose to fix the error margin and calculate the resulting adequate sample size, or to fix the sample size and then just calculate the resulting error margin. An example for the first scenario: suppose there is a populational quality with a variability of 50% and also that a confidence of 50% is set. Then, if the error margin is set at ±4%, then a sample size of 600 individuals is required.
Now to the second scenario. If variability is 50%, confidence is set to 95% and sample size is set to 400 individuals, then the resulting error margin will be ±4.90%.
Final notes
The error margin is a measure of the accuracy of the statistical analysis and report performed in a statistical research. Sometimes a specified error margin will be set so that the sample size will be calculated afterwards, and sometimes the sample size will be pre-set, so that the error margin will be calculated afterwards. Generally the choice is the former: the allowed error is specified and then the size of the sample is found.
AugustoRufasto
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