# Margin of error and sample size relationship planets One population proportion, p; Difference in two population proportions for independent When sample size increases, margin of error decreases. The confidence level. . “Do you think there is intelligent life on other planets? Note: Lower. So the margin of error for most polls is 2E with a confidence of 95%. Using E as your margin of error gives you a confidence interval of. In statistics, the two most important ideas regarding sample size and margin of error are, first, sample size and margin of error have an inverse relationship; and .

However, when the sample size is small and it is not given that the distribution is normal, then you cannot conclude anything about the normality of the distribution and neither z-score nor t-score can be used. When finding the critical value, confidence level will be given to you. Here are the steps for finding critical value: First, find alpha the level of significance.

Critical probability will depend on whether we are creating a one-sided confidence interval or a two-sided confidence interval. This will be the critical value. To find these critical values, you should use a calculator or respective statistical tables. Sample Standard Error Sample standard error can be calculated using population standard deviation or sample standard deviation if population standard deviation is not known.

For sampling distribution of means: Having looked at everything that is required to create the margin of error, you can now directly calculate a margin of error using the formula we showed you earlier: Confidence level and marginal of error As the confidence level increases, the critical value increases and hence the margin of error increases.

This is intuitive; the price paid for higher confidence level is that the margin of errors increases. If this was not so, and if higher confidence level meant lower margin of errors, nobody would choose a lower confidence level.

There are always trade-offs! Sample standard deviation and margin of error Sample standard deviation talks about the variability in the sample. The more variability in the sample, the higher the chances of error, the greater the sample standard error and margin of error. Sample size and margin of error This was discussed in the Introduction section.

It is intuitive that a greater sample size will be a closer representative of the population than a smaller sample size. Hence, the larger the sample size, the smaller the sample standard error and therefore the smaller the margin of error. Margin of Error Practice Problems Example 1 25 students in their final year were selected at random from a high school for a survey. Identify the sample statistic. Identify the distribution — t, z, etc.

Since population standard deviation is not known and the sample size is small, use a t distribution.

### Basics: Margin of Error | ScienceBlogs

The critical t value for cumulative probability of 0. Find the sample standard error. Find margin of error using the formula: Example 2 students in Princeton University are randomly selected for a survey which is aimed at finding out the average time students spend in the library in a day.

Among the survey participants, it was found that the average time spent in the university library was 45 minutes and the standard deviation was 10 minutes. The population standard deviation is not known, but the sample size is large.

Therefore, use a z standard normal distribution. The critical z value for cumulative probability of 0. Example 3 Consider a similar set up in Example 1 with slight changes. You randomly select X students in their final year from a high school for a survey. What should be the value of X in other words, how many students you should select for the survey if you want the margin of error to be at most 0. Find the critical value. Find the sample standard error in terms of X.

Find X using margin of error formula: Thus, a sample of students should be taken so that the margin of error is at most 0. Conclusion The margin of error is an extremely important concept in statistics.

In a way it is more uninformative because when you are using the uniform prior you assume a linear scale or linear measuring instrument whereas the Beta prior doesn't even make that assumption.

Uninformative priors for more complicated problems can be found using transformation groups to find invariance or symmetry and making sure that results are the same when we scale or change equivalent groups of models. More information can be found here or you can order the full book on amazon.

By "p" in the equation, presumably you mean the size of the sample, not the size of the population from which the sample was drawn. Earlier in the article you used "population" to mean the latter.

### survey - How are margins of error related to confidence Intervals? - Cross Validated

Log in to post comments By BenL not verified on 22 Jan permalink Yep, p is the same sample size according to wikipedia. Nonetheless, this is a great article. Perhaps something on Bayesian statistics in the future? The Beta 1,1 is a uniform distribution. I'm not convinced by it as a prior: For me, that's a more important property for an "objective prior" when doing applied statistics. And you're also right about using the uniform.

I use it all the time because it is simpler to understand. What's wrong with assuming some kind of linearity in models anyways? And as I said, results are pretty much the same except for very small samples in which statistics are all over the place and only display uncertainty anyways.

Even Jaynes said it, "A useful rule of thumb [ From the standpoint of principle, however, they are important and need to be thought about a great deal" The point of Jeffrey's prior is mostly a theoretical one, it shows how bayesianism can display complete objectivity.

In practice though it doesn't make much difference. I might be able to take some of this to use. Look at the comments you got and you see why people who do not have any background in stats are just buffaloed by the jargon. I always try to get across that Margin of error really has little to nothing to do with the accuracy or precision of the data used in a poll or other statistic.

Its actually kind of a best case thing in general, "this is as good as you are going to get with this many samples Log in to post comments By Markk not verified on 23 Jan permalink I'm reviewing this in my Econometrics class right now! Thanks for the reminders.

I study physics and have only seen confidence level used, but it seems to be measuring the same things: By MaxPolun not verified on 23 Jan permalink This article is really awful. There's no need to perpetuate the frequentist party line any longer. Bayesian inference is more powerful, and much simpler to boot. People get a mistaken idea that probability is very difficult, but that's only because of the messy non-Bayesian way it is taught.

See "Making Hard Decisions" by Robert Clemen for a basic introduction to probability and decision analysis. Log in to post comments By AnotherBob not verified on 23 Jan permalink Wow, I'm glad bayesianism is starting to pick up. Last time I had a probability discussion on this blog, it was me and maybe another guy agains a bunch of frequencists who dismissed us and told us we were using non objective, non scientific mathematics.

This time arround it seems like the Bayesians are setting the tone of the discussion. Log in to post comments By BenE not verified on 24 Jan permalink BenE - A problem with using an objective prior is that there are several available, so which one do you choose?

There were several articles about Subjective and Objective Bayesianism in Bayesian Analysis last year: I guess Mark is having second thoughts about writing a post about Bayesian methods now.

But it's nice to see so many crawl out of the woodwork. Log in to post comments By Bob O'H not verified on 24 Jan permalink Last time I had a probability discussion on this blog, it was me and maybe another guy agains a bunch of frequencists who dismissed us and told us we were using non objective, non scientific mathematics. I believe I was one of the participants, but if so I don't recognize the message. At least my message, which is that these are different conceptions of the concept of probability, with different best use.

One reason frequentist probability can be preferred in science is that it is easy to extract from most models and be verified by observations. As I understand it, bayesian inference can like modal logics be able to say anything about any thing. But of course when it comes down to predictions that are amenable to real data this should not be a problem.

And neither can frequentist models be automatically trusted. If the event happens or is expected to happen a few times, the result is of limited value. Another reason frequentist probability can be preferred in science is that it can handle theoretical probabilities over infinite spaces. Kolmogorov's axioms for frequentist probability vs Cox's axioms for bayesian.

As I understand it, there are real problems to define bayesian probabilities that can be used in common derivations in quantum field theory. But certainly bayesian methods have scientific uses. It is one method to address the question of parsimony. Bayesian measures are commonly used to choose between different models in cosmology and cladistics for example. So personally I don't think bayesian methods are non-scientific or of no value.

I'm less certain about its use in models which can't be tested, like in proofs for gods. Or when it is conflated with other probabilistic conceptions. True, in most cases the both conceptions can agree on frequencies. But in other cases there are differences. I'm reminded of Wilkins recent discussions of species http: Species probability is a phenomena, that can be described by a concept; "A species is any lineage of organisms that is distinct from other lineages because of differences in some shared biological property" "the extent to which something is likely to happen or be the case".

But for species perhaps also probabilities no single actualization, conception, can cover all uses and details. And when we look at them that way, it becomes clear why none of them are sufficient or necessary for all species: I could as well end this with Wilkins' words: Like it or not I understand that there are debates in the bayesian approaches.

These debates seem to be hard to solve because they are often more philosophycal than mathematical and are rooted in the way we think about science including all sorts of epistomological considerations. In my opinion the problem goes farther than mathematics. I would go as far as saying that we have to include the physical laws of the universe into the equation when we are considering the objectivity of different priors.

I would venture that the only way to find the best, most objective prior for any model or hypothesis that represent something that exists in this universe is to find the one that excludes any bias, not on a mathematical or number theoretic dimention, but on physical dimentions of the universe. I'm not convinced it is possible to solve this problem without thinking about the continuity of space, time, movements, and acceleration and the relative isolation these properties give to things and events and the effect this has on their probability of occuring or existing somewhere and sometime without physical disturbances to stop them.

This link with physics is a little bit like a return to the geometrical interpretation of mathematics used in ancient greece which was more grounded in concrete physical representations. Regardless of all this philosophical babling, the bayesian approaches seems to allow more objectivity and more robustness than the frequencist approaches while being simpler.

Null hypothesis testing comes to mind as a nonsensical consequence of the frequencist approach. The statistics it gives are counter intuitive and can usually be manipulated in saying anything. I think some day we will have a set of priors and applicability rules for hopefully all real world problems and these won't allow for any biases and number manipulation like frequencist theories.

Log in to post comments By BenE not verified on 24 Jan permalink Regardless of all this philosophical babling, the bayesian approaches seems to allow more objectivity and more robustness than the frequencist approaches while being simpler. Considering that observations must select which model is correct, I have personally much more trust in frequentist probability.

## Margin of Error: What to Know for Statistics

It models a specific and general characteristic that is easy to extract and verify. In bayesian terms, my prior can be set high compared to a particular bayesian inference. I remember this argument. You discuss a relative error. That is an experimental error, that has to be checked and controlled. And strictly speaking, it doesn't add up, its the variance of the populations that decreases. I see nothing special about relative errors and other model or experimental defects.

In this case with a very tight variance relative errors become extremely important and must be controlled.

Otherwise the firm limit looses its meaning, as you suggest. Did you have anything to add? Hypothesis testing, however it is done, is important to science. This method by contradiction from dataand falsification by denying the consequent from datais what makes us able to reject false theories. It is the basis that distinguish science from merely, well, suggesting models by inductive inference.

The detrimental effects of null hypothesis testing in real applications are very very common. I see it all the time! I've actually seen taught in statistics classes that you shouldn't use too many data points when doing these tests because you'll always end up finding something significant!

The exact number of data point to use is left as a personal choice to the researcher. What the hell is that! This means having a big and representative sample becomes a bad thing. It means that given that a researcher has enough resources to get enough data, it's his choice whether he makes his results significant or not! These tests are useless! University professors have a huge incentive to publish their job is at risk and because of the dumb trust in these statistical tests, papers that show statistical significance in rejecting null hypotheses get published even when the real effect is very small.

Researchers use this flaw to fish for results when there's really nothing interesting to report. It is especially easy to do in the social sciences where the nature of their tests makes all sorts of biases available to exploit through null hypothesis testing.

I am personally familiar with this in the world of psychology and I can tell you that academics who have become pros at this kind of manipulation are the ones that are hired by universities because these institutions often rate candidates by the number of publications they have. And it's a vicious circle since these people later become the ones who rate papers to be accepted for publication. Since it is of their tradition, they blindly accept papers that reject some null hypothesis even though the results are uninteresting and not useful.

You wouldn't believe the crap that is published in psychology journals based on rejected null hypothesis. Null hypothesis testing is one of the most widely exploited and blatant flaw in frequencist probabilities but there are other more subtle flaws you can read about in Jaynes book.

It's null hypothesis testing that is a symptom of the nonsense inherent in frequencism. Bayesian theory accepts that no theory is perfect they can always be rejected with frequencist techniques if you have enough data.

Log in to post comments By BenE not verified on 25 Jan permalink Regardless of all this philosophical babling, the bayesian approaches seems to allow more objectivity and more robustness than the frequencist approaches while being simpler. Since I don't know much about bayesian methods, this amounts to an argument from ignorance. And my personal trust have no bearing on the question.

I must plead guilty of posting on an empty stomach, IIRC, which usually leads to an empty head. Except that they void any trustworthyness null hypothesis tests might have I addressed the question of tight variances and relative errors in my comment. You can't extract more information than the errors in the experiment let you. And plotting data is essential in any good data analysis.