Confidence intervals how many standard deviations

However, the level of confidence MUST be pre-set and not subject to revision as a result of the calculations. For this example, let's say we know that the actual population mean number of iTunes downloads is 2. There is absolutely nothing to guarantee that this will happen. Further, if the true mean falls outside of the interval we will never know it.

We must always remember that we will never ever know the true mean. Statistics simply allows us, with a given level of probability confidence , to say that the true mean is within the range calculated. This is what was called in the introduction, the "level of ignorance admitted". Here again is the formula for a confidence interval for an unknown population mean assuming we know the population standard deviation:.

The Standard deviation of the sampling distribution is further affected by two things, the standard deviation of the population and the sample size we chose for our data. Here we wish to examine the effects of each of the choices we have made on the calculated confidence interval, the confidence level and the sample size. For a moment we should ask just what we desire in a confidence interval. Our goal was to estimate the population mean from a sample. We have forsaken the hope that we will ever find the true population mean, and population standard deviation for that matter, for any case except where we have an extremely small population and the cost of gathering the data of interest is very small.

In all other cases we must rely on samples. With the Central Limit Theorem we have the tools to provide a meaningful confidence interval with a given level of confidence, meaning a known probability of being wrong. By meaningful confidence interval we mean one that is useful.

Imagine that you are asked for a confidence interval for the ages of your classmates. You have taken a sample and find a mean of You wish to be very confident so you report an interval between 9. This interval would certainly contain the true population mean and have a very high confidence level. However, it hardly qualifies as meaningful.

The very best confidence interval is narrow while having high confidence. There is a natural tension between these two goals. The higher the level of confidence the wider the confidence interval as the case of the students' ages above. We can see this tension in the equation for the confidence interval.

A standard deviation can be obtained from the standard error of a mean by multiplying by the square root of the sample size:. When making this transformation, standard errors must be of means calculated from within an intervention group and not standard errors of the difference in means computed between intervention groups. Confidence intervals for means can also be used to calculate standard deviations.

Again, the following applies to confidence intervals for mean values calculated within an intervention group and not for estimates of differences between interventions for these, see Section 7.

The standard deviation for each group is obtained by dividing the length of the confidence interval by 3. This is where a choice must be made by the statistician. The analyst must decide the level of confidence they wish to impose on the confidence interval.

These numbers can be verified by consulting the Standard Normal table. Divide either 0. Then read on the top and left margins the number of standard deviations it takes to get this level of probability. Common convention in Economics and most social sciences sets confidence intervals at either 90, 95, or 99 percent levels. A good way to see the development of a confidence interval is to graphically depict the solution to a problem requesting a confidence interval.

This is presented in Figure for the example in the introduction concerning the number of downloads from iTunes. However, the level of confidence MUST be pre-set and not subject to revision as a result of the calculations. There is absolutely nothing to guarantee that this will happen. Further, if the true mean falls outside of the interval we will never know it.

We must always remember that we will never ever know the true mean. Statistics simply allows us, with a given level of probability confidence , to say that the true mean is within the range calculated. Here again is the formula for a confidence interval for an unknown population mean assuming we know the population standard deviation:. It is clear that the confidence interval is driven by two things, the chosen level of confidence, , and the standard deviation of the sampling distribution.

The Standard deviation of the sampling distribution is further affected by two things, the standard deviation of the population and the sample size we chose for our data. Here we wish to examine the effects of each of the choices we have made on the calculated confidence interval, the confidence level and the sample size. For a moment we should ask just what we desire in a confidence interval.

Our goal was to estimate the population mean from a sample. We have forsaken the hope that we will ever find the true population mean, and population standard deviation for that matter, for any case except where we have an extremely small population and the cost of gathering the data of interest is very small. In all other cases we must rely on samples.

With the Central Limit Theorem we have the tools to provide a meaningful confidence interval with a given level of confidence, meaning a known probability of being wrong.

By meaningful confidence interval we mean one that is useful. Imagine that you are asked for a confidence interval for the ages of your classmates. You have taken a sample and find a mean of You wish to be very confident so you report an interval between 9. This interval would certainly contain the true population mean and have a very high confidence level. However, it hardly qualifies as meaningful. The very best confidence interval is narrow while having high confidence.

There is a natural tension between these two goals. We can see this tension in the equation for the confidence interval. The confidence interval will increase in width as increases, increases as the level of confidence increases.

To estimate the probability of finding an observed value, say a urinary lead concentration of 4. The distance of the new observation from the mean is 4. How many standard deviations does this represent? Dividing the difference by the standard deviation gives 2.

Table 2 shows that the probability is very close to 0. This probability is small, so the observation probably did not come from the same population as the other children. To take another example, the mean diastolic blood pressure of printers was found to be 88 mmHg and the standard deviation 4.

One of the printers had a diastolic blood pressure of mmHg. The mean plus or minus 1. The For many biological variables, they define what is regarded as the normal meaning standard or typical range.

Anything outside the range is regarded as abnormal. Given a sample of disease free subjects, an alternative method of defining a normal range would be simply to define points that exclude 2.

This would give an empirical normal range. Thus in the children we might choose to exclude the three highest and three lowest values. The means and their standard errors can be treated in a similar fashion. This common mean would be expected to lie very close to the mean of the population. So the standard error of a mean provides a statement of probability about the difference between the mean of the population and the mean of the sample.

In our sample of 72 printers, the standard error of the mean was 0. The sample mean plus or minus 1. If we take the mean plus or minus three times its standard error, the interval would be This is the Confidence intervals provide the key to a useful device for arguing from a sample back to the population from which it came. With small samples - say under 30 observations - larger multiples of the standard error are needed to set confidence limits. These come from a distribution known as the t distribution, for which the reader is referred to Swinscow and Campbell Confidence interval for a proportion In a survey of people operated on for appendicitis 37 were men.