# Relationship between confidence level and standard deviation

### Confidence Interval for the Mean

The "2 sigma rule" where sigma refers to standard deviation is a way to to the distribution inside the population, whereas confidence intervals. You determine through the measures of central tendency, that mean systolic blood pressure for the treatment group was mmHg. Here is a graph with two. The Relationship Between Confidence Intervals and p-values The 95% confidence interval consists of all values less than standard errors CMV infected babies was g (standard deviation = g, standard error = g).

## Confidence Intervals

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. 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.

Confidence intervals 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 Because the normal curve is symmetric, half of the area is in the left tail of the curve, and the other half of the area is in the right tail of the curve.

This interval is only exact when the population distribution is normal. For large samples from other population distributions, the interval is approximately correct by the Central Limit Theorem.

In the example above, the student calculated the sample mean of the boiling temperatures to be As the level of confidence decreases, the size of the corresponding interval will decrease. This is because the standard deviation decreases as n increases. The margin of error m of a confidence interval is defined to be the value added or subtracted from the sample mean which determines the length of the interval: Suppose in the example above, the student wishes to have a margin of error equal to 0.

Confidence Intervals for Unknown Mean and Unknown Standard Deviation In most practical research, the standard deviation for the population of interest is not known. In this case, the standard deviation is replaced by the estimated standard deviation salso known as the standard error.

Since the standard error is an estimate for the true value of the standard deviation, the distribution of the sample mean is no longer normal with mean and standard deviation.

Instead, the sample mean follows the t distribution with mean and standard deviation. The t distribution is also described by its degrees of freedom. For a sample of size n, the t distribution will have n-1 degrees of freedom.

### Confidence Intervals

The notation for a t distribution with k degrees of freedom is t k. As the sample size n increases, the t distribution becomes closer to the normal distribution, since the standard error approaches the true standard deviation for large n.

Example The dataset "Normal Body Temperature, Gender, and Heart Rate" contains observations of body temperature, along with the gender of each individual and his or her heart rate.

This value is approximately 1.