# Relationship between standard error and confidence interval

### Standard error and confidence intervals | Health Knowledge The contrast between these two terms reflects the important distinction between The standard deviation (often SD) is a measure of variability. on both the standard deviation and the sample size, by the simple relation SE = SD/√( sample size). For a large sample, a 95% confidence interval is obtained as the values. The standard error (SE) of a statistic is the standard deviation of its sampling distribution or an Therefore, the relationship between the standard error and the standard deviation is such that, for a given sample size, statistic or the standard error for a particular regression coefficient (as used in, e.g., confidence intervals). Introduction Learning objectives: You will learn about standard error of a mean, standard error of a proportion, reference ranges, and confidence intervals. Thus the variation between samples depends partly on the amount of variation in the.

Confidence intervals The means and their standard errors can be treated in a similar fashion.

### Standard error - Wikipedia

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. The standard error for the percentage of male patients with appendicitis is given by: In this case this is 0. This is also the standard error of the percentage of female patients with appendicitis, since the formula remains the same if p is replaced by p. We can conclude that males are more likely to get appendicitis than females. This formula is only approximate, and works best if n is large and p between 0.

A better method would be to use a chi-squared test, which is to be discussed in a later module. There is much confusion over the interpretation of the probability attached to confidence intervals.

## Standard error and confidence intervals

To understand it, we have to resort to the concept of repeated sampling. Example Suppose a student measuring the boiling temperature of a certain liquid observes the readings in degrees Celsius He calculates the sample mean to be If he knows that the standard deviation for this procedure is 1.

In other words, the student wishes to estimate the true mean boiling temperature of the liquid using the results of his measurements. If the measurements follow a normal distribution, then the sample mean will have the distribution N. Since the sample size is 6, the standard deviation of the sample mean is equal to 1.

The selection of a confidence level for an interval determines the probability that the confidence interval produced will contain the true parameter value. Common choices for the confidence level C are 0. These levels correspond to percentages of the area of the normal density curve. 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.

## Standard error

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.