Confidence interval and standard error of the mean
Confidence interval and standard error of the mean
The confidence interval is a common and sometimes misunderstood principle in medical statistics.- a formal definition may be: a range of values for a variable of interest constructed so that this range has a specified probability of including the true value of the variable. The specified probability is called the confidence level, and the end points of the confidence interval are called the confidence limits*
- in simpler terms: a range of values within which the true effect of intervention is likely to lie
The likelihood of the true effect lying within the confidence interval is determined by the confidence level. For example a confidence interval at the 95% confidence level means that the confidence interval should contain the true effect of intervention 95% of the time.
How is the confidence interval calculated?
The standard error of the mean (SEM) is a measure of the spread expected for the mean of the observations - i.e. how 'accurate' the calculated sample mean is from the true population mean
Key point
- SEM = SD / square root (n)
- where SD = standard deviation and n = sample size
- therefore the SEM gets smaller as the sample size (n) increases
A 95% confidence interval:
- lower limit = mean - (1.96 * SEM)
- upper limit = mean + (1.96 * SEM)
The above formula is a slight simplification:
- if a small sample size is used (e.g. n < 100) then it is important to use a 'Student's T critical value' look-up table to replace 1.96 with a different value
- if a different confidence level is required, e.g. 90% then 1.96 is replaced by a different value. For 90% this would 1.645
Results such as mean value are often presented along with a confidence interval. For example, in a study the mean height in a sample taken from a population is 183cm. You know that the standard error (SE) (the standard deviation of the mean) is 2cm. This gives a 95% confidence interval of 179-187cm (+/- 2 SE).
*Last JM. A dictionary of epidemiology. Oxford: International Journal of Epidemiology, 1988
+PassMedicine Notes
- a formal definition may be: a range of values for a variable of interest constructed so that this range has a specified probability of including the true value of the variable. The specified probability is called the confidence level, and the end points of the confidence interval are called the confidence limits*
- in simpler terms: a range of values within which the true effect of intervention is likely to lie
The likelihood of the true effect lying within the confidence interval is determined by the confidence level. For example a confidence interval at the 95% confidence level means that the confidence interval should contain the true effect of intervention 95% of the time.
How is the confidence interval calculated?
The standard error of the mean (SEM) is a measure of the spread expected for the mean of the observations - i.e. how 'accurate' the calculated sample mean is from the true population mean
Key point- SEM = SD / square root (n)
- where SD = standard deviation and n = sample size
- therefore the SEM gets smaller as the sample size (n) increases
- lower limit = mean - (1.96 * SEM)
- upper limit = mean + (1.96 * SEM)
- if a small sample size is used (e.g. n < 100) then it is important to use a 'Student's T critical value' look-up table to replace 1.96 with a different value
- if a different confidence level is required, e.g. 90% then 1.96 is replaced by a different value. For 90% this would 1.645
Results such as mean value are often presented along with a confidence interval. For example, in a study the mean height in a sample taken from a population is 183cm. You know that the standard error (SE) (the standard deviation of the mean) is 2cm. This gives a 95% confidence interval of 179-187cm (+/- 2 SE).
*Last JM. A dictionary of epidemiology. Oxford: International Journal of Epidemiology, 1988