how to find standard error

7.7.3.2 Obtaining standard deviations from standard errors and confidence intervals for group means

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 7.7.3.3). Most confidence intervals are 95% confidence intervals. If the sample size is large (say bigger than 100 in each group), the 95% confidence interval is 3.92 standard errors wide (3.92 = 2 × 1.96). The standard deviation for each group is obtained by dividing the length of the confidence interval by 3.92, and then multiplying by the square root of the sample size:

For 90% confidence intervals 3.92 should be replaced by 3.29, and for 99% confidence intervals it should be replaced by 5.15.

If the sample size is small (say less than 60 in each group) then confidence intervals should have been calculated using a value from a t distribution. The numbers 3.92, 3.29 and 5.15 need to be replaced with slightly larger numbers specific to the t distribution, which can be obtained from tables of the t distribution with degrees of freedom equal to the group sample size minus 1. Relevant details of the t distribution are available as appendices of many statistical textbooks, or using standard computer spreadsheet packages. For example the t value for a 95% confidence interval from a sample size of 25 can be obtained by typing =tinv(1-0.95,25-1) in a cell in a Microsoft Excel spreadsheet (the result is 2.0639). The divisor, 3.92, in the formula above would be replaced by 2 × 2.0639 = 4.128.

For moderate sample sizes (say between 60 and 100 in each group), either a t distribution or a standard normal distribution may have been used. Review authors should look for evidence of which one, and might use a t distribution if in doubt.

As an example, consider data presented as follows:

Group	Sample size	Mean	95% CI
Experimental intervention	25	32.1	(30.0, 34.2)
Control intervention	22	28.3	(26.5, 30.1)

The confidence intervals should have been based on t distributions with 24 and 21 degrees of freedom respectively. The divisor for the experimental intervention group is 4.128, from above. The standard deviation for this group is √25 × (34.2 – 30.0)/4.128 = 5.09. Calculations for the control group are performed in a similar way.

It is important to check that the confidence interval is symmetrical about the mean (the distance between the lower limit and the mean is the same as the distance between the mean and the upper limit). If this is not the case, the confidence interval may have been calculated on transformed values (see Section 7.7.3.4).