# Standard deviation versus standard error

BMJ 2011; 343 doi: https://doi.org/10.1136/bmj.d8010 (Published 13 December 2011) Cite this as: BMJ 2011;343:d8010- Philip Sedgwick, senior lecturer in medical statistics

^{1}Centre for Medical and Healthcare Education, St George’s, University of London, Tooting, London, UK

- p.sedgwick{at}sgul.ac.uk

Researchers investigated the effectiveness of a weight loss programme in men with moderate to severe obstructive sleep apnoea. Men were eligible to join the programme if aged 30-65 years, had a body mass index between 30 and 40, had moderate to severe obstructive sleep apnoea (as measured by an apnoea-hypopnoea index ≥15 events/hour), and were being treated with continuous positive airway pressure. The weight loss programme lasted one year and consisted of a very low energy diet for nine weeks followed by a weight loss maintenance programme. A total of 63 men were recruited at an outpatient obesity clinic in a university hospital in Stockholm, Sweden.1

Outcome measures included the change in body weight after participation in the weight loss programme. At baseline the sample had a mean weight of 113.1 kg (SD=14.2 kg). The researchers reported that the weight loss programme resulted in a significant decrease in weight. The mean change in body weight at one year from baseline was a reduction of 12.1 kg (95% confidence interval 9.8 to 14.3) (SD=9 kg; SEM=1.13 kg).

Which of the following, if any, are true?

a) The standard deviation of body weight at baseline provides a measure of the spread of observations of weight in the sample before participants began the weight loss programme.

b) At baseline, approximately 95% of sample members had a body weight that was within two standard deviations of the sample mean.

c) The standard error of the mean change in body weight at one year provides a measure of precision of the sample mean as an estimate of the population parameter.

d) After one year on the weight loss programme, 95% of the population would have a reduction in body weight between 9.8 kg and 14.3 kg.

## Answers

Answers *a*, *b*, and *c* are true, whereas *d* is false.

Standard deviation and standard error of the mean are often confused. Standard deviation is used to describe the variation in measurements of a variable for members of a sample. Standard error of the mean describes the precision of the sample mean as an estimate of the population mean. The standard error of the mean (SEM), sometimes shortened to standard error (SE), is used to make statistical inferences about population parameters, through either statistical hypothesis testing or estimation by confidence intervals. One way of recalling when to use standard deviation and standard error is to remember that standard deviation is for description and standard error is for estimation.

The sample standard deviation of body weight at baseline provides a measure of the variation in weight—in particular, it provides a measure of how much on average the weights of the sample members deviated about the sample mean weight at baseline (*a* is true). The derivation of the sample standard deviation has been described in previous questions.2 3 For each man the difference between his weight at baseline and the sample mean was obtained; each difference was then squared and the results summed across all sample members. The sum of these squared differences was divided by the sample size minus one. The resulting quantity is called the sample variance, the square root of which is called the sample standard deviation. The sample standard deviation and variance are estimates of the population parameters. The population parameters are the values that would be observed if all members of the population had their weights measured before starting the weight loss programme. The population parameters are theoretical concepts: their values are not known but are estimated by the sample values.

The sample standard deviation of weight at baseline can be used to calculate a series of ranges in weight containing certain percentages of the sample members. Three ranges are typically derived. At baseline, approximately 68% of the sample had a weight that was no further than one sample standard deviation away from the sample mean—that is, from [113.1–14.2] kg to [113.1+14.2] kg, or 98.9 kg to 127.3 kg. Furthermore, approximately 95% of the sample had a weight at baseline that was no further than two sample standard deviations away from the sample mean (*b* is true)—that is, from [113.1–2(14.2)] kg to [113.1+2(14.2)] kg, or 84.7 kg to 141.5 kg. Finally, approximately 99% of the sample had a weight at baseline no further than three sample standard deviations away from the sample mean—that is, from [113.1–3(14.2)] kg to [113.1+3(14.2)] kg, or 70.5 to 155.7 kg. These three ranges can be derived for any variable measured on a continuous scale to describe the variation in the measurements.

The sample mean change in body weight at one year after being on the weight loss programme was a reduction of 12.1 kg, with a standard deviation of 9 kg for the change in weight from baseline. The sample mean change in weight is an estimate of the population parameter. The sample estimate is expected to be similar in magnitude to the population parameter, although it is unlikely to be exactly equal. Any inaccuracy in the sample estimate is because it was based on a sample of individuals from the population. The accuracy of the sample mean as an estimate of the population parameter is quantified by the standard error of the mean (*c* is true).4 The standard error of the mean is derived by dividing the sample standard deviation of the change in weight by the square root of the sample size. Therefore, as sample size increases, the magnitude of the standard error decreases. This may be intuitively deduced, because, as the sample size approaches that of the population, the sample mean will become closer in value to the population mean.

The standard error of the mean is used to derive a range of values known as the confidence interval, which quantifies the uncertainty in the sample mean change in body weight after one year as an estimate of the population parameter. A percentage is attached to a confidence interval, typically 95%. The 95% confidence interval for the population mean is derived as the interval 1.96 standard errors either side of the sample mean change in weight—that is, from [12.1−1.96(1.15)] to [12.1+1.96(1.15)], or 9.8 kg to 14.3 kg. The population mean is contained by these limits with 95% confidence (that is, with almost certainty). Therefore, it is estimated with 95% confidence that the mean weight loss after one year on the weight loss programme for the entire population could be as little as 9.8 kg or as much as 14.3 kg. The 95% confidence interval does not describe the variation in weight loss at one year in the population members (*d* is false). The standard error can be calculated for other sample estimates, including a proportion, a relative risk, or an odds ratio. The standard error of each estimate is used in a similar way as the standard error of the mean to calculate a 95% confidence interval for the population parameter.

## Notes

**Cite this as:** *BMJ* 2011;343:d8010

## Footnotes

Competing interests: None declared.