l stoodialinza
← All topics

Step 1 · Basics · Topic 3 of 17 · 7 min

Samples, SEM and confidence intervals

A sample can estimate the mean of a defined population. How precise is that estimate?

In short#

  • A sample can estimate parameters of a defined population. Different random samples give slightly different estimates.
  • For independent observations, SEM = SD / √n estimates the standard error of the mean. It describes uncertainty in the estimate, not spread among observations.
  • 95% CI of the mean under the t model: x̄ ± t(0.975; n − 1) × SEM. When assumptions hold, repeated use of the procedure covers the true mean in about 95% of cases.
  • If SD and confidence level stay fixed, SEM decreases as 1/√n and the CI narrows as n grows. SD describes observation-level spread and does not decrease just because the sample gets larger.

In plain words#

First define the population you want to learn about, such as products made by a specified process, material specimens or members of a defined group. Usually you measure a sample. Different random samples can have different means.

SEM estimates how much the mean would vary across independent samples from the same population. With other factors held constant, a larger n usually improves precision. SD describes spread among individual observations. If the population stays the same, increasing n does not by itself reduce that spread.

A confidence interval shows values of the population mean compatible with the data and chosen model. Its width depends on spread, sample size and confidence level. When assumptions hold, a narrower interval means a more precise estimate, but does not guarantee an absence of bias.

Try it yourself

Why sample means vary This synthetic population has mean μ = 10.97 and population standard deviation σ = 4.97. Each value is returned to the population after it is drawn, so it can be selected again. Watch how the sample means are distributed.
Bars show how often simulated sample means fall within the displayed range from 0 to 30. The vertical line marks the mean of the synthetic population.μ051015202530
Number of samples
0
Mean of the last sample
N/A
Standard deviation within the last sample
N/A
Standard deviation of sample means
N/A
σ / √n
1.57

As more samples are simulated, the standard deviation of their means usually approaches σ / √n. Larger n makes the means cluster more tightly. The standard deviation within each sample estimates σ and varies from sample to sample, especially when n is small.

In Linza#

  1. In descriptive statistics, Linza shows SD, SEM and a fixed 95% CI for the mean using the Student t distribution.
  2. For Bars and Dots and mean plots, choose SD, SEM or 95% CI error bars and state what they show in the figure caption. Box plot whiskers show the minimum and maximum.
Try it in Linza Opens in a new tab with synthetic data. Work in the original tab remains unchanged.

Common mistakes#

  • SEM used as if it showed the spread of observations. SEM describes the precision of the mean. Show SD or individual observations to display data spread.
  • Incorrect: “There is a 95% probability that the true mean lies in this calculated interval.” In frequentist terms, 95% describes the proportion of intervals that cover the true mean over repeated use of the procedure.

Check yourself#

The sample grows from 10 to 40 observations and SD stays the same. By what factor does SEM change?

Optional. Your answer is not saved.

Sources#

  • Altman D.G., Bland J.M. (2005) Standard deviations and standard errors. BMJ, 331. Open publication Opens in a new tab
  • Cumming G., Fidler F., Vaux D.L. (2007) Error bars in experimental biology. Journal of Cell Biology, 177(1). Search for publication Opens in a new tab

References are for further reading. A method’s suitability depends on the study design and its assumptions.