What Each Measures
Standard Deviation
Standard deviation measures the typical distance of individual observations from the mean. Large SD = data points are spread out. Small SD = data points cluster near the mean.
Standard Error
Standard error measures how much sample means would vary if we took many samples. It's the SD of the sampling distribution of the mean. SE = SD/√n, so it's always smaller than SD.
Direct Comparison
| Standard Deviation (SD) | Standard Error (SE) | |
|---|---|---|
| What it measures | Spread of individual values | Spread of sample means |
| Formula | √[Σ(x-x̄)²/(n-1)] | SD / √n |
| As n increases | Stays roughly same | Gets smaller |
| Purpose | Describe variability | Quantify uncertainty |
| Use in | Descriptive stats | Confidence intervals, tests |
SD and SE answer different questions. SD: 'How variable is my data?' SE: 'How precisely do I know the mean?' They're related by SE = SD/√n.
Effect of Sample Size
n Affects SE, Not SD
SD reflects population variability - it doesn't change much as you collect more data. SE reflects your precision in estimating the mean - it shrinks as n increases because SE = SD/√n.
Sampling Distributions at Different n
Wide SE
Medium SE
Narrow SE
These curves show the sampling distribution of the mean at different sample sizes. All come from the same population (same SD), but larger n produces tighter clustering of sample means around μ.
When to Report Each
Describing Your Data
• Characterizing a population or sample
• When readers want to know typical variation
• In descriptive statistics sections
This tells readers the typical spread of individual ages.
Use SD when you want to describe variability in your data. It answers: 'How different are individuals from each other?'
Making Inferences
• Constructing confidence intervals
• Comparing means across groups
• In inferential statistics (tests, CIs)
This tells readers how precisely we estimated the mean.
Use SE when you want to express uncertainty about an estimate. It answers: 'How confident can we be in this mean?'
Don't Confuse Them!
• Using SD in confidence interval formulas
• Confusing error bars (are they SD or SE?)
• Not specifying which you're reporting
Many papers confuse SD and SE, or don't specify which they're reporting. SE is always smaller (by factor of √n), so unscrupulous researchers might report SE to make variability look small. Always be explicit.
Error Bars in Graphs
What Do Error Bars Show?
Error bars in graphs can represent SD, SE, or confidence intervals. They look identical but have very different meanings. SE bars are much shorter than SD bars for the same data.
Same Data, Different Bars
(40-60)
(48-52)
(46-54)
For the same data, SD bars are 5× longer than SE bars! Always check the figure legend. If not specified, you can't properly interpret the graph.
Recommendations
There's no universal 'correct' error bar - it depends on your purpose. But you MUST always specify what they show. APA style recommends 95% CIs for comparing means.
Real-World Examples
Apply your knowledge to practical scenarios
Blood Pressure Study
A study measures blood pressure in n = 100 patients. Mean = 130 mmHg, SD = 20 mmHg.
Key Insights
- SD describes the patients; SE describes our certainty about the average
- More patients = same SD, smaller SE
- 95% CI would be 130 ± 1.96(2) = [126, 134]
SD vs SE Quiz
Test your understanding • Passing score: 70%