A sample mean is one number, but we also need to know how stable that number is.
An online store samples 100 customers and finds an average order value of $52.
A different sample of 100 customers would produce a slightly different mean. The standard error describes how much the sample mean would vary across repeated samples.
Key question
How stable is the current estimate of $52?
Measure the variation of an estimate
The standard error of the mean describes the spread of sample means across repeated samples.
Sample mean
$52
Standard deviation
$18
Standard error
$1.80
A small SE means sample means cluster tightly around the population mean. A large SE means the estimate is more sensitive to which observations enter the sample.
How is SE different from SD?
| Measure | Spread of what? | Question |
|---|---|---|
| Standard deviation | Individual order values | How different are customers from one another? |
| Standard error | Sample means | How much would the mean change across samples? |
Standard deviation describes the data. Standard error describes uncertainty in an estimate.
What sample size changes
| Sample size | SD | SE |
|---|---|---|
| 25 | $18 | $3.60 |
| 100 | $18 | $1.80 |
| 400 | $18 | $0.90 |
Multiplying the sample size by four cuts the standard error in half because SE decreases with the square root of n.
How should SE be read?
Individual order values can vary widely while the sample mean is estimated precisely from a large sample.
Read both numbers
SD = $18: customers differ substantially in order value
SE = $1.80: the mean order value is estimated fairly precisely
Change the sample size
The wide curve is how much individual orders differ. The narrow one is how much the average of n orders would move between samples.
Standard deviation
₩18,000
Standard error
₩1,800
SD ÷ SE
10.0×
With an SD of ₩18,000 and 100 customers, the standard error is ₩18,000 ÷ √100 = ₩1,800. Four times the customers would halve it, to ₩900.
Only the narrow curve responds to n. How different customers are from each other never changes.
Standard deviation describes the data; standard error describes the stability of an estimate.