Standard Error of the Mean Calculator
Compute the standard error of the mean (SEM) and visualize how it changes with sample size.
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How to Use This Calculator
The **Standard Error of the Mean Calculator** helps you determine how much a sample mean is likely to vary from the population mean.
- **Choose Your Tab:** Select the tab for either **"Known Population SD ($\sigma$)"** or **"From Sample Data ($s$)"**.
- **Enter Data:** Input your standard deviation ($\sigma$ or $s$) and the **Sample Size ($n$)**. Remember that the standard deviation must be positive and the sample size must be an integer of 2 or more.
- **Optional: Confidence Interval:** If you have a sample mean ($\bar{x}$) and want to calculate a confidence interval, enter the mean and choose a confidence level (90%, 95%, or 99%).
- **Set Precision:** Choose the number of **decimal places** for the final result.
- **Calculate:** Click the **"Calculate SEM"** button to see the results, including the SEM value, a dynamic chart, and a detailed table.
Formula Breakdown
The calculator uses one of two primary formulas depending on whether you know the population standard deviation ($\sigma$) or only the sample standard deviation ($s$).
When Population SD is Known:
$\text{SEM}=\dfrac{\sigma}{\sqrt{n}}$
When Only Sample SD is Known (Estimated SEM):
$\widehat{\text{SEM}}=\dfrac{s}{\sqrt{n}}$
The confidence interval for the mean is calculated using the formula:
$\bar{x} \pm \text{Critical Value} \cdot \text{SEM}$
**Note on Critical Value:** When the population standard deviation ($\sigma$) is known, we use a z-score as the critical value. When only the sample standard deviation ($s$) is known, we should technically use a t-score, which depends on the degrees of freedom ($n-1$). For simplicity and because t-scores converge to z-scores as the sample size increases, this calculator uses z-scores for both.
Frequently Asked Questions (FAQs)
What is Standard Error of the Mean (SEM)?
The Standard Error of the Mean (SEM) is a measure of the statistical accuracy of an estimate. It indicates how much the sample mean ($\bar{x}$) is expected to vary from the true population mean ($\mu$) if you were to take multiple samples from the same population. A smaller SEM means the sample mean is a more precise estimate of the population mean.
What's the difference between standard deviation and standard error?
Standard deviation measures the variability or spread within a single sample, telling you how much individual data points deviate from the sample mean. Standard error, on the other hand, measures the variability of the sample mean itself, telling you how much the sample mean is likely to vary from the population mean.
How does sample size affect SEM?
As the sample size ($n$) increases, the standard error of the mean (SEM) decreases. This is because larger samples are generally more representative of the population, leading to a more precise estimate of the population mean. The relationship is not linear; SEM decreases in proportion to the square root of the sample size.