What Is Sample Mean and Sample Standard Deviation?
The sample mean x̄ and the sample standard deviation s are the two most common statistics for describing a sample. The sample mean is the average of the values; s measures their spread and divides the sum of squared deviations by n − 1.
Enter your sample and choose a confidence level. The calculator returns x̄, s, the sample variance, the standard error s ÷ √n and a t-based confidence interval for the population mean, together with the full working table, a dot plot and an interval chart.
The article below explains the formulas, variables, why we divide by n − 1 and answers common questions.
How to Use the Sample Mean and Sample Standard Deviation Calculator
- Enter your sample values.
- Choose a confidence level.
- Read x̄, s, s² and the standard error.
- Use the confidence interval to estimate the population mean.
Sample Mean and Sample Standard Deviation Formula
Sample mean
Sample standard deviation
Confidence interval
Sample Mean and Sample Standard Deviation Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| x̄ | Sample mean | |
| s | Sample standard deviation | divides by n − 1 |
| SE | Standard error of the mean | s ÷ √n |
| t* | Critical t value | df = n − 1 |
Sample Mean and Sample Standard Deviation Worked Example
For the sample 4, 8, 6, 5, 7: x̄ = 30 ÷ 5 = 6. The squared deviations are 4, 4, 0, 1, 1, which sum to 10. So s² = 10 ÷ 4 = 2.5 and s = 1.581. The standard error is 1.581 ÷ √5 = 0.707.
With the example values loaded in the calculator above, the steps are:
- Sample mean x̄ = Σx ÷ n = 24.8375.
- Deviations are squared and summed: Σ(x − x̄)² = 16.33875.
- Sample variance s² = 16.33875 ÷ (n − 1) = 16.33875 ÷ 7 = 2.3341071.
- Sample standard deviation s = √2.3341071 = 1.5277785.
- SE = s ÷ √n = 0.54015127, so the 95 % interval is x̄ ± t*·SE = 24.8375 ± 1.27725.
Result: Sample mean x̄ = 24.8375; Sample standard deviation s = 1.5277785; 95 % confidence interval for the population mean = 23.56025 to 26.11475.
How to Read the Sample Statistics
For the sample 4, 8, 6, 5, 7 the sample mean is 6, the squared deviations sum to 10, the sample variance is 10 ÷ 4 = 2.5 and the sample SD is 1.581. The standard error is 1.581 ÷ √5 = 0.707.
The confidence interval for the population mean uses a t value with n − 1 degrees of freedom. With only 5 values the interval is wide, which is an honest reflection of how little a small sample tells you about the population.
Why divide by n − 1?
Deviations are measured from the sample mean, which sits closer to the data than the true population mean, so dividing by n would underestimate the spread. Using n − 1 (Bessel's correction) removes that bias in the variance.
Where Sample Mean and Sample Standard Deviation Is Used
Describing a sample
Get the mean, SD and variance in one step.
Estimating the population mean
Read the confidence interval with the t multiplier.
Standard error
Find how precisely the mean is estimated.
Working table
Check each deviation and its square.
Common Sample Mean and Sample Standard Deviation Mistakes to Avoid
- The sample SD divides by n − 1, which corrects for estimating the mean from the same data.
- The confidence level can be changed, and higher levels give wider intervals.
- Use the t distribution, not z, for small samples.
- Report n alongside the mean and SD.
Sample Mean and Sample Standard Deviation Terms and Definitions
- Sample statistic
- A number calculated from a sample, such as x̄ or s.
- Bessel's correction
- Dividing by n − 1 so the sample variance is unbiased.
Sample Mean and Sample Standard Deviation FAQs
How do I find the sample mean and sample standard deviation?
Add the values and divide by n for the mean. Then square each deviation from the mean, add them, divide by n − 1 and take the square root.
What is the difference between s and σ?
s is the sample standard deviation, found with n − 1. σ is the population standard deviation, found with n and used when you have the whole population.
What is the sample standard deviation used for?
It describes spread in a sample, and it is the basis of the standard error, confidence intervals and t-tests.
What is the difference between SD and standard error?
The SD describes the spread of the data. The standard error, s ÷ √n, describes how precisely the sample mean estimates the population mean.