1. What Are Standard Deviation and Standard Error?

Standard deviation (sd) measures the amount of variation or dispersion in a set of values. Standard error (se) estimates how far the sample mean of the data is likely to be from the true population mean.

2. How Does the Calculator Work?

The calculator uses these formulas:

Where:

• \( x_i \) — Individual values in the dataset
• \( \bar{x} \) — Mean of all values
• \( n \) — Number of values in the dataset
• \( \sum \) — Summation of all values

Explanation: The standard deviation quantifies how spread out the numbers are, while the standard error measures the precision of the sample mean as an estimate of the population mean.

3. Importance of These Statistics

Details: Standard deviation is crucial for understanding data variability, while standard error is essential for constructing confidence intervals and hypothesis testing.

4. Using the Calculator

Tips: Enter comma-separated numerical values (e.g., "1, 2, 3, 4"). The calculator will compute mean, standard deviation, standard error, and sample size. Requires at least 2 values.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between sd and se?
A: SD measures data variability, while SE estimates how close the sample mean is to the population mean.

Q2: When should I use n vs n-1 in the denominator?
A: Use n-1 for sample standard deviation (this is the default in most statistical software). Use n only when working with an entire population.

Q3: What does a large standard deviation indicate?
A: It indicates that the data points are spread out over a wider range of values.

Q4: How does sample size affect standard error?
A: Standard error decreases as sample size increases, following the square root law (SE = SD/√n).

Q5: Can I use this for non-normal distributions?
A: The formulas work mathematically for any distribution, but interpretation may differ for non-normal data.