Standard Deviation Calculator - Sample, Population & Variance

📊 Standard Deviation Calculator

Calculate Sample & Population Standard Deviation, Variance, and Mean

Enter Data Set (Numbers separated by commas or spaces):

Example: 12, 15, 18, 20, 25 or 12 15 18 20 25

Sample Std Dev (s)

0

Population Std Dev (σ)

0

Sample Variance (s²)

0

Population Variance (σ²)

0

Mean (Average)

0

Count (N) / Sum (∑)

N=0 | ∑=0

Understanding Standard Deviation in Statistics

Standard deviation is a crucial statistical measurement that quantifies the amount of variation or dispersion in a set of values. A low standard deviation indicates that data points tend to be close to the mean (average), whereas a high standard deviation indicates that numbers are spread out over a wider range.

Sample vs. Population Standard Deviation

Choosing between sample and population standard deviation depends on your data collection scope:

Type	Symbol	Formula	When to Use
Sample Standard Deviation	s	s = √[ ∑(x - x̅)² / (n - 1) ]	When analyzing a smaller subset or sample taken from a larger group.
Population Standard Deviation	σ (Sigma)	σ = √[ ∑(x - μ)² / N ]	When analyzing the complete set of every member in a total group.

Step-by-Step Formula Calculation

To calculate standard deviation manually, follow these mathematical steps:

1. Calculate the Mean (average) of all numbers in the dataset.
2. Subtract the Mean from each data point to get individual deviations.
3. Square each individual deviation result.
4. Sum all squared deviations together.
5. Divide by N (for Population) or n - 1 (for Sample) to find Variance.
6. Take the square root of the Variance to get Standard Deviation.

Frequently Asked Questions (FAQs)

Q: What is Variance?
A: Variance is the average of squared differences from the Mean. Standard deviation is simply the square root of variance.

Q: Why do we divide by (n - 1) for a sample?
A: Dividing by n - 1 (Bessel's correction) corrects the bias in estimating the true population variance from a smaller sample.