Statistics
Standard Deviation Solver
Analyze dataset variability and spread. Compute population and sample standard deviation, variance, and mean for any numeric dataset.
Dataset Input
Valid formats: 12, 15.5, 20 or newline separated lists. Non-numeric values are automatically excluded.
Summary Metrics
Sample Size (n)10
Min / Max Range[12, 30]
Summation (Σx)212
Arithmetic Mean
21.2
Median Value
21.5
Modal Value
22
Std. Deviation
5.2688
Population Variance
27.76
Total Range Spread
18
Frequency Distribution
Dataset visualization across unique values
Share & Embed
Share your exact result or embed this tool.
Solution Methodology
01
Input Dataset
Paste comma-separated, space-separated, or newline-separated numeric values.
02
Compute Mean
Calculate the arithmetic mean μ = Σxᵢ / N as the baseline reference point.
03
Accumulate Squared Deviations
For each value compute (xᵢ − μ)² and sum all results.
04
Divide & Root
Divide by N for population σ or by N − 1 for sample s, then take the square root.
05
Review Insights
Examine variance, range, and flagged outliers alongside the standard deviation.
Common Questions
What is standard deviation?
Standard deviation is a measure of how spread out the values in a dataset are relative to the mean. A low standard deviation means most values cluster tightly around the average; a high one means values are dispersed widely. It is the square root of the variance and shares the same unit as the original data, making it more interpretable than variance alone.
What is the difference between population and sample standard deviation?
Population standard deviation (σ) divides the sum of squared deviations by N — the total number of data points — and is used when your dataset represents the entire population. Sample standard deviation (s) divides by N − 1 (Bessel's correction) to produce an unbiased estimate when your data is a subset drawn from a larger population. Choosing the wrong variant will systematically underestimate or overestimate the true spread.
How do I interpret a high or low standard deviation?
A standard deviation near zero means the data points are almost identical. As σ grows relative to the mean, variability increases. A common rule of thumb for normally distributed data is the 68-95-99.7 rule: roughly 68 % of values fall within ±1σ of the mean, 95 % within ±2σ, and 99.7 % within ±3σ. Values beyond ±3σ are typically flagged as statistical outliers.
Can this tool detect outliers in my dataset?
Yes. Once the solver computes the mean and standard deviation, it flags any data point that lies more than 2σ (or a user-defined threshold) from the mean as a potential outlier. This z-score approach (z = (x − μ) / σ) is the standard method used in academic research, quality control, and financial risk analysis.