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16.8
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16.8
The Standard Deviation Calculator computes both the population standard deviation ($$\sigma$$) and sample standard deviation ($$s$$), along with their corresponding variances and the mean, for a dataset of five values.
Standard deviation is the most widely used measure of statistical dispersion, quantifying the average distance of data points from the mean.
When the dataset represents the entire population of interest:
$$\sigma = \sqrt{\frac{1}{N}\sum_{i=1}^{N}(x_i - \mu)^2}$$
For five values:
$$\sigma = \sqrt{\frac{(x_1-\bar{x})^2 + (x_2-\bar{x})^2 + (x_3-\bar{x})^2 + (x_4-\bar{x})^2 + (x_5-\bar{x})^2}{5}}$$
When the dataset is a sample drawn from a larger population, Bessel's correction is applied by dividing by $$n - 1$$ instead of $$n$$:
$$s = \sqrt{\frac{1}{n-1}\sum_{i=1}^{n}(x_i - \bar{x})^2}$$
This correction makes $$s^2$$ an unbiased estimator of the population variance $$\sigma^2$$. With 5 values, the denominator is 4 instead of 5.
Standard deviation is fundamental because of the empirical rule (68-95-99.7 rule) for normally distributed data:
This makes standard deviation directly interpretable as a measure of typical spread around the mean.
To compare variability across datasets with different means or units, the coefficient of variation (CV) normalizes the standard deviation by the mean:
$$CV = \frac{\sigma}{\mu} \times 100\%$$
In finance, standard deviation measures investment risk — higher standard deviation means higher volatility. In manufacturing, it quantifies process consistency for quality control. In experimental science, it indicates measurement precision. In education, it shows how test scores are distributed around the class average. In weather forecasting, it measures temperature variability across seasons.
Standard deviation is also central to hypothesis testing, confidence intervals, and regression analysis — forming the backbone of inferential statistics.
Enter five values. The calculator first computes the mean, then calculates the sum of squared deviations from the mean. Dividing by 5 gives the population variance (dividing by 4 gives the sample variance). The square root of each variance gives the corresponding standard deviation.
A small standard deviation (relative to the mean) indicates data points are clustered tightly around the mean. A large standard deviation indicates data is spread widely. Use population σ when your data includes every member of the group; use sample s when your data is a subset of a larger population.
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Results
The mean is 16.8. Population σ = 5.36, sample s = 5.99. The sample SD is larger due to Bessel's correction (dividing by 4 instead of 5).
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Results
Very low standard deviation (σ ≈ 0.63) relative to the mean of 50 indicates highly consistent data with CV ≈ 1.3%.
Population standard deviation ($$\sigma$$) divides by $$N$$ (total population size) and is used when you have data for the entire group. Sample standard deviation ($$s$$) divides by $$n - 1$$ (Bessel's correction) and is used when data is a sample from a larger population. The sample version is always slightly larger.
Dividing by $$n - 1$$ instead of $$n$$ is called Bessel's correction. It compensates for the fact that a sample mean underestimates the true variability. The sample mean is calculated from the same data, which introduces a downward bias; dividing by $$n - 1$$ corrects this, making $$s^2$$ an unbiased estimator of $$\sigma^2$$.
Standard deviation is expressed in the same units as the original data, making it directly interpretable. Variance is in squared units (e.g., if data is in meters, variance is in m²). Standard deviation is easier to visualize and communicate.
There is no universal good value — it depends on context. Use the coefficient of variation ($$CV = \sigma / \mu \times 100\%$$) to compare across contexts. A CV below 10% often indicates low variability; above 30% indicates high variability. What matters is whether the spread is acceptable for your specific application.
For normally distributed data, the empirical rule applies: ~68% of data falls within ±1σ of the mean, ~95% within ±2σ, and ~99.7% within ±3σ. Values beyond 3σ are considered extreme outliers. This relationship is the foundation of z-scores and many statistical tests.
Standard deviation is always ≥ 0. It equals zero only when all data values are identical (no variation). It can never be negative because it is the square root of a sum of squared terms, both of which are non-negative.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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