Standard Deviation Calculator
Calculate mean, variance, standard deviation, and other statistical measures for your data set
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Use sample if your data is a subset of a larger population
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What is Standard Deviation?
Standard deviation is a measure of how spread out numbers are from their average (mean). A low standard deviation indicates that values tend to be close to the mean, while a high standard deviation indicates that values are spread out over a wider range. It's one of the most commonly used measures of variability in statistics.
Sample vs. Population
Population Standard Deviation is used when you have data for an entire population. The formula divides by n (the total number of values).
Sample Standard Deviation is used when you have data for a sample of a larger population. The formula divides by (n-1) instead of n. This is called Bessel's correction and provides a better estimate of the population standard deviation from a sample.
Variance
Variance is the average of the squared differences from the mean. Standard deviation is simply the square root of variance. While variance is useful in many statistical calculations, standard deviation is often preferred for interpretation because it's in the same units as the original data.
Mean, Median, and Mode
Mean is the arithmetic average of all values. It's sensitive to outliers (extreme values).
Median is the middle value when data is sorted. It's more resistant to outliers than the mean.
Mode is the most frequently occurring value(s) in the dataset. A dataset can have no mode, one mode, or multiple modes.
The 68-95-99.7 Rule
For normally distributed data, approximately 68% of values fall within one standard deviation of the mean, 95% fall within two standard deviations, and 99.7% fall within three standard deviations. This empirical rule is useful for understanding the distribution of data and identifying outliers.
Real-World Applications
Standard deviation is used extensively in quality control (manufacturing tolerances), finance (measuring investment risk and volatility), education (standardizing test scores), weather forecasting (variability in predictions), and scientific research (assessing experimental reliability). Understanding variability is crucial for making informed decisions in virtually any field that deals with data.