Math

Standard Deviation Calculator

Calculate mean, median, mode, std dev, and more

One-Click Descriptive Statistics

Descriptive statistics summarise a dataset with a small set of numbers that capture its centre, spread, and shape. This calculator computes all key metrics in one step: mean, median, mode, variance, standard deviation, quartiles, IQR, range, and skewness. Just paste your numbers separated by commas, spaces, or new lines.

Measures of Central Tendency: Mean, Median, Mode

The three classic measures of the "centre" of a distribution each tell a different story about a typical value:

  • Mean (arithmetic average): (sum of all values) Γ· (count). Sensitive to extreme values β€” a single very large outlier pulls the mean substantially upward.
  • Median: The middle value when data is sorted. For an even count, it is the average of the two middle values. The median is robust to outliers β€” it does not change if the largest value doubles.
  • Mode: The most frequently occurring value. A dataset can be unimodal (one mode), bimodal, or multimodal. If no value repeats, there is no mode.

For nearly symmetric data (e.g., height, test scores), mean β‰ˆ median β‰ˆ mode. For right-skewed data (e.g., income, house prices), mean > median > mode, and the median better represents the typical value.

Measures of Spread: Variance, Standard Deviation, Range, IQR

Central tendency alone does not describe a distribution. Two datasets can have the same mean but very different spreads:

  • Range: max βˆ’ min. Simple but highly sensitive to outliers.
  • Variance (σ²): Average of squared deviations from the mean. Squaring emphasises large deviations but changes the unit (e.g., metresΒ² instead of metres).
  • Standard deviation (Οƒ or s): Square root of variance. In the original unit. The primary spread statistic for normally distributed data.
  • IQR (Interquartile Range): Q3 βˆ’ Q1. The range of the middle 50% of data. Robust to outliers β€” used in box plots and outlier detection rules.
  • Coefficient of Variation (CV): (Sample std dev Γ· |mean|) Γ— 100. A dimensionless measure of relative variability β€” useful for comparing spread across datasets with different units or scales.

Population vs. Sample Standard Deviation

The distinction matters whenever you are drawing conclusions about a larger group from a subset:

  • Population Οƒ: Divide sum of squared deviations by n. Use only when you have data for every member of your entire group of interest.
  • Sample s: Divide by n βˆ’ 1 (Bessel's correction). Use when your data is a sample and you want to estimate the true population standard deviation. This is the correct choice in almost all real-world research and data analysis.

When n is large (100+), the difference between Οƒ and s becomes negligible. For small samples (n < 30), using the wrong formula can meaningfully affect results.

Frequently Asked Questions

What is standard deviation?
Standard deviation measures how spread out the values in a dataset are around the mean (average). A low standard deviation means values are clustered closely around the mean; a high standard deviation means they are spread out widely. It is calculated as the square root of variance.
What is the difference between population and sample standard deviation?
Population standard deviation (Οƒ) is used when your data includes every member of the group you are studying. Sample standard deviation (s) is used when your data is a sample drawn from a larger population β€” it divides by (nβˆ’1) instead of n (Bessel's correction) to produce an unbiased estimate. In practice, use sample standard deviation unless you explicitly have the entire population.
What is the mean, median, and mode?
Mean is the arithmetic average: sum of all values divided by the count. Median is the middle value when the data is sorted β€” it is resistant to extreme outliers. Mode is the most frequently occurring value. For symmetric, normally distributed data, these three are equal. When data is skewed, they diverge β€” the median is often a more representative "typical" value than the mean in skewed distributions.
What is variance?
Variance is the average of the squared differences from the mean. It measures how far each value in the dataset is from the mean. Standard deviation is the square root of variance, expressed in the same units as the original data β€” which is why standard deviation is more commonly reported than variance.
What are quartiles and the IQR?
Quartiles divide a sorted dataset into four equal parts. Q1 (first quartile / 25th percentile) is the median of the lower half of data. Q3 (third quartile / 75th percentile) is the median of the upper half. The IQR (Interquartile Range) is Q3 βˆ’ Q1, and it measures the spread of the middle 50% of data. The IQR is widely used for outlier detection: values more than 1.5Γ— IQR below Q1 or above Q3 are typically considered outliers.
What does skewness mean?
Skewness measures the asymmetry of a distribution. A value near 0 indicates a symmetric distribution. Positive skewness (right-skewed) means the tail extends further to the right β€” income distributions are typically right-skewed. Negative skewness (left-skewed) means the tail extends to the left. The formula used here is the adjusted Fisher-Pearson moment coefficient of skewness.

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