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Standard Deviation Calculator

Calculate standard deviation, variance, mean, and other statistical properties for a set of data.

Standard Deviation Calculator

Enter a set of numbers to calculate statistical properties.

Understanding Standard Deviation

What is Standard Deviation?

Standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while a high standard deviation indicates that the values are spread out over a wider range.

Population vs. Sample

There are two main formulas for standard deviation, depending on whether your data represents an entire population or just a sample of it.

  • Population: Use this if your data includes every member of the group you are interested in.
  • Sample: Use this if your data is a random sample taken from a larger population. The sample formula uses n-1 in the denominator (Bessel's correction) to provide a better estimate of the population's standard deviation.

Formulas Used

  • Mean (μ or x̄): The average of all numbers.

    μ = (Σxᵢ) / N

  • Population Variance (σ²): The average of the squared differences from the population mean.

    σ² = Σ(xᵢ - μ)² / N

  • Population Standard Deviation (σ): The square root of the population variance.

    σ = √[Σ(xᵢ - μ)² / N]

  • Sample Standard Deviation (s): The square root of the sample variance.

    s = √[Σ(xᵢ - x̄)² / (n-1)]

What Is Standard Deviation and How Is It Calculated?

Standard deviation is a crucial concept in statistics that tells you how "spread out" your data is. A low standard deviation means your data points are clustered tightly around the average (the mean), while a high standard deviation indicates they are spread over a wider range. I built this calculator to provide a quick way to compute standard deviation and other key statistical properties for any set of numbers.

How to Use This Calculator

Simply enter your data set into the text area, separating each number with a comma or a space. The calculator will instantly update with the following information:

  • Count: The total number of data points.
  • Mean (Average): The sum of all numbers divided by the count.
  • Standard Deviation (Population & Sample): The measure of data spread.
  • Variance (Population & Sample): The square of the standard deviation.

Population vs Sample: Which Formula Should You Use?

This is the most important distinction when calculating standard deviation.

  • Population (σ): Use this if your data set includes every single member of the group you are interested in (e.g., the test scores of every student in a specific class).
  • Sample (s): Use this if your data is a smaller sample taken from a larger population (e.g., the test scores of 50 students chosen randomly from an entire school district). The sample formula uses n-1 in the denominator to provide a better, unbiased estimate of the true population standard deviation.

The Calculation Steps

  • 1. Find the Mean (μ): Calculate the average of all your data points.
  • 2. Find Squared Differences: For each data point, subtract the mean and square the result: (xᵢ - μ)².
  • 3. Calculate the Variance (σ²): Find the average of all the squared differences. This is the variance.
  • 4. Find the Standard Deviation (σ): Take the square root of the variance. This brings the value back into the original unit of measurement, making it more interpretable.

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