Variance & Standard Deviation Calculator
Calculate Population (σ²) and Sample (s²) Variance for your data set.
Enter Your Data
Calculation Results:
Understanding Variance & Standard Deviation
In statistics, **variance** measures the spread or dispersion of data points in a set around their average (mean). A low variance indicates that the data points tend to be close to the mean, while a high variance indicates that the data points are spread out over a wider range.
**Standard Deviation** is simply the square root of the variance. It’s often preferred because it’s expressed in the same units as the original data, making it easier to interpret the spread relative to the mean.
Population vs. Sample Variance
It’s crucial to distinguish between calculating variance for an entire **population** versus a **sample** taken from a population:
- Population Variance (σ²): Use this when your data set includes *all* members of the group you are interested in. The formula divides the sum of squared differences from the mean by the total number of data points (N).
- Sample Variance (s²): Use this when your data set is just a *sample* drawn from a larger population, and you want to estimate the variance of that larger population. The formula divides the sum of squared differences from the sample mean by the number of data points minus one (n-1). This is known as Bessel’s correction and provides a less biased estimate of the population variance.
Our calculator from Sustainable Digital Shop allows you to choose the appropriate type for your needs.
How to Use the Calculator
- Enter Data: Input your numerical data into the text area. You can separate numbers using commas (
,), spaces ( ), or new lines (pressing Enter). Non-numeric characters will be ignored. - Select Type: Choose whether your data represents an entire “Population” or a “Sample” using the radio buttons.
- Click “Calculate Variance”: The tool will process your data.
- View Results: The calculator will display:
- The number of data points (Count).
- The Mean (Average) of the data.
- The Variance (labeled as Population σ² or Sample s²).
- The Standard Deviation (labeled as Population σ or Sample s).
The results provide key insights into the central tendency (mean) and dispersion (variance, standard deviation) of your data set.
Formulas Used
Mean (μ for population, x̄ for sample):
Mean = (Σ xᵢ) / N (where Σ xᵢ is the sum of all data points and N is the number of data points)
Population Variance (σ²):
σ² = Σ (xᵢ - μ)² / N
Sample Variance (s²):
s² = Σ (xᵢ - x̄)² / (N - 1)
Standard Deviation (σ or s):
Standard Deviation = √Variance
Frequently Asked Questions (FAQ)
Q1: Is this Variance Calculator free?
Yes, this tool is 100% free to use, offered by Sustainable Digital Shop.
Q2: What’s the difference between population and sample variance again?
Use **Population Variance** if your data set includes every single member of the group you’re studying. Use **Sample Variance** if your data is just a subset (a sample) of a larger group, and you want to infer something about that larger group. The sample variance formula divides by n-1 instead of N to provide a better, unbiased estimate of the population’s variance.
Q3: How should I format my input data?
Enter numbers separated by commas, spaces, or new lines. For example: 12, 15, 11 or 12 15 11 or
12 15 11
The calculator will ignore non-numeric text.
Q4: What happens if I enter non-numeric data?
The calculator automatically filters out any input that cannot be interpreted as a number. Only valid numbers will be included in the calculation. It will show the count of valid numbers found.
Q5: Why do I need at least two data points for sample variance?
The sample variance formula divides by n-1 (number of data points minus one). If you only have one data point (n=1), the denominator becomes zero, and division by zero is undefined. You need at least two data points to measure spread within a sample.
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