How to Calculate Variance from Standard Deviation

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Variance from Standard Deviation Calculator

Result:

Enter a value above and click "Calculate Variance".

function calculateVariance() { var standardDeviationInput = document.getElementById("standardDeviation").value; var standardDeviation = parseFloat(standardDeviationInput); if (isNaN(standardDeviation) || standardDeviation < 0) { document.getElementById("varianceResult").innerHTML = "Please enter a valid non-negative number for Standard Deviation."; return; } var variance = standardDeviation * standardDeviation; document.getElementById("varianceResult").innerHTML = "

Result:

Variance (σ²) = " + variance.toFixed(4) + ""; }

Understanding Variance from Standard Deviation

In statistics, both variance and standard deviation are fundamental measures of data dispersion, indicating how spread out a set of data points are from their mean. While they both convey similar information, they do so in different units and are used in various contexts. This calculator helps you quickly determine the variance of a dataset if you already know its standard deviation.

What is Standard Deviation (σ)?

Standard deviation, often denoted by the Greek letter sigma (σ), measures the average amount of variability or dispersion in a dataset. It tells you, on average, how far each data point lies from the mean. A low standard deviation indicates that data points tend to be close to the mean, while a high standard deviation indicates that data points are spread out over a wider range of values. It is expressed in the same units as the data itself, making it easily interpretable.

What is Variance (σ²)?

Variance, denoted by σ² (sigma squared), is another measure of data dispersion. It is defined as the average of the squared differences from the mean. Squaring the differences serves two main purposes: it makes all differences positive, and it gives more weight to larger deviations. Unlike standard deviation, variance is expressed in squared units, which can sometimes make it less intuitive to interpret directly in the context of the original data. However, variance is crucial in many statistical tests and theoretical calculations because of its additive properties.

The Relationship: Variance = (Standard Deviation)²

The relationship between variance and standard deviation is straightforward: variance is simply the square of the standard deviation. Conversely, standard deviation is the square root of the variance. This direct mathematical link means that if you know one, you can easily calculate the other.

Formula:
Variance (σ²) = (Standard Deviation (σ))²

Why Calculate Variance from Standard Deviation?

While standard deviation is often preferred for its interpretability (being in the same units as the data), variance plays a critical role in various statistical analyses. For instance:

  • ANOVA (Analysis of Variance): This statistical test uses variance to determine if there are significant differences between the means of two or more groups.
  • Regression Analysis: Variance is used to assess the goodness of fit of a regression model.
  • Portfolio Theory: In finance, variance is used to measure the risk of an investment portfolio.
  • Theoretical Statistics: Many statistical theorems and derivations are based on variance due to its mathematical properties.

Therefore, being able to convert standard deviation to variance is a common requirement in statistical work.

How to Use This Calculator

  1. Enter Standard Deviation: In the input field labeled "Standard Deviation (σ)", enter the numerical value of the standard deviation you have.
  2. Click Calculate: Press the "Calculate Variance" button.
  3. View Result: The calculator will instantly display the calculated variance (σ²) in the "Result" area.

Example Calculation

Let's say you have a dataset representing the daily temperature fluctuations in a city, and you've calculated the standard deviation to be 3.5 degrees Celsius.

To find the variance, you would use the formula:

Variance = (Standard Deviation)²

Variance = (3.5)²

Variance = 3.5 × 3.5

Variance = 12.25

So, the variance of the temperature fluctuations would be 12.25 (degrees Celsius)².

This calculator simplifies this process, allowing you to quickly obtain the variance without manual calculation, ensuring accuracy and saving time.

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