Variance Calculator

Your Data
Select Population if the data contains all measurable values or all values you are interested in.
Select Sample if the data is a sample of a large or unlimited population and you wish to make a statement about the entire population.
Input the data separated by comma, space or enter. To calculate variance, you need at least two values in your dataset.
Choose the number of decimal places.

How to Find the Sample Variance: A Step-by-Step Calculation

Typically, statistical analysis software will automatically find the variance for you. You can also obtain it with our Variance Calculator. But knowing how to calculate variance by hand can help you better understand how the formula works.

To calculate the sample variance of a dataset, follow these steps:

  1. Calculate the sample mean of the dataset:

    X¯=i=1nXin

  2. Subtract the mean from each value in the dataset:

    (XiX¯)

  3. Calculate each observation’s squared deviation from the sample mean:

    (XiX¯)2

  4. Calculate the sum of all the squared differences to obtain the sum of squares (SS):

    i=1n(XiX¯)2

  5. Divide the sum of squared differences by n – 1 to obtain the variance (

    s2
    ):

    s2=i=1n(XiX¯)2n1

How to Find the Population Variance: A Step-by-Step Calculation

To calculate the population variance, follow these steps:

  1. Calculate the population mean of the dataset:

    μ=i=1nXin

  2. Subtract the mean from each value in the dataset:

    (Xiμ)

  3. Calculate each observation’s squared deviation from the population mean:

    (Xiμ)2

  4. Calculate the sum of all the squared differences to obtain the sum of squares (SS):

    i=1n(Xiμ)2

  5. Divide the sum of squared differences by n to obtain the population variance (

    σ2
    ):

    σ2=i=1n(Xiμ)2n

  6. Take the square root of the sum of the squared deviations to calculate the population standard deviation:

    σ=i=1n(Xiμ)2n

Variance Calculator

What Is Variance?

By statistical definition, variance measures how spread out a dataset is. In other words, it shows the distance of each individual data point from the mean. Higher variance indicates that the points have greater variability, thus are more spread out, while lower variance means that they are generally similar and more tightly clustered around the mean.

What Is the Importance of Variance in Statistics?

Variance is an important concept in statistics and data analysis because it provides a measure of how much the data points in a dataset differ from its mean (average). Understanding this measure is crucial for several reasons, such as:

  1. Quantifying dispersion – Variance measures the dispersion of observations in a dataset. Knowing the spread of data points can provide insights into the variability of the underlying phenomenon being studied. For example, in finance, knowing the variance of returns that a mutual fund generates allows you to determine the risk of investing in it. Variance analysis can also help you monitor and control the production in the manufacturing industry. By analyzing the variance between actual and expected results, manufacturers can identify where the processes are not operationally efficient and take corrective action. In marketing, meanwhile, the variance of customer response to different marketing messages can determine the success of a campaign.
  2. Evaluating reliability – Variance can be used to assess the reliability of the data or the predictions we derive from it. For example, high variance may suggest the presence of outliers or inconsistencies.
  3. Improving decision-making – Variance is often used in decision-making processes as it helps identify patterns and trends in the data. In this way, stakeholders can determine the risk associated with a particular approach, such as investments, policy changes, or business strategies.
  4. Comparing datasets – Variance can be used to draw comparisons between different datasets or populations. As a result, you can determine which has more variability or identify any significant differences.
  5. Model evaluation and optimization – Variance plays a crucial role in evaluating and optimizing models in the context of machine learning and statistical modeling. High-variance models may overfit the training data, while a low variance may underfit it.
  6. Hypothesis testing and confidence intervals – Variance is a key component in many statistical hypothesis tests, such as t-tests and analysis of variance (ANOVA). These formulas are used to compare the means of different groups or populations. Additionally, variance is used to calculate confidence intervals, which provide a range estimate of where a population parameter might fall.

To summarize, understanding and analyzing variance is important for data interpretation, informed decision-making, and model optimization in various scientific and real-world applications.

How to Calculate Variance

Our Variance Calculator is a helpful shortcut that speeds up your work. But knowing how to calculate it by hand will help you better understand and interpret it.

Variance can be calculated using different formulas depending on whether the data being analyzed is from a population or a sample.

Population variance, denoted by

σ2
, measures the entire population’s variability. It is calculated by taking the sum of the squared deviations of each data point from the population mean, and dividing it by the total number of data points.

Use the following population variance formula for the calculation:

σ2=i=1n(Xiμ)2n

Here u, represents the population mean and n is the number of observations in the dataset.

We calculate the sample variance, denoted by s2 , by dividing the sum of squared deviations from the mean by the number of data points (n) minus one. Known as Bessel’s correction, this equation reduces the bias due to the finite sample size. In practice, we often prefer to use sample variance as a measure of variability since it's generally easier to collect a data sample than to gather information on an entire population.

Here’s how to calculate sample variance:

s2=i=1n(XiX¯)2n1

Here

X¯
, represents the sample mean and n is the number of observations in the sample.

Another interesting observation is that we square the differences between each data point and the mean in both formulas. This step is vital because it ensures that the deviations—otherwise known as the differences between the mean and the individual points—are always non-negative.

It is generally agreed that the square of any real number is always non-negative; that’s why the sum of squared differences will also be non-negative. This is important because the dispersion—the extent to which a distribution is stretched or squeezed—cannot be negative. In essence, this metric concerns itself with distance, which cannot have a negative value. Thus, by squaring the differences, we can guarantee that we always obtain positive values.

Another reason why we square the differences when calculating variance is because it amplifies the effect of large differences. This means that data points farther from the mean have a greater impact on the variance than those that are closer.

Consider a dataset with a mean of 0 and an observation of 100. Without squaring, this observation would contribute a spread of only 100, which might not fully capture the magnitude of its deviation from the mean. However, squaring the difference of 100 from the mean returns a spread of 10,000—a more accurate representation of the observation’s impact on the overall distribution. This allows us to better understand the data’s variability and make more informed decisions based on the distribution’s characteristics.

What Are the Advantages and Disadvantages of Variance?

Variance is widely used in many fields, including manufacturing, finance, education, healthcare, and agriculture. By understanding the measure’s pros and cons, professionals can make more informed decisions about when and how to use it in their analyses.

Here are the advantages and disadvantages of using variance.

Advantages of Variance

  1. Variance is a powerful tool that can help us understand the degree of variation in a dataset. This knowledge can provide valuable insights into the way our data is distributed.
  2. Variance is an important component of statistical methodologies such as the t-test and ANOVA.
  3. Variance can be used to compare different datasets even if they have different scales or units of measurement.

Disadvantages of Variance

  1. Variance can be highly sensitive to outliers—observations that lie an abnormal distance from other data points. Squaring these outliers can add extra weight to them and skew the dataset.
  2. Variance can be difficult to interpret. When we calculate the variance of a dataset, we square the differences between each data point and the mean. This results in a different unit of measurement than the original data, making it difficult to compare the variance to individual data points. For example, let’s assume that the variance of a store’s monthly sales data is equal to $10,000; this doesn’t provide meaningful information about individual sales figures. But if we calculate the standard deviation of the data, the number will be
    10,000
    (or $100), thus the measure of variability will be in the same units as the original data. As a result, you can better understand the distribution of sales revenue and make informed decisions about your store's financial performance.

What Are the Differences Between Variance and Standard Deviation?

Variance and standard deviation are two of the most common measures of variability .

We define variance as the average of the squared deviations around the mean. But, since this measure is expressed in squared units, it can be difficult to interpret and compare across multiple datasets. Therefore, we need a way to translate it back to the original units of measurement.

This is where standard deviation comes in. We define it as the square root of the variance, and express it in the same units as the original data. This measure of spread is more straightforward than variance because of its direct interpretation in the same unit as the observations. Consequently, we use standard deviation more regularly, particularly when it comes to practical applications such as data analysis, reporting, and decision-making.

You can use our Variance Calculator and Standard Deviation Calculator to obtain both.