Variance Calculator

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Variance calculator is an online free tool to calculate the variation of each number in data set from the mean value of that data set. It let you calculate the variance very easily by entering the set of values in the input box. It is equally useful for students, teachers, researchers, and statisticians. If you are a teacher, you can use this pooled variance calculator to match the answers of your students. If you are a student, you can use this tool to understand and solve the complex and lengthy variance problems. This calculator offers the ease of use which makes it preferable as compared to other calculators. You can find variance and standard deviation for your statistics problems and assignments on just one click.

How to use the Variance calculator?

To calculate variance or standard deviation, enter the values of your data set in the given input box. Values should be separated by a comma, because the calculator will identify each value if values are separated by a comma. You can enter as many values as you want, and there is no restriction or limitation to use this calculator. After entering the values in the input box, click the "Calculate" button to get the result. 

You will see the result for four values as soon as you click the button. It will give you the number of samples, mean, standard deviation, and variance in one click. Below this result, you will also find the detailed calculation for mean, standard deviation, and variation which is given with the formulas and step by step procedure. This variation calculator elaborates each step in such a detail that any student can easily comprehend the whole process of variance and standard deviation calculation.

Check this covariance calculator if you need to calculate the covariance between two data sets.

What is Variance?

The variance is a measure of how far a data set from the mean of that data set is spread. It will have a high variance if the numbers are very widely distributed. It will have a very low variance if the numbers are very close to each other. A variation of 0 shows that all values in the data set are the same. Any variance other than zero is a positive one.

What is variance in statistics?

In statistics, the variance of a random variable is the average value of the squared distance from the average. It shows the distribution of the random variable by the mean value. A small variance indicates the distribution of the random variable close to the mean value. If the variance is greater, it shows that the random variable is far from the average value. For example, the narrow bell curve has a small variance in the normal distribution, and the wide bell curve has a large variance.

Formula for Variance

If you look at a set of 20 results and see only values of 8, 9, and 10 in the results, it is intuitively obvious that the average is about 9. As we have already discussed, the variance is a measure of how widespread are the points in a data set. The variance is the square of standard deviation and denoted by the σ2 which is a Greek letter “sigma.”

The population variance of a finite size N population is calculated using the following formula:

Variance \(=\sigma^2 = \dfrac{1}{N}\displaystyle\sum_{i=1}^n (x_i - \mu)^2 \) 
In this equation, σ2 refers to population variance, xi is the data set of population, μ is mean of the population data set, and N refers to the size of the population data set.

In this equation, σ2 refers to population variance, xi is the data set of population, μ is mean of the population data set, and N refers to the size of the population data set.

What are the different measures of variability?

You have certainly seen or heard the term "average," which is widely used in relation to a data set or set of numbers. You probably know what it means in our daily routines. For example, if someone tells you that the average age is 60 years in the United States, you can conclude that in the U.S, the typical age is 60 years for most of the people. This average number means that half of the people have age more than 60, and half of them have an age of 60 or less.

The average and the mean are mathematically exactly the same: you add every value into a set and divide it by the total number of those items in the data set. For example, for a test of 10 questions and 30 sets of results, the total score is 195. We can say that the average score is \(\dfrac{195}{30} = 6.5\)

The median is the midpoint of a set, and half of the values are above, and half are below that set. Normally it's close to the mean, but it's not the same.

How to calculate variance?

Calculating variance manually is a tedious task. You will need the mean of data set, arithmetic difference, and many additions and subtractions to find variance. You can also use the population variance calculator above to calculate variance for your set of data. 

Let's begin with a set of population data. The term "population" refers to the entire number of observations that are relevant. Analyzing Tokyo's residents' age for example, would include the age of every Tokyo resident in the population. Usually, for a large data set like this, you will create a larger sheet, but here is a smaller example.

Example: Suppose there are exactly five guest rooms in a hotel. Every room is accommodating the following numbers of people:

\(x_1 = 6, x_2 = 5, x_3 = 6, x_4 = 7, \text{and} x_5 = 4 \)

Let's use the formula for the population variance given above.

Variance Formula

Since there is all the information you need in a population, this formula gives you the exact population variance. Statisticians use different variables to distinguish it from sample variance. We will discuss sample variance later in the post.

For your ease, we will elaborate on this formula once more. In this equation, σ2 refers to population variance, xi is the data set of population, μ is mean of the population data set, and N refers to the size of the population data set.

Follow these steps to measure the variance for the given data set using this formula.

•    Find the mean of the data set

The symbol μ is the arithmetic mean when analyzing a population. To find the mean, sum up all the values in the data set and divide the sum by the total number of values in the data set. You may think of mean as the average, but the average is considered differently in various fields.

Mean \(= M = \dfrac{\sum x}{n} = \Big({6 + 5 + 6 + 7 + 4}{5}\Big) = \dfrac{28}{5} = 5.6\)

•    Subtract the mean value from every number in data set

The near to mean data points lead to a difference nearer to zero. Repeat each data point's subtraction problem and you might begin to understand how the data are spread. It is also called arithmetic difference.

\(x_1 - \mu = 6 - 5.6 = 0.4\)
\(x_2 - \mu = 5 - 5.6 = -0.6\)
\(x_3 - \mu = 6 - 5.6 = 0.4\)
\(x_4 - \mu = 7 - 5.6 = 1.4\)
\(x_5 - \mu = 4 - 5.6 = -1.6\)

•    Take the square of each arithmetic difference

Some of your numbers will be negative right now, and others will be positive. These two categories are numbers on the left of the mean and numbers on the right of the mean if your data is visualized in a number line. For calculating variance, this is not good since both groups are mutually exclusive. Make them positive by taking the square of each value.

Get \((x_i - \mu)^2\) for each value.

\((x_i - \mu)^2\) 
\((0.4)^2 = 0.16\)
\((-0.6)^2 = 0.36\)
\((0.4)^2 = 0.16\)
\((1.4)^2 = 1.96\)
\((-1.6)^2 = 2.56\)

•    Find the mean value of all of these values or use formula

You now have an indirect value for each data point, related to the distance between that data point and the mean. Take the mean by adding all these values and divide them by the number of values. Note that we have evaluated the terms which are in the formula step by step.

Variance = \( \sigma^2 = \Big\{\dfrac{0.16 + 0.36 + 0.16 + 1.96 + 2.56}{5}\Big\} = \dfrac{5.2}{5} = 1.04\)

What is Sample Variance?

The sample variation is denoted by s2 and is used to determine how different a sample is from the mean value. A data sample is a collection of data from a population in statistics. The population is typically very large, making it impossible to list all the values in the population. The solution is to collect a sample of the population and perform statistics on these samples. These samples then reflect the whole population.

What is the formula of Sample Variance?

The formula for sample variance is almost similar to the formula for population variance with a little difference. The following formula is used to calculate the sample variance.

\(= s^2 = \dfrac{1}{N-1} \displaystyle\sum_{i=1}^n (x_i - \bar{x})^2 \) 

In this equation, s2 is the sample variance xi is the sample data set x̄ is the mean value of a sample set of values, and N refers to the size of the sample data set.

How to calculate sample variance?

Statisticians can access only sample data for a population in most of the cases. For example, if a statistician wants to find the variance in the mileage of all bikes in China, he can find the mileage of a random sample of a few thousand bikes rather than the whole population, which is in billions. He can use this method to obtain a good approximation of the mileage, but it probably won't correlate exactly with the actual numbers.

Let's calculate the sample variance by using an example. First, write the sample data set that you have with you. Through analyzing the total numbers of apples sold in a store, we track the random results for seven days. The shopkeeper sold this number of apples every day for seven days: \(42, 48, 30, 36, 46, 53, 62.\) We will use this sample data to calculate the sample variance for the number of apples sold per day by a shopkeeper.

Write the formula for sample variance

The variance in a set of data shows how the data points are distributed. If the variance is closer to zero, it means that the points in a data set are close enough. Use the following formula to calculate sample variance when dealing with sample data sets.

We have explained all the terms in the formula above.

•    Compute the mean value for the sample data

The mean of a sample is denoted by x̅. Find the mean value of the sample taken from the shop by adding all values dividing it by the total number of days.

\(\bar{x} = \dfrac{\sum x}{n} = \dfrac{42 + 48 + 30 + 36 + 46 + 53 + 62}{7} = \dfrac{317}{7} = 45.28\)

The sample mean for the given values is 45.28 in this case. We will use this value in the next steps to complete the process. The mean can be considered as the central value of the sample data. If the data is around the average value or the mean value, there is a minimal variation. If it is distributed far from the mean value, the variance will be high.

•    Subtract the mean value from each number in the data set

Calculate \(x_i - \bar{x}\), where xi represents the values in the data set. In our example, xi is the number of apples sold each day. Each result of this calculation will describe how far it is from the mean value of the data set.

\(x_1 - \bar{x} = 42 - 45.28 = -3.8\)
\(x_2 - \bar{x} = 48 - 45.28 = 2.72\)
\(x_3 - \bar{x} = 30 - 45.28 = -15.8\)
\(x_4 - \bar{x} = 36 - 45.28 = -9.28\)
\(x_5 - \bar{x} = 46 - 45.28 = 0.72\)
\(x_6 - \bar{x} = 53 - 45.28 = 7.72\)
\(x_7 - \bar{x} = 62 - 45.28 = 16.72\)

Your job is easy to check because your answers should be zero if you these values. That is due to the concept of calculating average because the negative answers, which are the difference from average to smaller numbers, cancel precisely the positive answers.

•    Take a square of each result from the previous step

As discussed above, the sum of all deviations will be zero because of the nature of the mean. This means that the mean deviation is always zero, so that nothing tells how the results are distributed. Find the square of each resulted deviation to resolve this problem. Making all the deviations positive will ensure that summing up will not result in zero.

\((x_1 - \bar{x})^2 = -3.8^2 = 14.44\)
\((x_2 - \bar{x})^2 = 2.72^2 = 7.40\)
\((x_3 - \bar{x})^2 = -15.8^2 = 249.64\)
\((x_4 - \bar{x})^2 = -9.28^2 = 86.11\)
\((x_5 - \bar{x})^2 = 0.72^2 = 0.52\)
\((x_6 - \bar{x})^2 = 7.72^2 = 59.60\)
\((x_7 - \bar{x})^2 = 16.72^2 = 279.56\)

For each data point in your sample, now you have the value \((x_i - \bar{x}) 2\).

•    Calculate the sum of all values of \((x_i - \bar{x}) 2\)

In this step, we will evaluate the expression \(\sum (x_i - \bar{x})^2\), which is the numerator in the formula for sample variance. The \sum (upper case sigma) refers to the sum of the values. We have already calculated the \(\sum (x_i - \bar{x})^2\) expression, now add all the values of \(\sum (x_i - \bar{x})^2\) to get the sum.

\(\sum (x_i - \bar{x})^2 = 14.44 + 7.40 + 249.64 + 86.11 + 0.52 + 59.60 + 279.56 = 697.27\)

•    Divide the \(\dfrac{\sum (x_i - x)^2}{(n - 1)}\)

Statisticians had only divided the variance of the sample by n for a long time. It provides you the average squared deviation value, which corresponds flawlessly with the sample variance. However, remember that a sample is only a larger population estimate. You would have a diverse outcome if you grabbed another random sample and did the similar calculation. We found that dividing by n–1 instead of n provides a better approximation of the variance of the larger population.

There are seven values in the data set in the sample, so \(n = 7\).

Variance of the sample \(= s^2= \dfrac{697.27}{7 - 1} = 116.21\)

Remember that because the exponent was present in the variation formula, a squared unit of original data reflects the variance. Rather, the standard deviation is often useful. You did not waste the time, though, because the standard deviation is the square root of the variance. You can always use the sample variance calculator above to find the sample variance. This is why a sample variation is written as s2, and the standard sample deviation is s. Let's briefly discuss standard deviation before moving towards the advantages of variance.

Merits and drawbacks of variance

Statisticians and mathematicians use variance to see the relationship between the individual numbers in a data set instead of using extensive mathematical methods like quartile structure.

Variance, together with correlation, is one of the key asset assignment parameters. Calculating variance in profit on investments allows investors to have competitive portfolios by maximizing the exchange and risk fluctuations for each investment.

One benefit of variance is that all deviations from their mean value, irrespective of their direction, are treated as equivalent. The squared deviations cannot amount to zero and do not show any variability in the data.

One of the drawbacks is that it gives outliers additional weight. The data can be skewed by squaring the numbers. Variance can also be negative, and all of the values in a data set will be the same if the variance is zero.

One more disadvantage of variance is that it is difficult to understand. Variance applications use it often mainly to take the square root of its magnitude, showing the standard deviation of the data set.

Applications of Variance

Variance is a primary asset classification parameter. It can allow an investor to make a portfolio that enhances the profit ratio of investors if used along with correlations.

This means that the uncertainty or risk is often represented as SD rather than variances because the former is understood more easily.

For many various statistical purposes, the estimation of variance is significant and offers another way to compute our outcomes. We can have an average voltage or current value for electronics. With measured variance, we can determine the amount of variation that a certain voltage or current has from its average value. For various possible statistical purposes, this could be very useful.

Variance vs. Standard deviation

The variance is obtained by taking the mean of the data set, subtracting each point from the mean independently, squaring each and then taking the mean of the squares again, whereas standard deviation is obtained by taking the square root of the variance.

The variance helps to determine the size of the data in relation to the mean value. As the variance grows, there will be a greater variation in data values, and a wider distance will arise between each value in the data set. The variance will be less if the data values are near to each other. Nonetheless, this is difficult to understand, because variances are squared results which cannot be substantially represented with the original data set in the same graph.

Standard deviations are often easier to understand and implement. The standard variance is expressed in the same measuring unit as the data, which does not necessarily apply to the variance. The statisticians can determine whether the data has a normal curve or other mathematical relationship using the standard variation. 68 percentage of the data points come within a standard deviation from the mean data point when the data is collected in a normal curve. Greater variances lead to more data points going beyond the standard deviation. Lower variances give rise to average results.