Variance is the measure of how far a set of numbers are spread out from their average value. It is calculated by taking the differences between each number in the set and the average, and then squaring the result. Variance is often used to determine how accurately a sample represents a population. Knowing how to calculate variance of sample data can be an important skill for anyone working with statistical data sets.
Understand the Definition of Variance
The variance of a data set is a measure of its spread or dispersion. It is calculated by taking the differences between each number in the set and the average, squaring the differences and then taking the average of those squared differences. The squared differences are called squared deviations, or squared errors. The formula for calculating it is: V = (1/N) Σ(Xi – Xavg)², where V is the variance, N is the number of elements in the data set, Xi is the value of each element and Xavg is the average of the elements.
Calculate the Mean or Average of the Sample Data
The mean, or average, of a data set is the sum of all the elements in the data set divided by the number of elements. To calculate the mean, add up all the elements in the data set and divide the sum by the number of elements. For example, if the data set is {1,2,3,4,5}, the sum of the elements is 15 and the mean is 3, since 15/5 = 3.
Calculate the Squared Deviations
Once you have calculated the mean of the data set, you can calculate the squared deviations. To do this, subtract the mean from each element in the data set and then square the result. For example, if the mean of the data set {1,2,3,4,5} is 3, then the squared deviations will be (1-3)² = 4, (2-3)² = 1, (3-3)² = 0 (4-3)² = 1 and (5-3)² = 4.
Sum the Squared Deviations
The next step is to sum the squared deviations. In the example above, the sum of the squared deviations is 4+1+0+1+4 = 10. Once you have the sum of the squared deviations, you can calculate the variance of the sample data.
Calculate the Variance
The variance of a data set is calculated by dividing the sum of the squared deviations by the number of elements in the data set. In the example above, the variance of the data set is 10/5 = 2. This means that the data set is spread out to a distance of two units from its mean. If the variance is high, then the data set is widely spread out; if the variance is low, then the data set is closely clustered around its mean.
Find the Standard Deviation
The standard deviation is a measure of the spread of a data set that is related to its variance. It is calculated by taking the square root of the variance. In the example above, the standard deviation is the square root of 2, which is approximately 1.4. This means that the data set is spread out to a distance of 1.4 units from its mean.
Understand the Difference Between Population Variance and Sample Variance
It is important to understand the difference between population variance and sample variance. Population variance is calculated using all the elements in a population, while sample variance is calculated using a subset of a population. Sample variance is usually used to estimate population variance, since it can be difficult or impossible to measure the variance of an entire population.
Interpret the Results
Once you have calculated the variance of sample data, you can use it to interpret the data set. The higher the variance, the more spread out the data set is from its mean. Low variance indicates that the data set is closely clustered around its mean. Variance can also be used to compare two data sets, since it indicates how different one data set is from the other.
Conclusion
Variance is an important measure of how spread out a data set is from its mean. Calculating the variance of sample data is a simple process that can be used to understand and interpret data sets. Knowing how to calculate variance of sample data can be an important skill for anyone working with statistical data sets.