Variance is a statistical measure that quantifies the spread or dispersion of a set of data points. It provides information about how far each data point in a dataset is from the mean (average) and thus indicates the variability or diversity within the dataset. In simpler terms, variance measures how much the data points deviate from the average.
The concept of variance was introduced by the English statistician and geneticist Ronald A. Fisher in the early 20th century. Fisher developed the concept as part of his work on the analysis of variance (ANOVA) and the design of experiments. His contributions to the field of statistics revolutionized the way researchers analyze and interpret data.
Variance is typically introduced in middle or high school mathematics courses, depending on the curriculum. It is commonly taught in algebra or statistics classes, where students learn about data analysis and probability.
To understand variance, it is essential to grasp the following concepts:
The steps to calculate variance are as follows:
There are two main types of variance:
Sample Variance: This is used when the dataset represents a sample from a larger population. The formula for sample variance involves dividing the sum of squared deviations by the total number of data points minus one.
Population Variance: This is used when the dataset represents the entire population. The formula for population variance involves dividing the sum of squared deviations by the total number of data points.
Variance possesses several important properties:
To calculate variance, follow these steps:
The formula for sample variance is:
The formula for population variance is:
Where:
To apply the variance formula, substitute the values of the dataset into the formula and perform the necessary calculations. The resulting value will represent the variance of the dataset.
The symbol commonly used to represent variance is for population variance and for sample variance.
There are various methods to calculate variance, including:
Example 1: Calculate the sample variance for the following dataset: 5, 7, 9, 11, 13.
Solution:
Therefore, the sample variance is 10.
Example 2: Calculate the population variance for the following dataset: 2, 4, 6, 8, 10.
Solution:
Therefore, the population variance is 8.
Q: What is the purpose of calculating variance? A: Variance helps to understand the spread or dispersion of data points in a dataset. It is useful in comparing different datasets, identifying outliers, and making statistical inferences.
Q: Can variance be negative? A: No, variance is always non-negative. It can be zero if all data points are identical.
Q: How does variance relate to standard deviation? A: Standard deviation is the square root of variance. It provides a measure of dispersion in the original units of the dataset, while variance is measured in squared units.
Q: Is variance affected by outliers? A: Yes, variance is sensitive to outliers. Outliers, being extreme values, can significantly impact the variance value.
Q: Can variance be used for categorical data? A: No, variance is primarily used for numerical data. For categorical data, other measures like mode or chi-square tests are more appropriate.