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The Importance of Variance in Statistical Analysis: Understanding its Role and Applications


By gobrain

Jan 1st, 2024

In this article, we will examine in detail the variance, which has an important place in statistics and is often used together with different statistical measures. Let's first get started with what variance is.

WHAT IS VARIANCE?

Variance is a statistical measure used to assess the spread of data in a set, providing insight into the degree of deviation between the individual observations and the mean of the set. It is an important concept in statistics that is used to make inferences about a population based on a sample.

AREAS OF USE OF VARIANCE

There are numerous fields where variance is commonly utilized as a statistical measure. Let's take a closer look at some of them.

  • Finance: In finance, variance is used to measure the volatility of financial instruments, such as stocks or bonds. By calculating the variance of the returns of an investment portfolio, investors can assess the risk associated with the portfolio.

  • Economics: In economics, variance is used to measure the volatility of economic indicators, such as GDP or inflation. By calculating the variance of these indicators, economists can assess the stability of an economy and predict future trends.

  • Machine learning: In machine learning, variance is used to evaluate the performance of a model. By measuring the variance of the model's predictions, data scientists can assess if the model is overfitting or underfitting the data and adjust the model accordingly.

  • Psychology: In psychology, variance is used to measure the variability of the responses obtained from different participants in a study. By calculating the variance of the responses, psychologists can determine if there is a significant difference between the groups and draw conclusions about the underlying phenomena.

  • Education: In education, variance is used to evaluate the consistency of the scores obtained from different tests or assessments. By calculating the variance of the scores, educators can determine if there is any variability in the performance of the students and adjust the teaching methods accordingly.

How to Calculate Variance

As you realize, variance is an important measure in statistic but how easily calculate variance of a data set?

Let's follow the steps below.

  • Calculate the mean of the dataset
  • Subtract the mean from each value in set
  • Square the value found in the second step and sum all the values
  • Divide the sum by the number of values in set

For the steps above, formula is:

V = (Σ(x - μ)^2 / n)

Where:

  • x is a data in set,
  • μ is the mean,
  • n is the number of values in set
  • Σ is the sum of.

For example, A given [2,6,7] set

  • mean = (2 + 6 + 7) / 3 = 5
  • V = (2-5)^2 + (6-5)^2 + (7-5)^2 = 14
  • V = (14 / 3)
  • V = 4.66...

Variance: Sample vs Population

Since it is usually difficult to collect data for the whole population, a sample from the whole population is taken and the variance is calculated. In this case, there will be a small change in the formula as below

V = (Σ(x - μ)^2 / (n-1))

The new variance for the set above will be :

  • V = (14 / 2)
  • V = 7

Standart Deviation vs Variance

Variance and standard deviation are two related measures of dispersion or spread of data in a set of observations. There are, however, a few differences let's see them.

Result in Different Unit

As you will notice, in the standard deviation, the square root of the value is taken so that the values in the dataset have the same unit of the result, which can provide a better interpretation.

For Example:

  • [2cm,6cm, 7cm] -> SD = 2.160cm
  • [2cm,6cm, 7cm] -> V = 4.66cm^2

Deviation Value

In simpler terms, the standard deviation shows the general spread of the data set, while variance highlights the distribution of individual values (including outliers) within the data set.

Advantages and Disadvantages

Every statistical measure has advantages and disadvantages depending on a sample or population of a dataset. Let's see the advantages of disadvantages of variance:

Advantages:

  • Easy to calculate: Variance is a straightforward calculation that requires only a few basic arithmetic operations. As a result, it is easy to compute and understand.

  • Sensitive to differences: Variance is highly sensitive to differences in the data points, which makes it useful for detecting outliers or unusual values in the data set.

  • Provides a measure of spread: Variance provides a measure of how spread out the data points are from the mean, which can be useful for understanding the distribution of the data.

Disadvantages:

  • Sensitive to extreme values: Because variance is based on the squared differences between the data points and the mean, it is highly sensitive to extreme values or outliers in the data set. These outliers can distort the value of the variance and make it less useful as a measure of variability.

  • Not intuitive: Variance is not an intuitive measure of variability, and it can be difficult to interpret for non-statisticians.

  • Not robust: Variance is not a robust measure of variability, meaning that it can be heavily influenced by small changes in the data set. As a result, it may not be the best choice for data sets with a large number of outliers or extreme values.

Conclusion

To sum up, variance is a statistical measure of the spread or dispersion of a set of data values. It provides insight into the stability or heterogeneity of a data set and is used in many fields to make inferences about a population based on a sample. Finally, keep in mind that understanding the dispersion of a dataset requires a combination of statistical measures, not just variance alone.

Thank you for reading.