Question

What significance does variance hold?

Answer 1

We explain the significance of variance using an example from the general situation. The variance is used to determine how widely the observations in a data are spread out. Mathematically the variance is defined as the average of squared differences from the mean.
Here, we need to give the significance so we need to explain using the general life situation where it is used.

Complete step-by-step solution:
We are asked to explain about the significance of the variance.
We know that the Mathematical definition of variance is given as the average of squares of difference of individual data from mean data.
Here, we can see that this definition is used to calculate the variance for the given data but cannot explain the need to find it.
Let us take the general use of variance that is the variance is used to find how widely the observations of a data are spread out.
We know that the variance is mainly used in budget management.
Let us take an example of a data that a company produces over a year that is 12 months.
We know that the business cannot be constant, that is there may be profits in some month and loss in some other month. Let us assume that the overall cost of the product that the company produced as 10,000/- such that the overall sales of those products is 8,000/- which yields a variance difference of 2,000/-
Here, we can see that there is loss for the company. But the variance is given as $-2000/-$.
Let us assume that for the next month the same company sold same product of overall cost price 10,000/- to 14,000/- so that there is a profit and the variance is 4,000/-
Here, we can see that the variance is increased that yields a profit on the overall product. But the question is how did they do it since the product is the same and the company is the same?
Here, we can say the answer to this question is due to variance.
As they calculated variance for the first month they developed their strategies, they adjusted their goals to reach the profit to a considerable amount.
So, we can say that this variance is used to develop our methods to increase the productivity of our work on the same product.
Here, we explained very simply using two months. But we know that the company will take their productivity one in a year and compare it with next year.
So, the formula used to calculate the variance of the data is given as,
${{\sigma }^{2}}=\dfrac{\sum{\left( {{x}_{i}}-\bar{x} \right)}}{n}$
Where, $'{{x}_{i}}'$ is the observation in ${{i}^{th}}$ element, $'\bar{x}'$ is the mean of the observations and $'n'$ is the number of observations.
By using this formula to calculate the variance for the whole year of 12 months we can improve the productivity of the respected product.
Therefore, we can conclude that the variance is used to improve the productivity of a product in business. Not only in business we can use it to help a student increase his/her marks etc.

Note: We need to note that sometimes the variance of the data is too high in order to interpret the data. So, to avoid that much value of variance the standard deviation is formed which is given as the square root of the variance.
But interpreting the data using the variance is much better than that of using the standard deviation.
The variance can be used to compare with the budget directly. But in the case of standard deviation that cannot be done. For that we need to use Z score to interpret the data which will be difficult.