Question

What are the symbols for the sample variance and for the population variance?

Answer 1

Hint: In the above mentioned question we need to write down the symbols of sample variance and also for population variance for this we need to first understand what is the basic difference between these two types of variance i.e. population variance and sample variance.

Complete step by step answer:
As mentioned in the above question we will find out the symbol that represents population and sample variation but first we will understand the meaning of both the terms. Let us start with sample variance, in this variance is taken with regards to the sample that is provided, the way of calculation will be the same as before i.e. first calculating the mean of the sample variance then adding the square of difference of mean and all respective elements in the sample i.e. \[{{\sum{\left( x-\bar{x} \right)}}^{2}}\] and divide the whole term by the population size i.e. N. so we will get the final equation for population variance as:
\[{{s}^{2}}=\dfrac{{{\sum{\left( x-\bar{x} \right)}}^{2}}}{n-1}\] Where \[{{s}^{2}}\] is the symbol for sample variance and \[\left( n-1 \right)\] is the degree of freedom (where n is the size of the sample).
Now when we see population variance the only difference in the above two type of variance is the denominator, in sample we take the denominator to be degree of freedom which is one less than the size of the sample i.e. \[\left( n-1 \right)\] and in population variance the denominator is the size of the population i.e. \[{{\sigma }^{2}}=\dfrac{{{\sum{\left( x-\bar{x} \right)}}^{2}}}{N}\] where \[{{\sigma }^{2}}\] is the symbol for population variance and N is the population size.
So to summarize all the symbols for sample variance is \[{{s}^{2}}\] and as for population variance it is \[{{\sigma }^{2}}\].

Note: In the above mentioned question we needed to find the symbol of the two different variance even though both define variance there is a significant difference between the two due to which the symbols are different.