Basics of Probability and Statistics Week 1 Challenge

1)Why there is a difference in the formula of variance for population and sample?

This is the formula for population and sample variance.

`"POPULATION VARIANCE"=1/n*Sigma_(i=1)^"N" (x-m)^2`

`"SAMPLE VARIANCE"=1/(n-1)*Sigma_(i=1)^n (x-m)^2`

Sample variance is divided by n-1 to give an unbiased estimate of the true variance If we just divide by n we will get variance value which is far from the population variance. It all depends on the sample, If the sample's mean is close to the population mean then we can get the variance by dividing with n. But while sampling it is distinctly possible to consider numbers in such a way as to not get the population mean. In that case dividing by n-1 will give a variance closer to the population variance. We cannot divide by n-2 since it will overestimate the sample variance and n will underestimate the variance

2)Difference between stratified and clustered sampling?

Stratified sampling is sampling technique in which population is divided into different groups on basis of some common factors like age, gender, relligion,etc. Then simple random sampling is carried out individually in each group to create a single sample with equal number of samples from each group

Clustered sampling is also similar to Stratified sampling but the groups are formed based on existing groupings. There is no homogenity within each group.

Stratified Sampling	Clustered Sampling
Stratified sampling is one in which population is divided into homogenous groups and then random sampling is done in each group.	Clustered Sampling is in which members are chosen from randomly occuring clusters
There is homogenity within each group	Each group consists of random individuals
There is heterogenity between groups	There is heterogenity with each groups
Gives more precision and better representation of population	This is done to reduce cost and increase the efficiency of sampling but may not be a accurate representation of the population

3)How many different samples can be created out of population of size n?

There are to ways to sample a population . One is with replacement and other without.

With replacement

With replacement is that while sampling if a unit has already been included in a sample then it can be used again to be a part of another sample. Then formula is

`N^n`

Without replacement

Without replacement means once if an unit has been included in a sample it cannot be a part of another sample.

`(N!)/(n!(N-n)!)`

4)What is the probability of drawing 2 jacks from a pack of card?

There are four jacks in a pack of cards.

With replacement

Probability of getting one jack is`4/52`

Probability of getting 2 Jacks is `4/52 * 4/52 = 1/169`

Without replacement

Probability of getting 2 jacks is `4/52 * 3/51 = 1/221`

5)What is the probability that both the numbers on dice are same while rolling 2 dice?

There are 36 different combinations possible while rolling 2 dice and in that we get 2 numbers while rolling 6 times.

So probability of getting same numbers on both dice is `6/36 = 1/6`

6)Why standard deviation when variance can measure the dispersion in data?

Even though both give a measure of dispersion variance is calculate by adding the square of difference of data from mean. Since it is squared the units change will not give a clear interpretation numerically to the amount of dispersion. So square root of variance has to be taken to get standard deviation which is the same unit as of the original data points thus giviing a better numerical representation of amount of dispersion in data.