Basics of Probability and Statistics Week 1 Challenge
1. Why there is difference in the formula of variance for population and sample:
-> There is a difference in the fromula of variance for population and sample because we have to deal the inherent uncertainty when we working with sample data.
Population Variance:
Population variance denoted by sigma⊃2;;(σ⊃2;;),meassured by the avarage of squre of difference between every data point and population mean.
Actually it meassure the total spread or dispersion of the population as per its mean.
Formula:
σ⊃2;;=Σ(x-μ)⊃2;;/N
here:
x represents every data point in the population
μ represents the mean of overall population
N represents the total no. of data points in the population
Sample Variance:
When we work with a specific sample of data the we don't have all the data point so here uncertainty comes the in calculation.For this specific reason we use (N-1) instead of N(the total no. of data points in the population).Sample population denoted by (s⊃2;;).
Formula
s⊃2;;=Σ(x-μ)⊃2;;/N
here:
x represents every data point in the sample
μ represents the mean of the sample data set
N represents the total no. of data points in the sample.
Ultimately it gives us better estimate of true population variance although we are handling very less data rather than all population.
Finally for the introduction of uncertainty we or statistician have to change the formula.
2. Difference between stratified sampling and cluster sampling:
-> These two are the group of random sampling or probability sampling (any data point or data cluster or data set can be selcted as a sample).
stratified sampling:
For this sampling at first the complete data or population is divide in to different group or strata as per specific characteristic (e.g. age,gender,city etc). Then we take the random samples or data point from those groups and make a sample data set.
cluster sampling:
For this sampling at first the complete data or population randomly divided in groups or cluster and from them a group or cluster is selected for sampling.All data points of the particular group are selected in the sampling.
3. How many different sample created out of population size of n:
-> If there is replacement take place at the time of sample creation then we can create infinite sample because we can replace every data point indifinitely.
If else there is no replacement , then for population size of n and sample set size k , we can create this C(n,k) no. of sample.
C(n,k)=n!/k!(n-k)!
Consider n=0,1,2,3 for k = 0 to 3
we can found :
n0123∣∣∣∣∣∣∣∣01111k112321331
4. What are the probablity of draw two jacks from a pack of card:
-> First found the no. of total way to draw 2 cards from pack = C(52,2)
There is 4 jack in every pack of card and after drawing one jack we have choose from other 3 , so the no. of way we can get exactly 2 jacks is = C(4,2)
the probablity of draw two jack is = no. of way we can get exactly 2 jacks/no. of total way to draw 2 cards from pack
= C(4,2)/C(52,2)=6*2/52*51=3/663
5. What is the probablity of both the no. of dice are same when rolling to dice:
-> Total no. of possible outcome = 36
there no. of preferable event is = 6 , {(1,1),(2,2),(3,3),(4,4),(5,5),(6,6)}
the probablity of both the no. of dice are same when rolling to dice is= 6/36=1/6
6. Why stdev when variance can be meassure by despersion of the data:
-> Variance is a valuable measure of dispersion, but it has the limitation of being in squared units, which can be less intuitive. It gives the average squared deviation from the mean.
Standard deviation, on the other hand, addresses the issue of squared units by taking the square root of the variance. This provides a measure of dispersion in the same units as the original data, making it easier to interpret.
In summary, while variance measures dispersion by averaging squared differences, standard deviation is preferred when you want a more interpretable measure of dispersion that reflects the original units of measurement