Basics of Probability and Statistics Week 1 Challenge
1) Why there is a difference in the formula of variance for population and sample ?
The difference between the formula of variance in population and sampling is because :
The variance is calculated in five steps
- First, mean is calculated
- Then deviations from the mean
- deviations are squared
- Squared deviations are summed.
- Sum is divided by number of items.
It is in the fifth step that the difference is seen in the formula between population and sample
In case of the population we divide the whole sum by n, since n represents number of items in the population.
However in case of sample we consider n-1 items since as we lack the information about the population. also the variance calculated in case of sample is different than the population.
The mean of the population is :
`mu = 1/n sum_(i=1)^n (x i)`
the variance for population is:
`sigma^2 = 1/n sum_(i=1)^n (x i - mu ) ^2`
The formula for mean in the sample is:
`overline x = 1/N sum_(i=1)^N (x i )`
the formula for variance is:
`s^2 = 1/(N-1) sum_(i=1)^N (x i - overline x ) ^2`
2) Difference between stratified and clustered sampling
Stratified Sampling Method is type of probability sampling where the whole population is divided into mutual homogeneous subgroups which are termed as strata. After such a bifurcation, we select a random sampling from each group called as stratum which are together combined to form a single sampling. In general, a stratum is nothing but a homogeneous subset of a given population. All these stratums make together, the strata. The factors along which these separations are done are age, gender, income, race, religion etc.
Cluster Sampling is another method of population sampling in which the population is subdivided into already existing groups or clusters. The sample of the cluster is selected randomly from the population. It is a heterogenous process.
The differences between cluster sampling and stratified sampling:
| Cluster Sampling | Stratified Sampling |
| It is a Cost Reduction process | It is an enhanced process |
| We use randomly selected clusters or groups | We use random members from the strata developed |
| It results in more errors | it results in reduced errors |
| It is homogeneity externally | it is homogeneity internally |
3) How many different samples can be created out of a population of size n ?
There are two ways of selecting a sample from a population.
One is with replacement where the sample considered once is again replaced back to the population and again randomly selected from the population which might have the items from already selected sample.
The formula for such a sample is `N^n`
The other is without replacement where the selected items from the sample are reduced from the actual population and are not replaced back.
The formula for such a sample is : `(N!) / ((N-n)! n!)`
4) What is the probability of drawing 2 jacks from a pack of card
Since we are considering the fact that the drawn card is not replaced again the probability would be calculated using the permutation and combination technique.
There are actually 4 jacks from a pack of 52 cards.
The probability of drawing 2 jacks from a 4 jacks is `4c2` which is calculated as `(4!) / ((4-2)! 2!)` which results to `(4 xx 3) / (2 xx 1)`
The total probability of drawing 2 cards from a pack of 52 cards is given as `52c2`which is shown as `(52!) / ((52-2)! 2!` which results to `(52 xx 51) / (2 xx 1)`
So the total probability of drawing 2 jacks from 52 cards is `(4 xx 3 )/ (52 xx 51)`
5) What is the probability that both the numbers on dice are same while rolling 2 dice ?
Here the total sample of occurence while rolling 2 dices is `(6 xx 6)` = `36`
The occurence of same number on both the dices is `6`
So the total probability is `6 / (36)`
6) Why standard deviation when variance can measure the dispersion in data ?
Mean is the average of the given data values. A variance is used for measuring how far the data values is measured from the mean. Standard deviation is used generally to estimate the amount of dispersion from the mean. variance is generally considered as the average taken out of the squared deviations whereas standard deviation is considered as the root of mean squared deviation. Variance reflects the degree of spread in the data set. The more spread the data, the larger the variance is in relation to the mean. However standard deviation just explains the dispersion so it is the best way of calculating the dispersion. The reason for being the best method is because if observations are from normal distribution then 68% lie between mean `+-1` SD and 95% lie between mean `+-2`and 99.7% observations lie between `+-3`SD. Using these applicable ratios of observation we can determine the dispersion in data easily.