Basics of Probability and Statistics Week 1 Challenge
- Difference in formula of variance for population and sample
The calculation for variance of population and sample remains the same until the last step. It starts with calculating the mean, estimating the deviations from the mean, Square it and sum it up.
The calculation for variance in case of a Population of size 'N', the sum of the deviation squares is divided by 'N'. The value obtained captures the variance accurately for the entire population.
The calculation for variance in case of a sample of size 'n', the sum of deviation squares is divided by 'n-1'. Dividing the sum by 'n-1' increases the variance slightly and provides a better estimate of the population's variance. It accomodates for the fact that we are working with a sample and does not capture the entire data of the population. - Difference between clustered and stratified sampling
Clustered sampling divides the target population into multiple clusters and random clusters are selected to form a sample. The clusters are most likely naturally formed. This type of sampling is highly cost effective and less time consuming, however slection of clusters are random is prone to result in more errors.
Startified samples divides the target population into unique, homogenous strata. Data/members from each strata is selected randomly and combined together to form a sample. Elements of each strata has proportionate representation in the considered sample and results in a more precise result. This sampling technique is however very time consuming and strenous task to undertake as the population size increases.
As an example, Let us consider a researcher trying the estimate the education and career parameters among various ethnicities.
If the reaseracher chooses stratified sampling, The researcher can go ahead and collect the data from all the colleges and universities from target city/state. Proptionately divide them into stratas and select random members from each strata to form a sample. This will result in a very accurate and precise result. However the effort required to collect the data is strenous.
If the researcher chooses clustered sampling, The reasearcher can select 5 to 6 colleges at random which has a good representation of all ethnicities. The effort required is reduced and making the sampling technique cost effective. However the sample fails to capture the population data accurately and in an unbiased fashion. - Number of different samples that can be created out of Population of size 'n'
Type of sampling has not been specified. Assuming Random sampling technique,
'nC1' combinations of samples containing 1 data point.
'nC2' combinations of samples containing 2 data points.
'nC3' combinations of samples containing 3 data points.
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So on and so forth
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'nCn' combinations of samples containing n data points.
Total number samples = nC1 + nC2 + nC3+ ...... + nCn-1 +nCn
OR
Total number of samples = `sum_(i=1)^n nCi` - Probablity of drawing 2 jacks from a pack of 52 cards
Probablity = (Combinations of jacks)/(Combinations of deck)
In convention, A deck of 52 cards contain 4 jacks. The number of combinations containing 2 jack cards can be calculated as follows.
Combinations of jacks = 4C2 = 6
The number of 2 card combinations that can be made out of 52 cards can be calculated as follows
Combinations of deck = 52C2 = 1326
So the probablity of drawing 2 jacks (p) from a deck of 52 cards is
p = 6/1326 = 0.0045 - Probability that both the numbers on dice are same while rolling 2 dice
Probablity = (Combinations of same number sides)/(Combinations of 2 dices)
The combinations which will have same numbers on 2 dices : { (1,1),(2,2),(3,3),(4,4),(5,5),(6,6)} = 6
The number of combinations that can be made out of 6 sided dice can be calculated as follows
Combinations of deck = 6C2 = 15
So the probability that both the numbers on dice are same (p) while rolling 2 dice
p = 6/15 = 0.4 - Why standard deviation when variance can measure the dispersion in data
Standard deviation and Variance both measure the dispersion of data points from the mean. While calculating the variance, the deviations are sqaured and summed up to avoid the sum of deviations tending to zero. When the deviations of data points are to compared with respect to mean, the variance cannot be used as the units of mean and variance are not same. To compare dispersion of data points in terms of same units, standard deviation is required. Standard deviation captures the data dispersion from mean in same units.