Basics of Probability and Statistics Week 1 Challenge
1. Why there is a difference in the formula of variance for population and sample
Ans
Given below is the corresponding formulas to compute population variance and sample variance respectively. As we can see that the formula for sample variance is slightly different to that of the population variance.
To compute the variance of the variable say, X in a population of size N one takes the average of the squares of deviations of each variable from its Mean, M. This leads to an occurance of two peculiar scenarios:
1. The mean, M is known and given,
2. The mean M is not known.
With case 1, where M is known, the Variance is computed by taking the Sum of all sum of squares of difference between initial value and the mean and divided by the population size N to average out. This is the population variance.
In case 2, the M is unknown. That needs to be estimated and the sample mean m being the best unbiased estimate of the population mean, one competes that from a Random sample of size n drawn from the that population. The mean m is used instead of M to compute the total SS of squares, but only for the sample values, which are n in numbers. But, we have a situation in which we already had to estimate the M using the sample values, so that process has a price to pay. And that price is One Degree of Freedom( d.f.) That means we are restricted to taking the SS only for the sample and deviations from m. So, now the dividing quantity is ( n-1) and not n.
Using sample mean m, the best estimator of M, gives the unbiased estimate of the population variance too.
One can of course always divide by just n , but then the estimate is a biased estimate. And comparing the two kinds, biased and unbiased , one can find the amount of bias.
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2. Difference between stratified and clustered sampling
Ans
Definition of stratified sampling
Stratified sampling could be a style of probability sampling, within which first of all the population is bifurcated into various mutually exclusive, homogeneous subgroups (strata), after that, a subject is chosen randomly from each group (stratum), which are then combined to make one sample. A stratum is nothing but a homogeneous subset of the population, and when all the stratum are taken together, it's referred to as strata.
The common factors within which the population is separated are age, gender, income, race, religion, etc. a very important point to recollect is that strata should be collectively exhaustive in order that no individual is disregarded and also non-overlapping because overlapping stratum may lead to the rise within the selection chances of some population elements. The sub-types of sampling are:
Proportionate stratified sampling
Disproportionate stratified sampling
Definition of Cluster Sampling
Cluster sampling is defined as a sampling technique within which the population is split into already existing groupings (clusters), and so a sample of the cluster is chosen randomly from the population. The term cluster refers to a natural, but heterogeneous, intact grouping of the members of the population.
The most common variables employed in the clustering population are the region, buildings, school, etc. Heterogeneity of the cluster is a very important feature of a perfect cluster sample design. the categories of cluster sampling are given below:
Single-stage cluster sampling
Two-stage cluster sampling
Multistage cluster sampling
Major Differences Between Them:
1. In stratified sampling the individuals are randomly selected from all the strata, to constitute the sample. On the opposite hand cluster sampling, the sample is made when all the individuals are taken from randomly selected clusters.
2. In cluster sampling, population elements are selected in aggregates, however, within the case of stratified sampling the population elements are selected individually from each stratum.
3. In stratified sampling, there's homogeneity within the group, whereas within the case of cluster sampling the homogeneity is found between groups.
4. Heterogeneity occurs between groups in stratified sampling. On the contrary, the members of the group are heterogeneous in cluster sampling.
5. When the sampling method adopted by the researcher is stratified, then the categories are imposed by him. In contrast, the categories are already existing groups in cluster sampling.
6. Stratified sampling aims at improving precision and representation. Unlike cluster sampling whose objective is to enhance cost effectiveness and operational efficiency.
The total number of cards would be 52 in a fair pack of playing cards. Similarly, the number of jacks in a pack of cards is 4 (each family contain one jack each).
The general probability formula is given by
Here, there can be two possibilities here, first one being with replacement and second one being without replacement.
⦁ With replacement (the first card chosen is put back before choosing the second one)
In this case, the probability of choosing 2 jacks would be probability of choosing one jack for the first time, put the card back and choose a jack the second time.
Probability of choosing a jack for the first time = 4/52.
Probability of choosing a jack for the second time = 4/52.
Since, it says ‘and’, we multiply both and hence, the probability of getting a jack 2 times would be = 4/52 * 4/52 = 16/2704 or 1/169.
⦁ Without replacement (the first card chosen is not put back before choosing the second one)
In this case, the probability of choosing 2 jacks would be probability of choosing one jack for the first time, do not put the card back and choose a jack the second time.
Probability of choosing a jack for the first time = 4/52.
Probability of choosing a jack for the second time = 3/51 (one card removed).
Since, it says ‘and’, we multiply both and hence, the probability of getting a jack 2 times would be = 4/52 * 3/51 = 12/2652 or 1/221.
While rolling a pair of dice, there are 36 combinations possible which are listed out below;
The combination we want is that both should give the same number upon rolling together and the sample space would thus be the highlighted ones in the image.
Probability (A) = 6/36 = 1/6
Variance measures the dispersion of the data from the mean of the data points.
Say for example, we have a set of data points as listed below that shows the weight of 7 bags of rice sent to customers in a city every day.
(100+150+50+200+500+100+200)/7 = 1400/7 = 200 kg.
The variance of this data would be 20,714.28 kg2 units.
But this huge number does not make any sense with a different unit being kg2 which is not useful for a meaningful dispersion comparison.
With variance alone, it is not possible to find whether all 7 customers get equal amount of rice bags on an average. So, we convert the kg2 to kg by taking a root (arriving at standard deviation) that allows us to have a more clear inference from these values.
Standard deviation is square root of variance which is 143.92 kg.
The standard deviation is high meaning that the customers do not get equal amount of rice.
It can also be compared with the mean to find out whether there is abnormal data distribution or not.