Basics of Probability and Statistics Week 1 Challenge

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

ANS:

The true variance of a population could be estimated by considering the total population only. But this is not possible every time, especially if the population is very large. So, we resort to the measurement of variance of the population by calculating the variance of the sample. As we are not calculating the variance of the total population and estimating the variance from a small sample from the population it makes the variance “Biased”. As per Wikipedia https://en.wikipedia.org/wiki/Bias_of_an_estimator, this bias could be removed by dividing the square of deviations by n-1 instead of n while calculating the variance for a sample. This is the reason for the difference in the formulas.

2. What is the difference between stratified and clustered sampling?

ANS:

Stratified Sampling: In stratified sampling, we divide a population into sub-populations or subgroups and then select samples from each group through random sampling or systematic sampling. This ensures that objects from each subgroup are present in the sample.

The sub-groups may be formed on the basis of age, gender, level of education, socio-economic background etc. This enables the researcher to study particular subgroups in a population.

Example: If a company is having 200 male employees and 800 female employees then a sample of 100 people can be formed by taking 80 female employees and 20 male employees instead of selecting people randomly which may or may not lead to representation of both genders proportionally.

Clustered Sampling: This is used while evaluating a large population spread over a huge geographical area like a country/continent. This method is particularly useful if the population is too large for random sampling. A given geographical area is divided into regions separated by boundaries. All such regions are equally likely to be selected.

Example: Assessment of food habits of a people in South-East Asia can be carried out by selecting several countries like Vietnam, Singapore, South Korea instead of carrying out a survey for all the people form each and every country.

So in stratified sampling, we create sub-groups in a population to ensure representation of each group in the sample whereas in clustered sampling we create subgroups because the existing population is too large for random sampling.

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

ANS:

From a population of size n, we can form groups of size 1, 2, 3, …, n samples. So this is the same as the number of ways of choosing 1 object out of n, 2 objects out of n, 3 objects out of n,…, n objects out of n. We need to take the sum of all such ways which leads to the following series:

nC1 + nC2 + nC3 + nC4 +…+ nCn which is equal to 2ⁿ-1.

4. What is the probability of drawing 2 jacks from a pack of cards?

ANS:

Probability of drawing 2 jacks from a pack of cards.

5. What is the probability that both the numbers are the same while rolling 2 dice together?

ANS:

The probability that both the numbers are the same while rolling two dice

= `6/36=0.667`

6. Why do we have standard deviation if variance can measure the spread of data?

ANS:

Two quantities with different units can not be compared. Variance is having the unit of the square of the unit of a dataset whose sample we are analysing. Of course, variance gives an idea of the spread of the data we are analysing but if we would like to plot it on the same graph as that of the data then we need to bring it to the same units as that of the data being analysed. This is exactly the reason we have a standard deviation.