Basics of Probability and Statistics Week 1 Challenge

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

The formula used for determining variance for population is

`sigma^2 = sum(x_i - mu_x)^2/N`

Whereas, the variance for a sample is calculated using

`s^2 = sum(x_i - mu_x)^2/(N-1)`

It can be noted that the denominator for sample variance is reduced by 1. This results in a higher variance for the sample data as compared to the population.

For the cases where sampling techniques are used, it is very difficult to calculate the population statistics. For example, finding the statistics of weight of a population in a given region could be impossible as there exists huge number of people with varying range of weights. In such cases, we rely heavily on the sample data. To compensate for the lack of the information available regarding the population, we increase the variance of the sample data (by decreasing the denominator)

• Difference between stratified and clustered sampling?

Stratified sampling method involves forming stratas or groups from a population based on a distinct characteristics such as gender, age group, color etc. For determing the statistics, we take equal numbers of entities from each stratas. For examples, if there are 3 different colors of balls in the population, we segregate the balls based on the color. Later a definite number of balls are taken for each color and statistics are calculated. In stratified sampling, each strata is equally represented in the sample generated.

Clustered sampling method involves a similar approach of categorizing the population into clusters based on a distinct characteristic. However, cluster sampling is used for cases where the population is very large and it is difficult to form clusters representing all data points. In such case, the sample is generated from only a few clusters instead of all the clusters. This results in misrepresentation or biased representation of clusters in the samples generated.

• How many different samples can be created out of population of size n?

Let us consider a case where the size of population is 4

`X = {A, B,C,D}`

The different samples possible are : {A}, {B}, {C}, {D}, {A,B}, {A,C}, {A,D}, {B,C}, {B,D},{C,D},{A,B,C},{A,B,D},{A,C,D},{B,C,D},{A,B,C,D}

The total number of samples = 15 = `2^4-1`

Therefore, the total number of samples that can be generated from a population of size N = `2^N-1`

• What is the probability of drawing 2 jacks from a pack of card?

Number of jacks in a pack of card = 4

Considering without replacement, the probability of getting 2 jacks on picking 2 cards from a pack of card is

`P = 4/52 *3/51 = 1/221`

• What is the probability that both the numbers on dice are same while rolling 2 dice?

Outcomes when numbers are same on the dice twice = {(1,1), (2,2),(3,3),(4,4),(5,5),(6,6)}

Therefore, number of favorable outcomes = 6

Total number of outcomes = 36

Probability that both the numbers on dice are same while rolling 2 dice

`P= 6/36 = 1/6`

• Why standard deviation when variance can measure the dispersion in data?

The variance of a data is calculated by squaring the difference of the variable from its mean. Due to this, the unit of variance is not dimensionally same as that of mean.

To resolve this issue, standard deviation is introduced which is the square root of variance. Both mean and standard deviation are dimensionally equal.