Basics of Probability and Statistics Week 1 Challenge

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

The variance is a way to measure the spread of values in a dataset.

The formula to calculate population variance is:

σ² = Σ (x_i – μ)² / N

where:

• Σ: A symbol that means “sum”
• μ: Population mean
• x_i: The i^th element from the population
• N: Population size

The formula to calculate sample variance is:

s² = Σ (x_i – x)² / (n-1)

where:

• x: Sample mean
• x_i: The i^th element from the sample
• n: Sample size

Notice that there’s only one tiny difference between the two formulas:

When we calculate population variance, we divide by N (the population size).

When we calculate sample variance, we divide by n-1 (the sample size – 1).

When calculating the sample variance, we apply something known as Bessel's correction – which is the act of dividing by n-1.

This means that the sample variance tends to be lower than the population variance because it only takes into account the variability within the sample, and not the variability of the entire population.

2.Difference between stratified and clustered sampling

Stratified sampling is a sampling technique used to divide a population into homogeneous subgroups or strata, and then randomly selecting members from each strata to form a sample population. It is used to ensure that both strata are represented in the sample population.

Cluster sampling is a sampling technique in which the entire population is divided into clusters or groups, and then a sample of those clusters is selected. This is done to reduce the amount of time and cost associated with selecting an individual sample from the population. It is also used when it is not possible to access all members of the population.

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

The number of different samples that can be created out of a population of size n is equal to n choose r, where r is the size of the sample. This can be calculated using the formula n!/(r!(n-r)!).

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

The probability of drawing 2 jacks from a pack of cards is (4/52) x (3/51) = 1/221.

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

1/36

6.Why standard deviation when variance can measure the dispersion in data

The reason why standard deviation is so popular as a measure of dispersion is its relation with the normal distribution which describes many natural phenomena and whose mathematical properties are interesting in the case of large data sets