Basics of Probability and Statistics Week 1 Challenge

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

Variance is a measure of the spread of a set of data points around the mean.

Var(X) = Σ((Xᵢ - μ)²) / N

Xᵢ : individual data point in the population

μ : population mean

Σ :summation

N: number of data points in the population.

In sample variance is calculated using a modified formula to account for the fact that the data represent only a subset of the entire population.

Var(X) = Σ((Xᵢ - x̄)²) / (n - 1)

Xᵢ : Data point in the sample

x̄ : sample mean

Σ : summation

n : Total number of data points in the sample.

The difference is in the denominator. In the sample variance formula, the divisor is (n - 1) instead of N. This adjustment is to provide an unbiased estimate of the population variance based on the sample data. Dividing by (n - 1) instead of n ensures that the sample mean x̄ is used in the calculation, which introduces some uncertainty.

By dividing by (n - 1), it slightly increases the calculated variance compared to dividing by n, compensating for the sample mean which is likely to be closer to the data points than the population mean.

• Difference between stratified and clustered sampling?

Stratified sampling and clustered sampling are two distinct sampling techniques.Both methods is used for grouping elements within a population, they differ in how the groups are formed and how the sampling is conducted.

Stratified Sampling: It involves dividing the population into distinct, subgroups based on certain characteristics . The division is done in a way that elements within each group are relatively homogeneous, while the different group may have some degree of heterogeneity.

The objective of stratified sampling is to ensure that each subgroup is adequately represented in the sample.

Clustered Sampling: It involves dividing the population into clusters or groups.Clusters are formed based on geographical closeness or some other natural grouping condition.

cost-effective and logistically feasible.

used in surveys and studies

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

Total different samples=Combination taking1,2,3 .....n at a time

we know that nc0+nc1+nc2+......+ncn=2^n

so nc1+nc2+....+ncn=2^n-nc0

nc1+nc2+.....+ncn=2^n-1

so total different samples will be= 2^n-1

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

Number of ways to select 2 jacks = C(4, 2) =4!/2!*2! = 6

Total number of ways to select 2 cards = C(52, 2) = 52! / (2! * (52 - 2)!) = 1326

Therefore, the probability of drawing 2 jacks from a pack of cards is given by:

Probability = Number of favorable outcomes / Total number of possible outcomes = 6 / 1326 = 0.00452= 0.452%.

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

Total ways of getting doublet on both the dice=6

Total Sample Size=36

Probability = Number of favorable outcomes / Total number of possible outcomes = 6 / 36 = 1 / 6=16.67%

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

The standard deviation is expressed in the same units as the original data, making it more interpretable.

The standard deviation is more sensitive to outliers or extreme values in the data compared to the variance.