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 (xi – μ)2 / N

where:

Σ: A symbol that means “sum”
μ: Population mean
xi: The ith element from the population
N: Population size
The formula to calculate sample variance is:

s2 = Σ (xi – x)2 / (n-1)

where:

x: Sample mean
xi: The ith 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.

2.In cluster sampling, population elements are selected in aggregates, however, in the case of stratified sampling the population elements are selected individually from each stratum.
 In stratified sampling, there is homogeneity within the group, whereas in the case of cluster sampling the homogeneity is found between groups.

3.There are 10 possible samples and each of them has a probability of selection equal to 1 / 10.
 The selected sample will be any one of these 10 samples. The sample selected in this manner is also called a simple random sample.

4.If there are 52 cards, and 2 of them are Jacks, the probability of drwaing a Jack is 2 out of 52, because there are 2 Jacks out of the 52 cards you have total.

5.= 36

The Sample Space for rolling a die is given as,

{(1,1) ,(1,2) , (1,3) , (1,4) , (1,5) , (1,6) ,

(2,1) ,(2,2) , (2,3) , (2,4) , (2,5) , (2,6) ,

(3,1) ,(3,2) , (3,3) , (3,4) , (3,5) , (3,6) ,

(4,1) ,(4,2) , (4,3) , (4,4) , (4,5) , (4,6) ,

(5,1) ,(5,2) , (5,3) , (5,4) , (5,5) , (5,6) ,

(6,1) ,(6,2) , (6,3) , (6,4) , (6,5) , (6,6)}

Sample points of getting same number on both dice- (1,1) ,(2,2) ,(3,3) ,(4,4) ,(5,5) & (6,6).

Thus, the number of favourable outcomes = 6

Total number of outcomes = 36

P (getting same number on both dice) = 6/36

= 1/6

Hence, the probability of getting same number on both the dice is 1/6.


6.Variance: The average squared deviation from the mean of the given data set is known as the variance.
 This measure of dispersion checks the spread of the data about the mean.
 Standard Deviation: The square root of the variance gives the standard deviation. 
 Thus, the standard deviation also measures the variation of the data about the mean.