Basics of Probability and Statistics Week 1 Challenge

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

Answer 1: Population can be considered the whole data points and the sample can be considered as a small part of the population. While the statistical calculation of a population can be very time-consuming and labor-intensive therefore a sample from the population is taken for an efficient method of calculation.

Population Variance: σ² = Σ (x_i – μ)² / N

Sample Variance: s² = Σ (x_i – x)² / (n-1)

Here, the reason why sample variance has a different equation has to do with the concept of degree of freedom. which means the number of variables is free to vary. For example, let's say we have a sample of 100 items, the sample mean for them is 5. So the sum of all the values must be 500, as a result, once we know the value of 99 items, the value of the last item is determined on its own and thus it is not free to vary. so in order to prevent this we use n-1 to reduce bias.

Question 2: Difference between stratified and clustered sampling

Answer 2: Cluster sampling is a probability sampling method in which a population is turned into different clusters based on different characteristics or traits of the population. For example, a population in a region can be turned into different clusters based on age, industry in working, gender, etc. While stratified sampling is similar to a cluster in which each strata is made from the population, the difference in stratified is that each unit is randomly selected from each strata while in the clustering method, every unit of the randomly selected cluster is used in sampling.

Question 3: How many different samples can be created out of a population of size n

Answer 3: The number of possible samples from a population n that can be derived is n-1 of the population.

For example, consider a population {1,2,3,4} here the population size is 4 and the samples that can be taken from here are a sample size of "1" eg {1},{2} another sample size is "2" eg {1,2},{3,4} and another sample size is "3" eg {1,2,3},{2,3,4}. we cannot take sample size 4 as it is the population itself. Therefore the sample size in a population n will always be n-1.

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

Answer 4: The number of cards in a deck=52

Number of jacks in a deck=4

C(n,r)=P(n,r)/r!

The chances of picking 2 jacks out of 4= C(4,2) = (4 * 3)/2

The chances of picking 2 cards out of the 52 cards deck= C(52,2) = (52 * 51)/2

The probability of drawing 2 jacks from a pack of card= C(4,2)/C(52,2) = [(4 * 3)/2]/[(52 * 51)/2]

= (4 * 3)/(52 * 51)

= 1/221 =0.004

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

Answer 5:Probability = No. of the events/ Total number of events

Number of totals events = 6**2 = 36

Number of events in which the same number occurs = 6

The probability that both the numbers on the dice are the same while rolling 2 dice = 6/36 = 1/6 = 0.16

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

Answer 6:The variance provides the measure of dispersion that is how much the data points vary from the mean but the measure it gives will be the square of its values in the dataset. For example, if we are finding the variance of the dataset containing speed the variance will be Km2/h2 whereas the standard deviation measures how apart the numbers are in the dataset in the original metrics i.e as per the above example in Km/h.

Standard deviation is the root of variance.