Basics of Probability and Statistics Week 1 Challenge

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

Variance: Variance is a numerical value that shows how widely the individual figures in a set of data distribute themselves about the mean. That is how far each number is from the mean, and thus from each other.

• A variance of zero value means all the data are identical.
• More the variance, more are the values spread out about mean, hence from each other.
• Less the variance, less are the values spread out about mean, hence from each other, and variance can’t be negative.

Population variance refers to the value of variance that is calculated from population data, and sample variance is the variance calculated from sample data. Due to this value of denominator in the formula for variance in case of sample data is ‘n-1’, and it is ‘n’ for population data. As a result both variance and standard deviation derived from sample data are more than those found out from population data.

Q2. Difference between stratified and clustered sampling

Factors for Comparison	Stratified Sampling	Cluster Sampling
Definition	It is a sampling technique in which the population is divided into subgroups or strata. Here the samples are then extracted randomly from every group created.	It is a sampling technique in which the population is not divided manually into any groups. Here the samples are randomly picked from the naturally formed groups termed as clusters.
Purpose	To reduce cost and improve efficiency.	To increase precision and representation.
Divergence	Done by the researcher or group of researchers.	The clusters occur naturally forming subgroups.
Sample Selection	The sample is taken up randomly from all the manually created subgroups or strata.	The sample is taken up randomly from all over the naturally formed population clusters.
Population Assortment	The population elements are selected individually in a stratified sampling method.	In the cluster sampling method unlike stratified sampling, the elements of the population are selected collectively.
Heterogeneity	The samples are taken from between the manually created strata.	The samples are taken within the naturally developed group or cluster.
Homogeneity	The samples are taken from within the artificially created subgroups.	The samples in cluster sampling are taken from different natural clusters.
Uses	Diversification in Population.	No diversification in Population.
Subtypes	Proportionate Stratified Sampling, Disproportionate Stratified Sampling.	Single-stage Cluster Sampling, Double-stage Cluster Sampling, Multistage Cluster Sampling.
Pace	Stratified sampling is slower.	Cluster sampling is relatively faster.
Examples	• For a population of a state, male and female strata • For people working in a city, resident and non-resident strata • For students in a college, white, black, Hispanic, and Asian strata • For an audience of a debate regarding theology, Protestant, Catholic, Jewish, Muslim strata	• A researcher trying to understand the smartphone usage in Germany, here the cities of Germany will form clusters and select further towns with the highest population and also filter those using mobile devices. • A survey for evaluating the involvement of high school students in extracurricular activities. Here, selecting a class as the samples for the survey is cluster sampling. Then every member of the class is interviewed. In this case, classes are clusters of the student population.

Q3. How many different samples can be created out of population of size n

A simple random sample is a sample of size n drawn from a population of size N in such a way that every possible sample of size n has the same chance of being selected. Therefore

The number of different sample that can be created = Ncn
nCr = n! / (r!(n-r)!)
Since N=n, we can substitute n for n in the formula.
nCn = n! / (n!(n-n)!)
Simplify.
nCn = n! / (n! 0!)
n! over n! is 1. By definition 0! is also 1. So nCn = 1 when n=n

The size of the sample to be taken is equal to the size of the population. This means that the number of different random samples that can be taken is 1

Q4. What is the probability of drawing 2 jacks from a pack of card

Getting two cards out of 52 is

52C2=(52x51/2x1)=1326

Getting 2 jacks out of 4 jacks is

4C2=(4x3/2x1)=6

probability of getting 2 jacks out of 52 cards is given by

P=favorable outcomes/total outcomes

=6/1326

=1/221

Q5. What is the probability that both the numbers on dice are same while rolling 2 dice

When 2 dice are thrown simultaneously,

The number of possible outcomes are 36.

If getting the same number on both the dice is taken as event

For 1^st event: number of outcomes are 6

For 2^nd event

Number of outcomes/ Total number of outcomes.

6/(6*6) =6/36 = 1/6

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

Q6. Why standard deviation when variance can measure the dispersion in data

The mean is the average of the range of given data values, a variance is used to measure how far the data values are dispersed from the mean, and the standard deviation is the used to calculate the amount of dispersion of the given data set values.

The measures of dispersion can be categorised into two different types, namely absolute measure of dispersion and the relative measure of dispersion. When we consider the variance and standard deviation, both fall under the absolute measure of dispersion.