Basics of Probability and Statistics Week 1 Challenge

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

• In the world of Statistics, sampling always refers to selection of a section holistic of total statistical data with the main purpose of obtaining requisite values about the dataset. The whole of statistical information on a particular feature of all the elements covered by the process is called ‘population’ or ‘universe’.

  The part selected from the whole population which is to be used to obtain the characteristics of the population or universe is ‘sample’. The population is understood to be made of individual units or elements, and some of these elements are added in the sample. SO total number of elements of the population is stated to be population size, and for the sample it is called sample size.

  Both population and sample can be finite or infinite and similarly they can be both existent or hypothetical.

  Now if we consider variance, it is actualy a numerical value that explains how widely the individual figures in a dataset are distributed from the mean. To explain a little further, it is about how far each value is from the mean, and from each other. If variance is zero then it means all the data present are identical to each other. So more the variance, higher are the values spread out from the mean, also from each other. Also, lesser the variance, less are the values spread out from the mean, also from each other.

  And point to be noted is variance can’t be negative.

  The basic difference between population-variance and sample-variance actually relates to calculation of the variance. So to go a little further variance is calculated in five steps, firstly mean is calculated, then we have to calculate the deviations from the mean(first step), and thirdly the deviations need to be squared, fourthly the squared deviations are summed up and finally this sum is divided by number of items for which the variance is being calculated.

  Population variance represents the value of variance that is actually calculated from the 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.

  Now, if we look a little further the variance which is to be calculated from population data has n which is equal to the total number of items. So if variance in cholesterol level of all the 1000 people is to be calculated from data on cholesterol level of all the 1000 people, then here n = 1000. However when the variance is calculated from sample data a value of 1 is deducted from n before dividing the sum of the squared deviations. Thus in the above example if sample data have 100 items, the denominator would be 100 – 1 = 99.

  So as per the above noted example, the value of variance calculated from sample data is higher in comparision to the value that of population data. The logic behind this concept is to compensate our lack of knowledge about the population data. For instance it is implausible to find out the variance of heights in human beings, for our absolute lack of information about heights of all living human beings, not considering the future. Even if we take one moderate example, like population data on heights of all the living men in India(for say) and let's assume that this is physically possible, but the cost and time involved in this process would nullify our purpose of it's calculation. So keeping this reason in mind sample data are taken for most of the statistical purposes, and this is accompanied by lack of knowledge about majority of data. In order to compensate this, the value of variance and standard deviation, which is squared root of variance are higher in case of sample data than variance from population data.

  • Difference between stratified and clustered sampling.

  Stratified Sampling is a type of probability sampling which is conceptualized on dividing a population of data into strata, and elements for the sample are selected randomly from these strata. In stratified sampling, the strata must be homogenous in nature and also collectively exhaustive i.e; they cover all of the probability space, the chance of any one of them occurring is definite. Also strata must define a part of the population and the elements of the sample must be unique. So every element must be a part of one and only one strata in the population. This would mean that the entire population requires to be a part of the samples. So to ensure precision and reduce any sampling error, simple random sampling is employed in each stratum.

  Cluster Sampling is a technique used when homogeneity is external but heterogeneity is an internal within clusters or groupings. So this is a segregation process of population into multiple groups or clusters. Cluster Sampling is usually used as a decrementing factor for the number of interviews and the cost, to reach the desired accuracy. When the most part of heterogeneity is internal to the group in fixed sample size, so the chance of random error is decreased. So in cluster sampling there are single-stage and two-stage or multiple-stage sampling methods. These two methods are dependent upon the number of steps required to create the desired sample. Here the total population is divided into clusters. A simple random sample of the cluster is chosen and the elements in each of these clusters are then sampled. Briefly we can state that One-stage or Single-stage Cluster Sampling: A case when all elements in each cluster sample are sampled. And for Two-stage Cluster Sampling: When from each of these clusters a simple random subsample of elements is chosen.

  A tabulated differentiation is as follows:

  KEY INDICATORS	STRATIFIED SAMPLING	CLUSTER SAMPLING
  Sample	Randomly selected individuals are taken from all the strata.	All the individuals are taken from randomly selected clusters.
  Selection of Population Elements	Individually	Collectively
  Homogeneity	Within group	Between groups
  Heterogeneity	Between groups	Within group
  Bifurcation	Decided by the researcher	Naturally occurring groups
  Objective	To increase precision and representation.	To reduce cost and improve efficiency.

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

  This is one of the most common questions that is asked by people doing surveys in various development process. That “how big should my sample size be?”. While there are many sample size calculators and statistical guides available, but for those who never did statistics at university (or don't remember it at all) may find this process intimidating to use. Firstly I will explain how to choose a sample size for a basic survey without any of the complicated formulas.
  Most of the statisticians agree upon the minimum sample size to fetch any kind of meaningful result should be 100. And if your population is less than 100 then you really need to survey all of them.

  If we are considering blanket cases the a good maximum sample size is usually 10% as long as it does not exceed 1000. For example, in a population of 5000, 10% would be 500. In a population of 200,000, 10% would be 20,000. This exceeds 1000, so in this case the maximum would be 1000.

  Even in a population of 200,000, sampling 1000 people will normally give a fairly accurate result. Sampling more than 1000 people won’t add much to the accuracy given the extra time and money it would cost.
  If we are considering basic surveys such as feedback forms, requirement assessments, opinion surveys, etc. conducted as part of a program or surveys that use random sampling we can consider the above analogy.

  In case of complex research study, sample size is a research term used for defining the number of individuals included in a research study to represent a population. The sample size references the total number of respondents included in a study, and the number is often broken down into sub-groups by demographics such as age, gender, and location so that the total sample achieves represents the entire population. Determining the appropriate sample size is one of the most important factors in statistical analysis. If the sample size is too small, it will not yield valid results or adequately represent the realities of the population being studied. On the other hand, while larger sample sizes yield smaller margins of error and are more representative, a sample size that is too large may significantly increase the cost and time taken to conduct the research.
  In this scenario we can calculate the sample size using a sample calculation formula known as the Andrew Fisher’s Formula. And the steps include:
  1. Determine the population size (if known).
  2. Determine the confidence interval.
  3. Determine the confidence level.
  4. Determine the standard deviation (a standard deviation of 0.5 is a safe choice where the figure is unknown)
  5. Convert the confidence level into a Z-Score. This table shows the z-scores for the most common confidence levels:

  Confidence level	z-score
  80	1.28
  85	1.44
  90	1.65
  95	1.96
  99	2.58

  Sample Size = [(Z-score)^2 * Std-Dev * (1-Std-Dev)]/(Confidence Interval)^2

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

  Probabilty = 2/52

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

  Probability = 1/36

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

  Standard deviation and variance are types of statistical properties that measure dispersion around a central tendency which is most commonly the arithmetic mean. They are descriptive statistics that measure variability around a mean for continuous data. The greater the standard deviation and variance of a particular set of scores, the more spread out the data points are around the mean.

  Standard deviation and variance are closely related descriptive statistics, though standard deviation is more commonly used because it is more intuitive with respect to units of measurement whearas variance is reported in the squared values of units of measurement, whereas standard deviation is reported in the same units as the data.
  They each have different purposes. The Standard deviation is usually more useful to describe the variability of the data while the variance is usually much more useful mathematically.

  Standard deviation is the best measures of dispersion, because it posseses most of the characterstics of an ideal measure of dispersion.
  1. Most of the statistical theory is based on Standard Deviation. It helps to make comparison between variability of two or more sets of data. Also, Standard Deviation helps in testing the significance of random samples and in regression and correlation analysis.
  2. It is based on the values of all the observations. In other words, Standard Deviation makes use of every item in a particular distribution.
  3. Standard Deviation has a precise value and is a well-defined and definite measure of dispersion. That is, it is rigidly defined.
  4. It is independent of the origin.
  5. It is capable of further algebraic treatment.