Introduction to statistics
- 1. Introduction to Statistics The science of collectiong, organizing, presenting, analyzing, and interpreting data to assist in making more effective decisions Statistical analysis – used to manipulate summarize, and investigate data, so that useful decision-making information results. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of surveys and experiments. Uses and Functions of Statistics (1) Statistics helps in providing a better understanding and exact description of a phenomenon of nature. (2) Statistics helps in the proper and efficient planning of a statistical inquiry in any field of study. Statistics helps in collecting appropriate quantitative data. (4) Statistics helps in presenting complex data in a suitable tabular, diagrammatic and graphic form for easy and clear comprehension of the data. (5) Statistics helps in understanding the nature and pattern of variability of a phenomenon through quantitative observations. (6) Statistics helps in drawing valid inferences, along with a measure of their reliability about the population parameters from the sample data. Types of statistics Two main statistical methods are used in data analysis: descriptive statistics, which summarize data from a sample using indexes such as the mean or standard deviation, and inferential
- 2. statistics, which draw conclusions from data that are subject to random variation (e.g., observational errors, sampling variation). Descriptive statistics – Methods of organizing, summarizing, and presenting data in an informative way A descriptive statistic (in the count noun sense) is a summary statistic that quantitatively describes or summarizes features of a collection of information, while descriptive statistics in the mass noun sense is the process of using and analyzing those statistics. Descriptive statistics is distinguished from inferential statistics (or inductive statistics), in that descriptive statistics aims to summarize a sample, rather than use the data to learn about the population that the sample of data is thought to represent. Descriptive statistics are most often concerned with two sets of properties of a distribution (sample or population): central tendency (or location) seeks to characterize the distribution's central or typical value, while dispersion (or variability) characterizes the extent to which members of the distribution depart from its center and each other. Inferences on mathematical statistics are made under the framework of probability theory, which deals with the analysis of random phenomena. Inferential statistics – The methods used to determine something about a population on the basis of a sample Statistical inference is the process of using data analysis to deduce properties of an underlying probability distribution. Inferential statistical analysis infers properties of a population, for example by testing hypotheses and deriving estimates. It is assumed that the observed data set is sampled from a larger population. Inferential statistics can be contrasted with descriptive statistics. Descriptive statistics is solely concerned with properties of the observed data, and it does not rest on the assumption that the data come from a larger population. Non-Probability and Probability Sampling Sampling is the method of selecting a representative subset of the population called sample. Sampling makes research more accurate and economical. It’s the sampling method which
- 3. actually determines the generalizability of the research findings. In simple words, the process of choosing a sample of the population to study is called sampling. The sample should represent the population in all the respects. So, the question here is how to draw a sample? Through a sampling scheme, the researcher can choose a sample drawn by assigning selection probabilities with each draw. There are mainly two types of sampling techniques; probability sampling and non-probability sampling. Probability Sampling In probability sampling, each sample has an equal probability of being chosen. We can say, a probability sample is one in which each element of the population has a known non-zero probability of selection. This method of sampling gives the probability that our sample is representative of a population. Some probability sampling methods are as follows; Simple Random Sampling Stratified Random Sampling Systematic Random Sampling Cluster Sampling Multi-stage Systematic Sampling 1.1 Simple Random Sampling Simple random sampling is a completely random method of selecting a sample in which each element and each combination of elements in the population have an equal probability of being selected as a part of the sample. Being one of the simplest forms of random sampling, this method is a fair way to select a sample. As each member of the population has an equal probability of being selected, simple random sampling is the best-known probability sample. Even though it may not be considered an ideal method of choosing the sample, still result obtained through this method has high external validity or generalizability as compared to some other method of sample selection. Simple random sampling can be done by using a number of techniques such as: (a) Tossing a coin. (b) Throwing a dice.
- 4. (c) Lottery method. (d) Blindfolded method. Random sampling can be conducted in two ways; with and without replacement. In case, a certain element is selected and has the chance to be selected again after the required variables are measured is called sampling with replacement. 1.2 Systematic Random Sampling Systematic sampling is an improvement over the simple random sampling. This method requires the complete information about the population. In this sampling method, we select one unit from the sampling frame and then calculations to draw following units are done on the basis of the interval size. Systematic sampling being a very easy method to do, you actually choose every “nth” participant from a complete list. Even though each element has an equal probability of selection, but unlike as in simple random sampling, a combination of elements has different probabilities in systematic random sampling. In other words, the population must be listed in a random order and every element must be chosen from the sequence framed, as above. One of most widely used sampling methods, systematic random sample and simple random sample have same error rate if the list of the population is in random order. 1.3 Stratified Sampling Stratified Random Sampling is an improvement over systematic sampling. In this method, the population elements are divided into strata on the basis of some characteristics and from each of these smaller homogeneous groups draws at random a predetermined number of units. Stratified random sampling can be of two types (1) proportionate stratified sampling and (2) disproportionate stratified random sampling. When the size of the sample is proportionate to the size of the unit, it is called proportionate stratified sampling. 1.4 Cluster Sampling Cluster sampling is one of the efficient methods of random sampling in which the population is first divided into clusters, and then a sample is selected from the clusters randomly. Unlike the above, in pure cluster sampling, the whole cluster is sampled. In contrary to stratified
- 5. sampling, there should be heterogeneity within the clusters and homogeneity between the clusters. The more homogeneity among the clusters, lesser will be the margin of error or vice- versa. The method is mostly feasible in case of diverse population spread over different areas. 1.5 Multi-Stage Sampling To draw the sample, this method actually uses a combination of various techniques. In this method, the population is divided into groups at various levels. A group within a group, within a group and so on. The sample is finally drawn from the smallest group among all the groups. 2. Non-probability Sampling Unlike probability sampling method, non-probability sampling technique uses non- randomized methods to draw the sample. Non-probability sampling method mostly involves judgment. Instead of randomization, participants are selected because they are easy to access. One of the major shortcomings of the non-probability sampling is that the findings established through this method lack generalizability. Even though findings obtained through this method apply mostly to the group studied, it may be wrong to extend these findings beyond that particular sample. Some non-probability methods of sampling are as follows: Convenience Sampling Purposive Sampling Quota Sampling Snowball sampling 2.1 Convenience Sampling: In this type of sampling, researchers prefer participants as per their own convenience. The researcher selects the closest live persons as respondents. In convenience sampling, subjects who are readily accessible or available to the researcher are selected. For example, you will choose your classmates and friends for the study as per your convenience. 2.2 Purposive Sampling:
- 6. In this type of sampling, the researcher chooses the participants as per his/her own judgment, keeping back in mind the purpose of the study. It uses the judgment of an expert in selecting cases or it selects cases with a specific purpose in mind. This type of sampling is used in exploratory research or in field research. For example: For studying attitude toward any national issue, a sample of journalists, teachers and legislators may be selected for the study. 2.3 Quota Sampling: To understand this type of sampling, all we have to do is to understand the literal meaning of quota. In this sampling method, we pre-plan the number of participants in specified categories (For example; 100 literates, 100 illiterate). You select your sample according to some fixed quota. We allot shares to different groups (for example 100 men, 100 women). 2.4 Snowball Sampling: Also called "chain referral sampling,” in this method, the sample is actually collected in various stages. Snowball sampling which is a non-probability sampling method is basically socio-metric in nature. Although snowball sampling is considered to be a form of accidental sampling by some, this method is appropriate when the members of a special population are difficult to locate for example homeless people, migrant workers etc. It begins by the collection of data from one or more contacts usually known to the person collecting the data. At the end of the data collection process (e.g., questionnaire, survey, or interview). This process goes on till the purpose of the researcher is achieved. Numerical scale ofmeasurement: Nominal – consist of categories in each of which the number of respective observations is recorded. The categories are in no logical order and have no particular relationship. The categories are said to be mutually exclusive since an individual, object, or measurement can be included in only one of them. Ordinal – contain more information. Consists of distinct categories in which order is implied. Values in one category are larger or smaller than values in other categories (e.g. rating-excelent, good, fair, poor)
- 7. Interval – is a set of numerical measurements in which the distance between numbers is of a known, sonstant size. Ratio – consists of numerical measurements where the distance between numbers is of a known, constant size, in addition, there is a nonarbitrary zero point.