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Fundamentals of Statistical Analysis. Dr. Surej P John. Definition of Variables. A variable is an attribute of a person or an object that varies. Measurement are rules for assigning numbers to objects to represent quantities of attributes. Back to Table of Content. Definition.

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## Fundamentals of Statistical Analysis

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**Fundamentals of Statistical Analysis**Dr. Surej P John**Definition of Variables**• A variable is an attribute of a person or an object that varies. • Measurement are rules for assigning numbers to objects to represent quantities of attributes. Back to Table of Content**Definition**• Datum is one observation about the variable being measured. • Data are a collection of observations. • A population consists of all subjects about whom the study is being conducted. • A sample is a sub-group of population being examined.**What Is Statistics?**• Statistics is the science of describing or making inferences about the world from a sample of data. • Descriptive statistics are numerical estimates that organize and sum up or present the data. • Inferential statistics is the process of inferring from a sample to the population.**Five Types of Statistical Analysis**• Descriptive analysis – data distribution • Inferential analysis – hypothesis testing • Differences analysis – hypothesis testing • Association analysis – correlation • Predictive analysis – regression**Descriptive vs. Inferential Statistics**A Hypothesis: • A statement relating to an observation that may be true but for which a proof (or disproof) has not been found • The results of a well-designed experiment or data collection may lead to the proof or disproof of a hypothesis**Inferential Statistics**Samples Sub-samples Population**For example, Heights of male vs. female at age of 25.**Our observations: male H > female H; it may be linked to genetics, consumption and exercise etc. Is that true for male H> female H? i.e. Null hypothesis: male H ≤ female H Scenario I: Randomly select 1 person from each sex. Male: 170 Female: 175 Then, Female H> Male H ? Scenario II: Randomly select 3 persons from each sex. Male: 171, 163, 168 Female: 160, 172, 173 What is your conclusion then? Which is the better Scenario?**Important messages here:**• Sample size is very important and will affect your conclusion • Measurement results vary among samples (or subjects) – that is “variation” or “uncertainty”. • Variation can be due to measurement errors (random or systematic errors) and inherent within samples variation. For example, at age 20, female height varies from 158 to 189 cm. Why? • Therefore, in Statistics, we always deal with distributions of datarather than a single point of measurement or event.**Moments of a Normal Distribution**Each moment measures a different dimension of the distribution. 1. Mean (1st moment) 2. Standard deviation (2nd moment) 3. Skewness (3rd moment) 4. Kurtosis (4th moment)**Mean**mean Mean (µ) is equal to the sum of n number of observation divided by the number of observations (sample size) Mean = Sum of values/n = Xi/n e.g. length of 8 fish larvae at day 3 after hatching: 0.6, 0.7, 1.2, 1.5, 1.7, 2.0, 2.2, 2.5 mm mean length = (0.6+0.7+1.2+1.5+1.7+2.0+2.2+2.5)/8 = 1.55 mm 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 mm**Standard deviation**• The standard deviation (SD) (represented by the Greek letter sigma, σ) shows how much variation or dispersion from the average exists. • A low standard deviation indicates that the data points tend to be very close to the mean (also called expected value); a high standard deviation indicates that the data points are spread out over a large range of values. • The formula is easy: it is the square root of the Variance.The Variance is defined as: the average of the squared differences from the Mean.**Skewness**• In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. The skewness value can be positive or negative, or even undefined.**Kurtosis**• The coefficient of Kurtosis is a measure for the degree of peakedness /flatness in the variable distribution. Kurtosis > 0 Kurtosis = 0 Kurtosis <0**Frequency Distribution**• In statistics, a frequency distribution is an arrangement of the values that one or more variables take in a sample. Each entry in the table contains the frequency or count of the occurrences of values within a particular group or interval, and in this way, the table summarizes the distribution of values in the sample. • Frequency distribution tables can be used for both categorical and numeric variables.**Cross Tabulation**• A cross-tabulation (or cross-tab for short) is a display of data that shows how many cases in each category of one variable are divided among the categories of one or more additional variables. • In a cross-tab, a cell is a combination of two or more characteristics, one from each variable. • If one variable has two categories and the second variable has four categories, for instance, the cross-tab will have 6 cells, each with a number specific to that category**Comparing Means**• We need to compare the means of groups in Inferential statistics. • T-tests and ANOVA (Analysis of Variance) are the methods commonly used for comparing means. • Independent T tests • Independent T tests are used for testing the difference between the means of two independent groups. For Independent T-tests, there should be only one independent variable but it can have two levels. There should be only one dependant variable. • Ex: gender (male and female) • How male and female students differ in academic performance?**Anova (Analysis of Variance)**• Anova is used as the extension of Independent t-tests. • This is used when the researcher is interested in whether the means from several ( >2) independent groups differ. • For Avova, only one dependant variable should be present. There should be only ONE independent variable present (but it can have many levels unlike in independent t-tests)**Statistical Errors in Hypothesis Testing**• Consider court judgments where the accused is presumed innocent until proved guilty beyond reasonable doubt (I.e. Ho = innocent)**Statistical Errors in Hypothesis Testing**• Similar to court judgments, in testing a null hypothesis in statistics, we also suffer from the similar kind of errors:

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