Fundamental of Statistics and Types of Correlations
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This document provides an overview of key concepts in statistics including parametric vs non-parametric statistics, descriptive vs inferential statistics, types of errors, significance levels, correlation, and different correlation coefficients. Parametric statistics rely on assumptions of the normal distribution while non-parametric do not. Descriptive statistics describe data and inferential statistics draw conclusions. Type I and II errors occur when the null hypothesis is incorrectly rejected or not rejected. Significance levels like 0.05 are used to determine statistical significance. Correlation measures the relationship between variables from -1 to 1. Different coefficients like Pearson, Spearman, and Kendall's Tau are used depending on the scale of measurement and data distribution.
Fundamental of Statistics and Types of Correlations
- 1. Statistics Dr Rajesh Verma Assistant Professor in Psychology Govt. College Adampur, Hisar (Haryana)
- 2. Parametric and Non-Parametric Statistics. Parametric Statistics – The data that meets the assumptions of Normal Probability Curve (NPC) is analyzed through parametric tests. Based on parameters of normal curve. Assumption – The data should be normally distributed and Sample size > 30. Nonparametric Statistics – Not based on the parameters of the normal curve. The data violates the assumption of NPC and have limited sample sizes (e.g., n < 30).
- 3. Descriptive and Inferential Statistics. Descriptive Statistics – The statistical procedure of describing fundamental characteristics, features and properties of the collected data with the help of certain measures: - (i) Mean/median/mode (Central Tendency) (ii) Range, standard deviation, variance (dispersion) and (iii) Standard Error. The data is graphically (Graphs, Pie charts, tables), represented so that the trends it shows can be grasped easily.
- 4. Inferential Statistics – Inference means a conclusion reached on the basis of evidence and reasoning. A type of statistics in which conclusion are drawn from sample and generalized to the population from which sample was drawn. The data collected from sample is put to analysis techniques and inferences are drawn. The Analysis of Variance is the most often used method that tends to determines the probability of characteristics of the sample.
- 5. Error – The hypothesis is tested using inferential statistics based on probability theory therefore, no hypothesis test is foolproof. It means there are always chances of making errors while making conclusions. Hence there are chances of occurrence of two type of errors i.e. type I error and type II error.
- 6. Type I Error – Rejecting the true null hypothesis (False positive or incorrect rejection). It means there is significant difference between two groups or relationship between two variables when in fact there is not. E.g. a pathological test shows a disease when there is none. The probability of making a type I error is the level of significance set for the hypothesis test usually it is .05 and denoted by α (alpha). To decreased the type I error lower the level of significance or increase the sample size.
- 7. Type II error – Not rejecting the false null hypothesis (False negative or incorrect not rejection). It means there is no significant difference between two groups or no relationship between two variables when in fact there is. E.g. a pathological test fails to detect a disease when there is one. The probability of making a type II error is β, which depends on the power of the test. To decrease the chances of type II error is increase the sample size.
- 8. Setting Up the Level of Significance In English, "significant" means ‘important’, while in Statistics "significant" means probably true (not due to chance). A research finding may be true without being important. When statisticians say a result is "highly significant" they mean it is very probably true. They do not (necessarily) mean it is highly important. Popular Level of Significance in Social Sciences .05 and .01. The confidence with which an researcher rejects or retains a null hypothesis depends upon the level of significance.
- 9. Meaning of Levels of Confidence Le vel Amount of confidence Interpretation .05 95% If the experiment is repeated a 100 times, only on five occasions the obtained mean will fall outside the limited µ ± 1.96 SE .01 99% If the experiment is repeated a 100 times, only on one occasions the obtained mean will fall outside the limited µ ± 2.58 SE
- 11. Prior to the experiment for testing the hypothesis the researcher has to decide about the level of confidence or significance (i) .05 or 5% level of significance – A criteria for rejecting a null hypothesis (If a hypothesis is rejected at the 5% level it is said that the chances are 95 out of 100, that the hypothesis is not true and only 5 chances out of 100 that it is true). (ii) .01 or 1% level of significance – A criteria for rejecting a null hypothesis (If a hypothesis is rejected at this level, the chances are 99 out of 100, that the hypothesis is not true and that only 1 chance out of 100 is true).
- 12. Setting up Levels of Confidence (i) Sample Size – N < 30 or N > 30 (ii) Type of test – One tailed or two tailed test (iii) State the null hypothesis and the alternative hypothesis. (iv) Set the criteria for a decision i.e. .05 or .01 (iii) Level of significance for the hypothesis test is the probability used to define the very unlikely sample outcomes, if the null hypothesis is true. (iv) Critical Region - This is the region which is composed of extreme sample values that are very unlikely outcomes if the null hypothesis is true. The boundaries for the critical region are determined by the alpha level. If sample data fall in the critical region, the null hypothesis is rejected. (v) Collect data and compute sample Statistics.
- 13. Product Moment Coefficient Correlation (i) Definition - Correlation is a measure of association between to variables. The relationship between two variables can be - Linear - The relationship between two variables that can be plotted as a straight line. - Non-linear – The relationships that cannot be plotted as a straight line.
- 14. (ii) Direction of Relationship – Positive or Negative. (iii) Strength of Relationship – Range of correlation is -1 to +1 (perfect). The correlation between two variables is expressed in terms of a number and called as correlation coefficient that is denoted by r. The near the correlation coefficient to -1 or +1, stronger it is.
- 15. The strength is calculated in terms of percentage of r i.e. 𝒓 𝟐 𝐱100 for example if r = .73 then .5329x100 = 53.29%. The correlation coefficient is a description of association between two variables in the sample hence, it is a descriptive statistics. (ii) Measurements of Correlation – - Pearson Product Moment Method (Data must be in interval or ratio scale), - Partial Correlation, - Biserial Correlation, - Point Biserial Correlation, - Tetrachoric, and Phi coefficient, - Spearman Rank Order (Data in ordinal scale), and - Kendall’s Tau (Data in ordinal scale),
- 16. Pearson Product Moment Method XxY XxY X Sqr Y Sqr 11x13 143 121 169 13x16 208 169 256 16x14 224 256 196 9x10 90 81 100 6x8 48 36 64 17x16 272 289 256 7x9 63 49 81 12x12 144 144 144 5x7 35 25 49 14x15 210 196 225 1437 1366 1540
- 17. Pearson Product Moment Method Example (your book)
- 18. Significance of Correlation Coefficient When the Pearson’s Correlation coefficient is computed as an index of description of relationship between two variables in the sample, the significance testing is not required. The interpretation of correlation from the value and direction is enough. However, when correlation is computed as an estimate of population correlation, obviously, statistical significance testing is required.
- 19. Other Types of Correlation (Phi-Coefficient) Other types means other than Pearson r correlations. They are of three types: - (i) Special type Pearson Correlations (Point-Biserial Correlation and Phi coefficient), (ii) Non-Pearson Correlations (Biserial and Tetrachoric), and (iii) Rank Order Correlations (Spearman’s rho and Kendall’s tau).
- 20. Special type of Pearson Correlations (a) Point-Biserial Correlation (rPB) (PBS) – When variables are dichotomous i.e. the variables that have two mutually exclusive categories e.g. male-female, rural-urban etc. They are not continuous hence Pearson r can not be applied. For example – If correlate gender, then let male = 0, and females = 1 or any allot any other number to males and females. So PBS is used when one variable is Dichotomous and other continuous.
- 21. Example of Point Biserial Correlation
- 23. Phi Coefficient (ϕ) When both the variables are dichotomous the Pearson Correlation is called as Phi Coefficient (ϕ). Formula and other calculations will remain same.
- 25. Significance of Phi Coefficient The value of Phi Coefficient is squared and multiplied with n which is equivalent to Chi Square (χ²), the resultant value is checked in the Chi Square table. If the computed value is less than the tabulated value we retain the null hypothesis and vice-versa. As we know the relationship between Chi Square and Phi Coefficient is
- 26. Biserial Correlation Coefficient (rb) The Biserial Correlation Coefficient is calculated when both the variables are continuous but one of them is measured dichotomously. For example correlation between Academic Achievements (AA) and IQ where subjects are measured discreetly as those with Low IQ and High IQ. Take AA as Y and IQ as X Low IQ = 0 and High IQ = 1 Formula to calculate rb
- 27. Y1 = Mean of data correspondence to X = 1 Y0 = Mean of data correspondence to X = 1 Sy = Standard Deviation of Y Po = Proportion of data when X = 0 (Po/no) P1 = Proportion of data when X = 1 (P1/n1) H = Height of Normal Distribution Curve that divides Po and P1. Suppose Therefore X Y Y1 = 75 +90+48=213/3=71 0 80 Y0 = 80+64 = 144/2=72 1 75 P0 = (n0 = 2, total=5) 1 90 therefore P0 = 2/5 0 64 P1 = (n1 = 3, total=5) 1 48 therefore P1 = 3/5
- 28. Tetrachoric Correlation When both the variables are continuous but measured as Dichotomous then the technique used to assess the correlation between such variables is Tetrachoric Correlation. In other words a correlation between two dichotomous variables that have underlying continuous distribution. The Tetrachoric Correlation coefficient rtet tells the degree of the association between ratings for two raters. “0” indicates no agreement while “1” indicates a perfect agreement.
- 29. Tetrachoric Correlation Example from IGNOU Book For example, attitude towards females and attitude towards liberalization are the two variables that are to be correlated. In this case we collect the data dichotomously i.e. Positive or Negative Attitude. So, 0 represents ‘negative attitude’ and 1 ‘positive attitude’ on both the variables. The correlation between these two variables will be computed using Tetrachoric correlation.
- 30. Continued……… Check Cos value in the Cosine table or through internet. Therefore correlation between attitude towards Women and Liberalization is moderately strong and positive.
- 31. Rank Order Correlations When the data is in ordinal scale, rank order correlations are used. It means the data is presented in the rank such as first rank, second rank, third rank etc. In other words the individuals are assigned ranks. The rank-order correlations are applicable when the relationship between two variables is not linear but monotonic. Monotonic relationship means the values in the dataset consistently increases and never decreases or consistently decreases and never increases. We have two methods under Rank Order Correlations i.e. (i) Spearman’s Rank order Correlation or Spearman’s Rho ρ (rs). (ii) Kendall’s Tau (τ )
- 32. Spearman’s Rank order Correlation or Spearman’s Rho ρ (rs). Charles Spearman developed this technique in 1904. It is computed when the data is in Rank Order or when two judges for inter-judge agreement. Range is -1 to +1 and interpretation is dependent upon the sign and value of the coefficient. Where, D = Difference between ranks of X and Y n = Number of pair of ranks
- 33. Example Subject Rank (X) Rank (Y) Difference (D) D² 1 2 6 -4 16 2 4 8 -4 16 3 6 3 3 9 4 7 1 6 36 5 8 9 -1 1 6 1 2 -1 1 7 9 10 -1 1 8 10 4 6 36 9 3 5 -2 4 10 5 7 -2 4 Total 124 = 1 − 6(124) 10 102 − 1 = 1- 744/990 = 1-0.752 =0.248
- 34. Spearman’s Rho ρ (rs) for Tied Ranks If the two or more subjects get the same score than allot the ranks to all subjects. Then check the scores and add up the ranks of same score. Calculate the average of the ranks. For example if two subjects have similar score allotted with rank 6 and 7 then their respective ranks will be 6+7=13/2=6.5.
- 35. Spearman’s Rho ρ (rs) Formula for Tied Ranks
- 36. Spearman’s Rho ρ Example for Tied Ranks from IGNOU Book
- 37. Significance Testing of Spearman’s Rho ρ Check the table for significance (remember to consider the whether the test is two tailed or one tailed). If the calculated value is more than the tabulated value reject the Null hypo and vice-versa. In our case the calculated value is 0.902 which is higher than the table value (0.648 at .05 significance level and 0.794 at .01 significance level for two tailed tests). Hence, we reject the Null hypothesis and accept the alternative hypothesis.
- 38. Kendall’s Tau (τ) It was developed by Kendall in 1938 and alternative to Spearman rho. The range is -1 to +1. The coefficient is interpreted on the basis of sign and value. It is based on the concordance and discordance of two set of ranks of variables X and Y.
- 39. Example to Calculate Kendall’s Tau (τ) Suppose we asked two booksellers to rank 8 subjects as per demand of their books among the students. C is concordance (The number of scores below a particular score greater than that particular Score). D is Discordance (The number of scores below a particular score smaller than that particular score). Book Seller (X) Book seller (Y) C D Psychology 1 3 5 2 Maths 2 2 5 1 Geography 3 4 4 1 History 4 6 2 2 Phy Edu 5 1 3 0 Hindi 6 7 1 1 English 7 8 0 1 Biology 8 5 Total 20 8
- 40. Example to Calculate Kendall’s Tau (τ) Formula to calculate τ = 𝐂−𝐃 𝑪+𝑫 Where, C is concordance = 20 D is Discordance = 8 Therefore, τ = 𝟐𝟎−𝟖 𝟐𝟎+𝟖 = 𝟏𝟐 𝟐𝟖 = 𝟎. 𝟒𝟐𝟖𝟔 The correlation between two booksellers is moderate and positive. For testing the significance of τ check the table at .05 or .01 if calculated value is lesser than the table value the Null Hypothesis is rejected and vice versa.
- 41. References: 1. Retrieved on 12 November to 23 November 2019 from http://egyankosh.ac.in/
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