psychological statistics

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1 psychological statistics B Sc. Counselling Psychology 011 Admission onwards III SEMESTER COMPLEMENTARY COURSE UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION CALICUT UNIVERSITY.P.O., MALAPPURAM, KERALA, INDIA

2 UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION STUDY MATERIAL B Sc. COUNSELLING PSYCHOLOGY III Semester COMPLEMENTARY COURSE PSYCHOLOGICAL STATISTICS Prepared by: Dr.Vijayakumari.K, Associate Professor, Farook Teacher Training College, Farook College.P.O. Feroke Scrutinised by: Prof.C.Jayan, Department of Psychology, University of Calicut. Layout & Settings Computer Section, SDE Reserved Psychological Statistics Page

3 CONTENTS MODULE I CORRELATION MODULE II PARAMETRIC AND NON-PARAMETRIC TESTS 14-0 MODULE III MEDIAN TEST 1-6 Psychological Statistics Page 3

4 MODULE I CORRELATION Objectives: The student will be - acquainted with knowledge of correlation, types of correlation and different methods of calculating correlation; - able to calculate coefficient of correlation using different methods. - able to draw scatter diagram and interpret it. Correlation In Psychology, there are situations where two or more variables are involved. Some of them will be related to each other and some will be independent. To know whether these variables are related or not, one has to be thorough with the concept of correlation. Correlation is a statistical technique that is used to measure and describe relationship between two variables. Two variables are said to be related, if for a change in one variable, there is a change in the other. That is, the two quantities vary in such a way that movements in one are accompanied by movements in the other. For example, the scholastic achievement and general intelligence of a child will be related to each other. The extent of relationship and the nature of relationship will be measured using a correlation coefficient. If an increase in one variable brings a corresponding increase in the other variable or a decrease in one variable brings a corresponding decrease in the other variable, the two variables are said to be positively related. For example, it is supposed that as height of individuals increase weight also increases. Hence the two variables height and weight are positively related. If increase (or decrease) in one variable brings a corresponding decrease (or increase) in the other variable the two variable are negatively related. The relation between performance of an individual in a test and test anxiety will be negative as for an increase in test anxiety, there will be a decrease in the performance or vice versa. If there is no corresponding change in one variable for the changes in the other variable, the two variables are said to be not related or there is zero correlation between the variables. The height of an individual and his intelligence are independent (or not related to each other) as there will not be any changes in intelligence accompanied with changes in height or vice versa. It is to be noted that correlation gives idea about the relation between the variables, but not the causation. That is, the change in one variable is not the cause for change in the other variable, but due to some reasons, the two variables vary together. Psychological Statistics Page 4

5 The relation between two variables can be linear or curvilinear. If the relation between two variables can be represented graphically by a straight line, the relation is linear. If the graph is not a straight line, but a curve, the relation is curvilinear. In other words, if the amount of change in one variable tends to bear constant ratio to the amount of change in the other variable, then the correlation is said to be linear. If the change in one variable does not bear a constant ratio to the amount of change in the other variable, the relation is non-linear or curvilinear. In common purposes, usually linear correlation is calculated, and it reveals how the change in one variable is accompanied by a change in the other variable. If two variables are linearly related, the degree of relationship and the nature of relation can be measured using coefficient of correlation. It is a ratio which expresses the extent to which changes in one variable are accompanied by changes in the other variable. A coefficient of correlation ranges from -1 to +1. Two important methods of computing coefficient of correlation are 1. Pearson's Product Moment method and. Spearman's Rank different method. Pearson's Product Moment Method This method is the most common method of calculating the linear relationship between the variables (usually with interval or ratio measure) Pearson's Coefficient of Correlation r = X X Y Y X X Y Y Where X & Y are the arithmetic means of the two sets of values X and Y respectively. Machine formula for Pearson's coefficient of correlation is r N XY X Y N X X N Y Y Spearman's Rank difference method Pearson's product moment correlation is used when the variables are in interval or ratio scale and if the relation between them is linear. But in practice one may face situation when the data are not in interval or ratio scale or the expected relationship does not fit a linear form. Spearman's method is one alternative to Pearson's method when the variables are in ordinal form (When X and Y values are ranks) or when the expected relationship is not linear. (When one wants to measure the consistency of a relationship between X and Y, independent of the specific form of the relationship). Psychological Statistics Page 5

6 Spearman's coefficient of correlation n 6 D 1 Where D is the difference between X rank and the Y rank for n 1 each individual and n-number of individuals. If X and Y are not given as ranks, convert X and Y into ranks. Rank can be assigned by taking either highest value as 1 or the lowest value as 1, but the same method must be followed while ranking the other variable. If two or more entries are equal, each entry is given an average rank. For example if two scores are ranked equal at 6 th place, they are each given the rank , giving the 8 th rank for the next item. If three scores are at 6 th place, the average rank 7 will be given to the three values, the next item with rank 9. 3 Interpretation of correlation co-efficient A correlation coefficient tells whether there is any relation between the variables and if a relationship exists between variables, is it positive or negative. It also indicates the degree of relationship, ie., closeness of relationship. Thus a correlation coefficient can be interpreted with respect to 1) The sign : If the coefficient of correlation computed is positive, there is a positive relationship between the variables and if it is negative, the relationship between the variables is negative. ) The magnitude: The value ranges from -1 to +1. One way of interpreting the correlation coefficient is, Illustration 0 - zero relation or no relationship ± Negligible to low relation ±0.0 - ±.40 - Low to moderate relation. ± ±.60 - Moderate to high relation. ± ±.80 - High to very high relation ± ± 1 - Very high to perfect relation. Pearson's Method Following are the marks obtained by five students in two tests. Pearson's Product moment coefficient of correlation and interpret it. Sl. No. Test 1 Test Calculate Psychological Statistics Page 6

7 This can be written as X Y X X Y Y X X Y Y X XY Y X Y 5 r = X XY Y X X Y Y = x r = Negative sign of 'r' indicates a negative correlation between the two sets of test scores. That is, for an increase in the first test score, there will be a decrease in the second test score or vice versa. The magnitude is which is in between ±0.0 and ± 0.40 and hence the relationship can be considered as moderate or substantial. Thus, the two sets of test scores are negatively related and the relationship is moderate. Using Machine formula X Y X Y XY Psychological Statistics Page 7

8 r = N XY X Y N X X N Y Y = 5x74 6x5 5x x = Illustration Spearman's Rank difference method 74x The ranks given by two judges for ten items of a Thurston's scale of attitude are given below. Find the correlation between the two ranks. Item Judge 1 Judge Item Item 8 6 Item Item 4 1 Item Item Item 7 1 Item Item Item R 1 R D=R 1 -R D Psychological Statistics Page 8

9 p = 6 n n 1 D 1 6x4 144 p = x = Here n 10 The correlation coefficient obtained is The positive sign indicates the direction of correlation, that is the two sets of ranks are positively related. The magnitude is which denote a high correlation between the two sets of data. Hence the two judges have agreement in the ranks given to the items. Illustration 3 X Y R R 1 D=R 1 -R D Here the value 5 repeats two times in X-set and the rank for the two values will 5 6 be five and six. Hence the average ie 5. 5 is given as rank for both the values. The next lower value 4 is given the rank '7'. Similarly, '3' repeats twice and the corresponding ranks are 8 and 9, the rank 8 9 given is 8. 5 for the 3's. The next lower value is given the rank 10. In the second set of values (y - values), '10' occurs three times occupying the positions 4, 5 and 6. Hence the average 5 is given as rank for 10, the next 3 lower value '9' getting the rank '7'. Similarly, '8' has got the rank 8.5 n 6 D p = 1 n 1 6x = ) 990 = = -0.0 Psychological Statistics Page 9

10 The obtained value shows that the two sets of values are negatively related but the relationship is negligible. Scatter diagram Scatter diagram or scatter plot is the graphical representation of correlation between two variables. It gives the pattern of relationship between the two variables at a glance itself. To construct a scatter diagram the following steps can be followed. Draw the axes and decide which variable goes on which axis. Two perpendicular axes are drawn and one variable is taken on the horizontal axis and the other on the vertical axis. Determine the range of values for each variable and mark them on the axes. Usually the point at which the axes meet is taken as 'zero' and the points are marked on the axes by taking an appropriate scale. The lowest and highest values of both variables must be included in the respective axes. It is conventional that the scatter diagrams are roughly square in shape and hence the scales must be taken so that the horizontal and vertical axes have almost the same length (1:1 ratio). Mark a 'dot' for each pair of scores. Plot points on the paper by taking the score of the variable on horizontal axis as the X-co-ordinate and that of the vertical axis as the Y-co-ordinate. Each pair of scores can be represented as a point by following this method. Examples The scores obtained by 10 individuals in two tests are given below. Draw a scatter diagram. Individual Test A Test B Score Test B Scatter diagram Score Test A Psychological Statistics Page 10

11 Here the points plotted tend to lie on a straight line and hence the two variables are linearly related (or the relation is linear). The points are from left bottom to right top which indicate a positive relationship between the variables. In the following diagram the points are arranged from the left top to right bottom indicating a negative relationship between the variables. Individuals. X Y A 4 B 3 3 C 3 D 1 4 E Y X If the variables are not related or if there is zero relation between the variables, there will not be any pattern in the distribution of points, the points will be scattered in the plane. Indls. X Y A 1 B 4 C 3 1 D 4 3 E 5 Psychological Statistics Page 11

12 Y X But scatter diagram do not give any clear picture about the strength of the relation as the correlation coefficient give. Any how, whether the variables are highly related or the relationship is low can be identified by looking into how do the points fall to a straight line. That is, if the points more tend to a straight line, the variables are highly related. If they fall far away from the straight line, the relationship is zero or very low. The correlation is moderate, if the pattern of dots is somewhere between a low and a high correlation. If the points lie exactly on a straight line (as shown in the figure, the correlation is perfect). Indls. X Y A 1 1 B C 3 3 D 4 4 E 5 5 Y X Psychological Statistics Page 1

13 Scatter diagram showing positive perfect linear relationship Indls. X Y A 1 5 B 4 C 3 3 D 4 E Y X Scatter diagram showing negative perfect linear relationship Scatter diagrams are important because they clearly exhibit the nature of relationship, whether linear or curvilinear. If the variables are found to be linearly related, then only, coefficients of correlation can be calculated. When the relation is curvilinear, calculating coefficient of correlation and interpreting it will be misleading or will be an error. Psychological Statistics Page 13

14 Objectives MODULE PARAMETRIC AND NON-PARAMETRIC TESTS The student will be able to - discriminate parametric and non-parametric tests. - use chi-square test and c-coefficient of contingency in appropriate situations. Introduction In order to understand this module, a more knowledge about statistics is needed. First of all you should be familiar with the descriptive and inferential statistics. Descriptive statistics describes the data as does arithmetic mean or median or standard deviation or correlation. But statistics is mainly used to predict or infer the unknown. This function of statistics is done by inferential statistics. Two terms related to inferential statistics are parameter and statistics. Usually we are conducting research on a sample, not on the entire population. But the purpose of the study will be to know about the population and not the sample. The value calculated from the sample is called statistic and the corresponding value for the population is parameter. For example, if arithmetic mean of a sample is, it is a statistic. The corresponding arithmetic mean of the population can be taken as µ, the parameter. Statistics tries to predict the unknown parameter from the known statistics. In the process of predicting or inferring population values, hypotheses are formulated and tested. A hypothesis is a tentative solution to the problem which is tested through the study. Hypotheses are stated as affirmative sentences. Inferential statistics tests these hypotheses using the data available through the study of sample, a representative group of the population. Hypotheses can be directional or non directional. The statement, Boys are better than girls in mechanical aptitude is a directional hypothesis as there is a clear indication of direction of change. But the statement, Boys and girls differ in their mechanical aptitude is a non directional hypothesis as there is no indication of direction of change. Statistical procedures are there to test hypothesis in which no change or no effect is stated. Such hypothesis is known as null hypothesis denoted as H0. H0 is tested against the alternative hypothesis, H1. An alternative hypothesis is the hypothesis that is accepted when the null hypothesis is rejected. Example: H0: There is no significant mean difference in mechanical aptitude between boys and girls.(µ1= µ). H1: There is significant difference in mechanical aptitude between boys and girls. ( µ1 µ). While testing a hypothesis, there is a possibility of occurrence of two types of errors. One error is rejecting H0 when it is true and the other is accepting H0 when it is not true. The first error is known as Type I error and the second is Type II error. That is any decision regarding hypothesis testing may include these two types of errors. Probability of Type I Error ie., rejecting H0 when it is true is known as Level of Significance. In hypothesis testing level of significance is determined by the researcher in advance itself. In behavioural sciences, it is taken commonly as.05 or.01. As the probability of type I error decreases, probability of type II error increases and vice versa. Psychological Statistics Page 14

15 Decisions regarding hypothesis testing are made based on the concept of a Normal distribution. Normal distribution is a theoretical model of distribution of quantitative data. A normal probability curve is a bell shaped curve which is symmetric with respect to the ordinate at mean. It is non-skewed, lepto kurtic curve for which mean, median and mode coincide. It has wide application in statistics. The two tails of the curve determines the chance of accepting or rejecting a null hypothesis. The area for which the hypothesis is rejected is known as rejection area or critical region. (More about normal curve is included in fourth semester). Parametric and Non-parametric Tests The researches in behavioural science need different types of statistical tests to test hypotheses about specific population parameters (Values corresponding to population are called parameters and those of sample are statistics). A number of statistical techniques like 't' tests and ANOVA make assumptions about the shape of the population distribution and about other population parameters. As these tests concern parameters and require assumptions about parameters, they are called 'Parametric tests'. Parametric tests require a numerical score for each individual in the sample. The scores then are added, squared, averaged and otherwise manipulated using basic arithmetic. That is, parametric tests require data from an interval or ratio scale. If some assumptions are not met, it may not be appropriate to use a parametric test. Violation of assumptions of a test may lead to erroneous interpretation of the data. In such situations, there are several hypothesis testing techniques that provide alternatives to parametric tests. These tests are known as 'non-parametric' tests. Non-parametric tests usually do not state hypotheses in terms of a specific parameter and there may few (if any) assumptions about population distribution. (Because of this reason, non-parametric tests are also known as distribution free tests) Chi-square test, Sign test etc are non-parametric tests. For non-parametric tests, the subjects (sample elements) are usually classified into categories. (These measures are on nominal or ordinal scales and they do not produce numerical values that can be used to calculate means and variances). The data for many non-parametric tests are frequencies. Non-parametric tests are not as sensitive as parametric tests. (i.e., nonparametric tests are more likely to fail in detecting a real difference between two treatments). Hence whenever there is a chance to use parametric test, it must be preferred than a non-parametric test. Chi-Square Test Chi-square ( ) test is a non-parametric test with wide applications in the field of social sciences. (The name of the test is related with the Greek letter '' 'Kye' used to identify the test statistic). Chi-square test is used when the data is in the form of frequencies (i e., the measurement is on nominal scale). Mainly the test is used for two purposes- testing goodness of fit and to test the independence between two nominal variables. Psychological Statistics Page 15

16 The Chi-square test for Goodness of fit Here Chi-square test is used for testing hypothesis related to sample proportions with respect to the corresponding population proportions. The Chi-square test for goodness of fit determines how well the obtained sample proportions fit the population proportions specified by the null hypothesis. Formula for calculating is ( O E E ) where O is the observed frequency and E is the expected frequency (as per the null hypothesis) Usually in the test of Goodness of fit, the null hypothesis will fall into one of the following categories. 1. No preference: The null hypothesis states that there is no preference among the different categories. ie., Ho states that the population is divided equally among the categories.. No difference from other population The null hypothesis can state that the frequency distribution for one population is not different from the distribution that is known to exist for another population. The null hypothesis for the goodness of fit test specifies an exact distribution for the population. So the alternative hypothesis (H1) state that the population distribution has a different shape from that specified in Ho. For example, if Ho states that the population is equally divided among three categories, the alternative hypothesis will say that the population is not divided equally. measures the discrepancy between the observed frequencies (the data) and the expected frequencies (Ho). When the calculated value of is large, observed and expected frequencies differ highly and hence Ho is rejected. If value is small, the difference between the observed and expected frequencies are less, then Ho is accepted. To determine whether the value is large or small, the theoretical distribution of is used. The table of gives values for different levels of significance and degrees of freedom(df). In goodness of fit test, the degree of freedom of is C-1 where C is the number of categories. If there are three groups or categories in the specified population, the degrees of freedom will be 3-1=. If there are five categories, the df will be 5-1=4. In a table, the first column lists df, the top row lists proportions of area in the extreme right hand tail of the distribution (or level of significance). The numbers in the body of the table are the critical values of Chi-square. To test the hypothesis the critical value of for given df and level of significance is located from the table. Then the calculated value is compared with the tabled value. If the calculated value is greater than the tabled value, Ho is rejected and if it is less than the tabled value, Ho is accepted. Psychological Statistics Page 16

17 Illustration 1 Forty eight subjects were asked to express their response to an item in an attitude scale in three categories Agree, Undecided and Disagree. Of the members in the group, 4 marked 'agree', 1 'undecided' and 1 'disagree'. Do these results indicate a significant difference among groups? Here the null hypothesis Ho can be stated as Ho: There is no significant difference among groups H1: The group differ significantly. Category O E Agree 4 16 Undecided 1 16 Disagree 1 16 Since the assumption is that the groups do not differ in their size, the total 48 can be equally divided into three groups. So the expected frequency of each group is Next step is finding by summing ( O E E ) (4 16) = 16 (1 16) = = 6 16 (1 16) 16 ( O E) E With degree of freedom c-1=3-1=. Fix the level of significance as From table of, the value of ( df) 0.05 level is Since the calculated value is greater than the tabled value of, Ho is rejected. That is one cannot expect equal preference among the subjects at 0.05 level of significance or the subjects showed a significant difference in the response to the given item. Illustration- In a study to know if high-performance over powered cars are more likely to be involved in accidents than other types of cars, the investigator classified 50 cars as high performance, sub compact, mid-size, or full size. The observed frequencies are given below. Psychological Statistics Page 17

18 High performance Subcompact Mid size Full size It is assumed that only 10% of the cars in the population are the highperformance variety. The subcompact, mid size and full size cars are 40%, 30% and 0% respectively. Can the researcher conclude that the observed pattern of accidents does not fit the predicted values (Test with =0.05). Ho: In the population, no particular type of car shows a disproportionate number of accidents. (or the observed frequencies are in agreement with the expected frequencies) H1: In the population, a disproportionate number of accidents occur with certain types of cars (There is disagreement with O and E). O E (O-E) (O-E) /E Here expected frequencies are calculated according to the assumption. So the 10 expected frequency of the first group is 50 x = 5x1= The E of second group is 0 (50 x ) The E of third group is 50 x = and that of fourth group is 50 x = ( O E) = E degrees of freedom = C-1=4-1=3 The tabled value of (3 df) at 0.05 level is Since the calculated value is greater than the tabled value of for 3df at 0.05 level, Ho is rejected. That is a disproportionate number of accidents occur with certain types of cars. Psychological Statistics Page 18

19 Chi-square test of independence Chi-square test can be used to test whether two nominal variables (values expressed as frequencies) are independent or not (associated or not). The formula for calculating is same as that of Goodness of fit. ( O E E ) Degrees of freedom in the case of Test of Independence is (C-1) (R-1) where C is the number of columns and R number of rows. The steps involved in the test of independence are Calculate the expected frequencies. Expected frequency for each cell can be calculated by RTxCT E N E- Expected frequency RT- CT- The row total for the row containing the cell The column total for the column containing the cell N- The total number of observations. Take the difference between observed and expected frequencies and obtain the squares of these differences, i.e., obtain the values of (O-E). Divide the values of (O -E) obtained in step by the respective expected ( O E) frequency and obtain the total E The calculated value of is compared with the table value. Example: Two hundred students were classified by personality and colour preference as given below. Test whether colour preference and personality are dependent for the student population. Red Yellow Green Blue Introvert Extrovert N=00 Here the null hypothesis is Ho: Colour preference and personality of students are independent. The expected frequencies of the cells are calculated by multiplying the corresponding row total and column total and then dividing it by total number of participants. Psychological Statistics Page 19

20 The expected (theoretical) frequencies will be Red Yellow Green Blue Introvert 50x =5 50x0 = x40 = x40 =10 00 Extrovert 150x =75 150x0 00 =15 150x40 00 =30 150x40 00 =30 ( O E E ) substituting the values (10 5) (3 5) (18 15) + 15 = 37.4 Degrees of freedom (df) is (15 10) + 10 (R-1) (C-1) = (-1) (4-1) = 1x3=3 ( 10) + 10 (90 75) + 75 The table value for 3 df at 0.05 level is and at 0.01 level is (17 15) + 15 (5 15) + 15 Since the calculated value of is greater than the tabled value for significance at 0.01 level, the null hypothesis is rejected. That is, the hypothesis "Colour preference and personality of students are independent' is rejected. Hence the two variables colour preference and personality are associated or dependent. Note: If two variables are found to be associated using test of independence, the strength of association can be calculated using c-coefficient of contingency. C N Basic Assumption Chi-square test is a non-parametric test and hence is distribution free. That is, as in parametric tests, variables need not be normally distributed. But, the variables are to be measured in a nominal scale. That is, the data is in the form of frequencies. Psychological Statistics Page 0

21 Objectives MODULE 3 MEDIAN TEST Introduction The student will know about various non-parametric tests like Man-Whitney U test, Sign test, Wilcoxon Matched pairs signed ranks test. Non-parametric tests can be used in situations where - the sample size is quite small (i.e., when N=5 or 6) - assumptions like normality of the distribution of values in the population is not ensured. - When the measurement of data is either in ordinal (ranks) or nominal (frequencies). As non-parametric tests are less powerful than the parametric tests, when ever possible, parametric tests are to be used. But one cannot neglect the nonparametric tests because researchers in psychology need such tests for statistical inferences in many situations. Some major non-parametric tests are discussed below. 1. Mann-Whitney U Test To test the significance of difference between two independent means, usually t- test (parametric) is used. But when the normality of the variables are questioned, t-test is not applicable. Then Mann-Whitney U test can be used, but it tries to find out the difference between population distributions, and not the population means. It is a powerful test to test the difference between two independent samples having uncorrelated data. The basic assumption for Mann-Whitney U test is given below: "If two sets of data differ significantly, generally, the scores in one sample will be larger than the scores in the other. If the two samples are combined and all the scores are placed in rank order on a line, then the scores from one sample will concentrate at one end of the line, the scores of the other sample will concentrate at the other end. But if there is no significant difference between the groups, the large and small values will be mixed evenly in the two samples, values of one sample not concentrating at one side of the line. Step 1 The null hypothesis Ho can be written as there is no tendency for the ranks in one group to be systematically higher or lower than the ranks in the other group. Fix the level of significance as 0.05 or Step Two samples are there sample A and sample B. Let na be the number of subjects in sample A and nb be the number of subjects (sample size) of sample B. Then combine the scores of the two groups and the na+ nb scores are ranked. Psychological Statistics Page 1

22 Step 3 The ranks in each group are summed up to get RAand RB (RA+ RB = N (N+1)/ where N= na+ nb) Step 4 Step 5 Then the Mann-Whitney U for the two sets are calculated using the formula n (n + 1) U = n n + R n (n + 1) U = n n + R Take the Mann-Whitney U as the smaller of UA and UB. Refer the table of critical values of Mann-Whitney U for the tabled value of U corresponding to the obtained UA and UB and the level of significance fixed. If the tabled value is less than the obtained U value, Ho can be accepted and if the obtained U value is less than the critical value Ho is rejected. Example: The time in seconds recorded for 13 children (5 boys and 8 girls) in a blockmanipulation task are given below. Test whether there is consistent difference between boys and girls in the time required for completing the task. Step 1 Boys Girls The null hypothesis Ho: There is no consistent difference between boys and girls in the time needed for completion of the task. Alternate hypothesis: H1: There is a systematic difference. Take level of significance as =0.05. Step na = 5 nb = 8 na + nb = 13 Boys Ranks Girls RA = RB = 69 Psychological Statistics Page

23 U = n n + n (n + 1) R U = = = 40-7 = 33 U = n n + 5(5 + 1) n (n + 1) = ( ) 69 = = 7 R The least value of UA and UB is 7 hence U=7 The critical value is 6 (for na =5 & nb =8) at 0.05 level of significance. Since the obtained U value is greater than the tabled value, accept Ho. At the 0.05 level of significance, the data do not provide sufficient evidence to conclude that there is significant difference between boys and girls in time needed for completing the task.. The Sign Test Sign test is the most simple non-parametric test. It is used for comparing two correlated samples, i.e., two parallel set of scores which are paired off in some way (dependent samples). This test uses plus and minus signs instead of quantitative values and hence the name 'sign test'. Sign test is useful when 1. Normality and homogeneity of variance of the variables are not sure, but the variable considered has a continuous distribution.. All the subjects are not from the same population. 3. Two correlated samples are to be compared, with the null hypothesis the median difference between the pairs is zero. 4. There are two sets of measurements which can be matched (paired) with respect to relevant extraneous variables. 5. When the data is not in interval or ratio scale, especially when the direction of change is given, not the quantitative measure. If the number of individuals in the single sample is less than or equal to 5, the table of probabilities associated with values as small as observed values of x (number of fewer signs) in the Binomial test. If N is greater than 5, the normal approximation to the binomial distribution or may be used; Here Psychological Statistics Page 3

24 = ( ) 1/. With Yate's correction this formula can be rewritten as = ( ±. ) / where x+.5 is taken when <,.5 is taken when > ' ' is the total of '+' or '-' signs. Example: A researcher tests the effect of a diet plan on body weight. A sample of 50 people is selected. The observation made at the start of the diet and one month later revealed that 37 people lost weight, 1 gained and one had no change in weight. Test whether the treatment leads to gain in weight. Number of positive signs (initially high value, second one smaller, i.e., loss in weight) = 37 Number of negative signs (gain in weight) =1 Here N = Number of positive signs + Number of negative signs =49 (not the sample size 50). Ho: The diet has no effect; H1: The diet has some effect. ( ± 0.5) / = x can be taken as 37 or 1 leading to the same numerical value of z. The only change will be in the sign. = (. ) / =.. =. = 3.43 Since this value is greater than 1.96, the critical value for significance at 0.05 level, Ho is rejected. That is the assumption that 'the diet has no effect on weight' is rejected. (If N is small, instead of calculating z, the value of N and x is taken for finding the value of P from a table of probabilities associated with values in the Binomial test. If it is greater than 0.05, the null hypothesis is accepted, if it is less than 0.05, the null hypothesis is rejected). 3. Wilcoxon Matched-pairs Signed Ranks Test Wilcoxon Matched- Pairs Signed Ranks Test is more powerful than the sign test because it tests not only direction but also magnitude of differences within pairs of matched group. This test is used to test the difference between two related (dependent) samples/matched pairs of individuals and is not applicable to independent groups. Method Step 1 Let d1 be the difference of scores for any matched pair, representing the difference between pair's scores under two treatments A and B. Psychological Statistics Page 4

25 Step Step 3 Delete all such pairs for which d1 = 0. Rank all the s with out regard to sign, giving rank 1 to the smallest d1, rank to the next smallest and so on. Step 4 Indicate which ranks arose from negative d 1s and which ranks arouse from positive d ' 1 s by giving sign of difference to each rank. Step 5 Sum the ranks for the positive differences and sum the ranks for the negative differences. Under the null hypothesis, the two sums are expected to be equal. Z is calculated using the formula ( ) = where ( )( ) N- Number of pairs related. T- Sum of the ranks of the smaller, of the like signed ranks Illusteration The local Red Cross has conducted an intensive campaign to increase blood donations. This campaign has been concentrated in 10 local businesses. In each company, the goal was to increase the percentage of employees who participate in the blood donation program. Using Wilcoxon test to decide whether these data provide evidence that the campaign had a significant impact on blood donations. The data from each company has listed in a rank order to the absolute value of the difference scores. Company Percentage of participation Rank Before After Difference A B C D E F G H I J Psychological Statistics Page 5

26 Step 1: The null hypotheses state that the campaign has no effect. Therefore, any differences are due to chance, and there should be no consistent pattern. Step : the two companies with difference scores of zero are discarded, and n is reduced to 8 and α=.05, the critical value for the Wilcoxon test is T=3. A sample value that is less than or equal to 3 will lead us to reject Ho. Step 3: For these data, the positive differences have ranks of 3, 4, 5.5, 5.5, 7, and 8; R+ = 33 The negative differences have ranks of 1 and ; = 3 The Wilcoxon T is the smaller of these sums, so T = 3. Step 4: T vlue from the data is in the critical region. This value is very unlikely to occur by chance (p<.05); therefore, we reject Ho and conclude that there is a significant change in participation after Red Cross campaign. (source: Gravetter, F. J., & Wallnau, L. B. (000). Statistics for the Behavioral Sciences (5 th ed) Psychological Statistics Page 6