The fundamentals of statistics, methods of sampling, types of data, levels of measurement, measures of central tendency, measures of dispersion, and hypothesis testing are covered in Chapters 1, 2 and 3. Chapter 4 deals with (chi-square) tests that can be applied for nominal (categorical) and ordinal data. Categorical data that are commonly collected in studies to investigate relative risk and odds ratio are dealt with in Chapter 5. Statistical tools for handling and analyzing interval data are elaborated in Chapter 6 (t-test and Z-test), Chapter 7 (Analysis of Variance / ANOVA, comparing averages of two and more sources) and Chapter 8 (Correlation, Regression Analysis). The steps to be taken in analyzing quantitative data are summarized in Appendix I. Information about some useful software packages, references and website links are provided in Appendix II. Readers can resort to statistical tables or Microsoft Excel software to obtain certain statistical indices and values. The relevant Microsoft Excel commands are summarized in Appendix III. Some residual plots related to regression analysis are presented in Appendix IV. The statistical tables for(Chi-square) distribution, t-distribution, Standard Normal distribution, and F-distribution are provided in Appendix V, VI, VII, and VIII respectively

Phua, Kai Lit

Wong, Kam Cheong

University of Queensland eSpace

  
 
 Preface 
             
 
We decided to write this book because we noticed that many people find statistics 
to be confusing and mystifying. They also find standard textbooks on statistics to be 
difficult to understand and to use. We are optimists and believe that it is not difficult to 
learn basic statistics and apply statistical tests to analyze data and test hypotheses. This 
can be done if the use of mathematical and statistical notation and jargon are minimized 
and if attempts are made to explain things as simply and clearly as possible for those who 
want to use statistics as a tool rather than ponder over the intricacies of statistical theory. 
The authors believe that this book will help you to understand, appreciate and apply 
statistics (happily!) in your work and studies.  
  
The fundamentals of statistics, methods of sampling, types of data, levels of 
measurement, measures of central tendency, measures of dispersion, and hypothesis 
testing are covered in Chapters 1, 2 and 3.  Chapter 4 deals with χ2 (chi-square) test that 
can be applied for nominal (categorical) and ordinal data.  Categorical data that are 
commonly collected in studies to investigate relative risk and odds ratio are dealt with in 
Chapter 5.  Statistical tools for handling and analyzing interval data are elaborated in 
Chapter 6 (t-test and Z-test), Chapter 7 (Analysis of Variance / ANOVA, comparing 
averages of two and more sources) and Chapter 8 (Correlation, Regression Analysis).  
The steps to be taken in analyzing quantitative data are summarized in Appendix I.  
Information about some useful software packages, references and website links are 
provided in Appendix II.  Readers can resort to statistical tables or Microsoft Excel 
software to obtain certain statistical indices and values.  The relevant Microsoft Excel 
commands are summarized in Appendix III.  Some residual plots related to regression 
analysis are presented in Appendix IV.  The statistical tables for χ2 (Chi-square) 
distribution, t-distribution, Standard Normal distribution, and F-distribution are provided 
in Appendix V, VI, VII, and VIII respectively. 
 
Dr Wong is a statistician trained in a Master of Science (statistics) program at the 
National University of Singapore.  He received his Bachelor of Medicine and Bachelor of 
Surgery (MBBS) from the University of Queensland, Australia.  Dr Phua is a social 
scientist trained in a PhD program (Johns Hopkins University, United States of America) 
which emphasizes the acquisition of statistical skills.  Dr Phua is currently a senior 
lecturer in Community Medicine at the International Medical University in Kuala 
Lumpur, Malaysia.  Both authors like statistics and believe that it is a useful and elegant 
tool for making sense of data. Welcome to the world of statistics! 
 
 
Dr. Wong Kam Cheong  Dr. Phua Kai Lit 
MBBS, BE, MSc.   BA, PhD.       
 
March 2006              http://www.btm.upm.edu.my/penerbit/ 
  
Authors’ Background 
 
Dr. Wong Kam Cheong received his Bachelor of Medicine and Bachelor of 
Surgery (MBBS) from the University of Queensland (Australia), and Bachelor of 
Engineering (Mechanical) and Master of Sciences (Statistics) from the National 
University of Singapore. He has worked as a statistician and engineer in Singapore and 
Malaysia, and as a medical research associate in the Centre for Chronic Disease at the 
University of Queensland (Australia). He is currently working as a medical doctor in 
Australia. His past awards include a research scholarship from the Dartmouth Medical 
School (Dartmouth College in the United States of America) which is one of the Ivy 
Leagues universities. 
Dr. Phua Kai Lit received his BA in Public Health and Population Studies from 
the University of Rochester (United States of America) and PhD in Sociology (Medical 
Sociology) from Johns Hopkins University (United States of America). He has worked as 
a research statistician with the Maryland Department of Health and Mental Hygiene in 
Baltimore, USA. Dr. Phua is currently a senior lecturer with the Community Medicine 
section of the International Medical University in Kuala Lumpur, Malaysia. His past 
awards include an Asian Public Intellectual Senior Fellowship from the Nippon 
Foundation.   
  
 
 
 
 
 
 
 
The following is a chapter extracted from the book 
 
“Statistics Made Simple for Healthcare and Social Science 
Professionals and Students” 
 
Authors: Dr. Wong Kam Cheong, Dr. Phua Kai Lit. 
 
 
 
International Standard Book Number (ISBN) = 983-3455-03-4. 
 Chapter 4 
• Introduction to the χ2 Test (Chi-square Test) 
• Assumptions of the χ2 Test  
• Using the χ2 Test (Worked Example) 
 
4.1 Introduction to the χ2 test (Chi-square Test) 
In this chapter, you will be introduced to a very useful test called the χ2 test (also known as the 
“chi-square test of association” or “chi-square test of independence”). It is used to determine if 
there is a statistically significant association between two variables measured at the nominal level. 
It can also be used to test for association between two variables measured at the ordinal level. 
 
4.2 Assumptions of the χ2 Test: 
• The data should be nominal data or ordinal data 
• The samples should be random samples 
• The ‘expected value’ of each cell should be at least 5 (if not, the categories can be combined 
to overcome this problem). The ‘expected value’ will be explained later on in this chapter.   
 
4.3 Using the χ2 Test (Worked Example) 
Here is an example to illustrate how the χ2 test is used.  Suppose you wish to test if there is an 
association between gender (“sex”) and anorexia nervosa in the case of Malaysian teenagers.  
Anorexia nervosa is an eating disorder characterized by refusal to maintain a normal minimal 
body weight. Sufferers believe that they are overweight and will refuse to eat properly even if 
they are actually seriously underweight. You could select a random sample of 100 male 
Malaysian teenagers from the population of all male Malaysian teenagers and a second random 
sample of 100 female Malaysian teenagers from the population of all female Malaysian teenagers. 
Next you would check how many of the female teenagers suffer from anorexia nervosa vis-à-vis 
how many of the male teenagers suffer from anorexia nervosa. Then you would use the χ2 test to 
see if the difference in prevalence of anorexia nervosa between the males and the females is 
statistically significant or due to chance alone. The following denotations are used when we refer 
to a ‘cell’ in a table prepared for the χ2 test: 
 
      “Cell 1-1” 
 
      “Cell 1-2” 
 
     “Cell 2-1” 
 
      “Cell 2-2” 
 
The steps involved in analyzing data using the χ2 test are as follows: 
[Step 1] Place the data in a 2 X 2 “contingency table” (two by two contingency table) of 
OBSERVED VALUES (Table 4.1). 
 Table 4.1: Gender and occurrence of anorexia nervosa, sample of 100 male Malaysian teenagers 
and sample of 100 female Malaysian teenagers (OBSERVED VALUES)  
 
                                                    Anorexia Nervosa 
                                                   Yes                     No 
 
                           Male                                     
            Gender 
                           Female                               
 
If there is a relationship between gender and anorexia nervosa, e.g. if females are more 
likely to suffer from the disease than males, then the number in cell 1-1 (top left hand corner cell) 
would be small and the number in cell 2-1 (bottom left hand corner cell) would be large (Table 
4.2). 
 
Table 4.2: Female teenagers are more likely to suffer from anorexia nervosa              
                                      Anorexia Nervosa 
                                                  Yes                      No 
                            
                            Male                                     
              Gender 
                            Female                               
 
 
 
3 
 
97 
 
10 
 
90 
Small 
1 
 
99 
Large 
11 
 
89 
            If there is a relationship between gender and anorexia nervosa in the other direction, i.e. if 
males are more likely to suffer from the disease than females, then the number in cell 1-1 (top left 
hand corner cell) would be large and the number in cell 2-1 (bottom left hand corner cell) would 
be small (See Table 4.3). 
 
Table 4.3: Male teenagers are more likely to suffer from anorexia nervosa  
 
                                                          Anorexia Nervosa 
                                                      Yes                            No 
                            
                           Male                                     
            Gender 
                           Female                               
 
            
 If there is NO relationship between gender and anorexia nervosa, e.g. if both females and 
males are equally likely to suffer from the disease, then the number in cell 1-1 (top left hand 
corner cell) would be large and the number in cell 2-1 (bottom left hand corner cell) would also 
be large (Table 4.4). 
 
Large 
13 
 
87 
Small 
2 
 
98 
 Table 4.4: Both male and female teenagers are equally likely to suffer from anorexia nervosa 
 
                                                     Anorexia Nervosa 
                                                   Yes                    No 
                            
                           Male                                     
            Gender 
                           Female                               
 
 
 
            If there is NO relationship between gender and anorexia nervosa in the other direction, i.e. 
if both males and females are equally unlikely to suffer from the disease, then the number in cell 
1-1 (top left hand corner cell) would be small and the number in cell 2-1 (bottom left hand corner 
cell) would be small (Table 4.5). 
 
Table 4.5: Both male and female teenagers are unlikely to suffer from anorexia nervosa 
                                                            Anorexia Nervosa 
                                                   Yes                    No 
                            
                            Male                                     
             Gender 
                            Female                               
 
Large 
12 
 
88 
Large 
11 
 
89 
Small 
3 
 
97 
Small 
2 
 
98 
[Step 2] After drawing the table of OBSERVED VALUES (Table 4.1), we need to 
construct a table of EXPECTED VALUES (See Table 4.6). 
 
Table 4.6: Gender and occurrence of anorexia nervosa, sample of 100 male Malaysian teenagers 
and sample of 100 female Malaysian teenagers (EXPECTED VALUES)  
 
                                                      Anorexia Nervosa 
                                                    Yes                    No 
                                         
                         Male                                                          Row 1 total 
            Gender 
                           Female                                                          Row 2 total 
 
                                 Column 1 total     Column 2 total 
 
The EXPECTED VALUE of “a” in cell 1-1 would be 
                          (Row 1 total) X (Column 1 total)  
                                                n 
where n = sum of sample sizes, i.e.100 males + 100 females = 200 
Similarly, the EXPECTED VALUE of “b” in cell 1-2 would be 
                          (Row 1 total) X (Column 2 total)  
                                                n 
The EXPECTED VALUE of “c” in cell 2-1 would be 
                          (Row 2 total) X (Column 1 total)  
                                                n 
 
 
a 
 
b 
 
c 
 
d 
 The EXPECTED VALUE of “d” in cell 2-2 would be 
                          (Row 2 total) X (Column 2 total)  
                                                n 
 
Thus, based on the numbers given in Table 4.1, the EXPECTED VALUES for the entire table 
would be: 
 
Table 4.7: Gender and occurrence of anorexia nervosa, sample of 100 male Malaysian teenagers 
and sample of 100 female Malaysian teenagers (EXPECTED VALUES)  
 
                                                              Anorexia Nervosa 
                                                 Yes                              No 
                            
                           Male                                     
 
            Gender 
                           Female                               
 
 
 
[Step 3] Calculate an ‘index’ called the χ2 for each cell and add them all up to get the cumulative 
χ2.  The formula for the cumulative χ2 is:  
 
5.6
200
13100
=
×
 
 
5.93
200
187100
=
×
 
 
5.6
200
13100
=
×
 
 
5.93
200
187100
=
×
 
( )
∑
−
=
E
EO 22χ
 where O = observed value found in a particular cell in the table 
E = expected value associated with a particular cell in the table 
( ) ( ) ( ) ( ) 031.4
5.93
5.9390
5.6
5.610
5.93
5.9397
5.6
5.63 22222
=
−
+
−
+
−
+
−
=χ  
The observed and expected values in the above equation are obtained from Table 4.1 and Table 
4.7 respectively. 
 
[Step 4] Once we get the cumulative χ2, we should compare this value to the corresponding 
“critical value” in a χ2 table to see if it is statistically significant. If it is statistically significant (it 
equals or exceeds the “critical value”), we would reject H0 and accept H1 
 
H0 : There is no association between gender and suffering from anorexia nervosa. Any 
association seen is due to chance alone. 
H1 : There is a statistically significant association between gender and suffering from anorexia 
nervosa. 
 
           To find the “critical value” of χ2 for a 2 X 2 contingency table at the 0.05 level, we would 
look at the “critical value” for degree of freedom = 1 and probability level of 0.05. The table 
(Table 4.8) shows that the critical value is 3.841 {The “degree of freedom” is equal to (number of 
rows minus 1) X (number of columns minus 1) = (2 – 1) X (2 – 1) = 1}. 
 
 Table 4.8: χ2 table (truncated and simplified) 
 
   Degree of Freedom    Probability of 0.05   Probability of 0.01 
               1           3.841              6.635 
               2           5.991             9.210 
               3           7.815           11.345 
               4           9.488           13.277 
               5         11.070           15.086 
               6         12.592           16.812 
 
            Alternatively, the critical χ2 value can be obtained by using statistical software such as 
Microsoft Excel.  Type the following command in a field in a Microsoft Excel spreadsheet: 
=CHIINV(probability, degree of freedom) (Appendix III part 1a). For this example, the 
command is =CHIINV(0.05, 1)  A value of “3.841” will be returned. 
 
Decision Rule of Less Stringent Test:  
Reject H0 and accept H1 if the calculated cumulative χ2 is greater than or equal to the 
critical value of 3.841 when testing at the 0.05 level with degree of freedom of 1 (where the value 
0.05 refers to the probability of H1 occurring by chance). 
Alternatively, reject H0 and accept H1 if the p-value is less than or equal to 0.05. The p-
value can be obtained by typing the following command in a field in a Microsoft Excel 
spreadsheet: =CHIDIST(computed cumulative chi-square value, degree of freedom) (Appendix 
III part 1b).  In this example, the cumulative chi-square value is computed by adding up the χ2 
(chi-square) value for each cell. This is found to be 4.031. Hence, the command for obtaining p-
value from Microsoft Excel is =CHIDIST(4.031, 1).  A value of “0.045” will be returned. 
 
Decision Rule of More Stringent Test:  
Reject H0 and accept H1 if the calculated cumulative χ2 is greater than or equal to the 
critical value of 6.635 when testing at the 0.01 level with degree of freedom of 1 (where the value 
0.01 refers to the probability of H1 occurring by chance). The critical value “6.635” is obtained 
from Table 4.8 or by Microsoft Excel command =CHIINV(0.01, 1). 
Alternatively, reject H0 and accept H1 if the p-value is less than or equal to 0.01. Thus, in 
the example given in Table 4.1, the cumulative χ2 is found to be 4.031. When this cumulative chi-
square value is used for the less stringent test (α= 0.05 at degree of freedom = 1), it is found to 
exceed the critical value of 3.841. Also, the p-value (0.045) is less than 0.05.  Therefore, we reject 
H0 and accept H1 and conclude that there is a statistically significant association between gender 
and suffering from anorexia nervosa. When we reject H0 and accept H1, the probability that H1 
has occurred by chance is less than 0.05 or less than 5%.  
However, when we use the cumulative χ2 value of 4.031 for the more stringent test (α= 
0.01 at degree of freedom = 1), it is found to be less than the critical value of 6.635. Also, the p-
value (0.045) is more than 0.01.  Therefore, it has passed the less stringent test but failed the more 
stringent test. Hence, when we reject H0 and accept H1, the probability that H1 has occurred by 
chance is less than 5% but more than 1%. 
 
 
 
 
 
 
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