1. Chi – square
Dr. Anshul Singh Thapa

2. An Introduction
The Chi – square is one of the simplest and most widely used non – parametric test in the statistical work.
The Chi – square describes the magnitude of the discrepancy between the theory and observation.
The Chi – square test was first used by Karl Pearson in the year it is defined as :
Chi – square (X2) = Σ [(O – E)2 /E].
Where O refers to the observed frequencies and E refers to the expected frequencies.

3. Chi – square performs two types of functions:
Goodness of fit: A common use is to assess whether a measured/observed set of measures follows an expected pattern. The expected frequency may be determined from prior knowledge (such as previous year’s exam results) or by calculation of an average from the given data. Measure of Independence: The chi-square test can be used in the reverse manner to goodness of fit. If the two sets of measured are compared, then we can determine whether it align or do not align.

4. Steps to determine Chi – square
Write the observed frequency in column O.
Figure the expected frequency and write them in column E.
Expected Frequencies:
expected frequency for chi-square can be find in three ways:
When we hypothesized that all the frequencies are equal in each category. In such cases the expected frequency is calculated by dividing the of the sample by the numbers of categories.
We can also determine expected frequencies on the basis of some prior knowledge. In such cases we can find frequency distribution by multiplying the sample size by each of the hypothesized proportions.
Goodness of fit test

5. RT = The row total for the row containing the cell
Measure of independence
Calculate the expected frequencies. In general the expected frequency for any call can be calculated from the following equation:
E = RT X CT/ N
E = Expected Frequency
RT = The row total for the row containing the cell
CT = The column total for the column containing the cell
N = The total number of observation
Take the difference between observed and expected frequencies and obtain the square of these differences. i.e., obtain the values of (O – E)2
Divide the values of (O – E)2 obtain in the second step by the respective expected frequency and obtain the total Σ [(O – E)2 /E].
this gives the value of Chi – Square which can range from zero to infinity. If Chi – square is zero it means the observed and expected frequencies completely coincide. The greater the discrepancy between the observed and expected frequencies, the greater shall be the value of Chi – square

6. The calculated value of Chi – square is compared with the table value of Chi – square for given degree of freedom at certain level of significance. If at the stated level (generally 5% level is selected), the calculated value of Chi – square is more than the table value of Chi – square, the difference between the theory and observation is considered to be significant.
If the calculated Chi – square value is less than the table value than the difference between the theory and observation is not considered as significant, i.e., it is regarded as due to fluctuation in sampling.

7. In an anti malaria campaign in a certain area, quinine was administered to 812 persons out of a total population of 3,248. the number of fever cases is shown below:
Treatment
Fever
No fever
Total
Quinine 20
792
812
No Quinine
220
2216
2436
240
3008
3248
let us take the hypothesis that Quinine is not effective in checking malaria
Applying Chi – Square:
Find out Expected frequency by applying the formula = E = RT X CT/ N = 240 X 812 / 3248 = 60
Expected frequency corresponding to the first row and column is 60. therefore the table for expected frequency shall be: 60
752
812
180
2256
2436
240
3008
3248

8. O E
(O – E)2
(O – E)2 / E
Fever
Quinine and No Quinine 20 60
1600
26.667
220
180
8.889
No Fever
792
752
2.128
2216
2256
0.709
Σ [(O – E)2 /E] =
Degree of freedom (v) = (r – 1) (c – 1) = (2 – 1) (2 – 1) = 1 v = 1 level of significance = the table valve of Chi – Square = 3.84 The calculated value of Chi – Square is greater than the table value. The hypothesis is rejected. Hence Quinine is useful in checking malaria