Definition :
Definition :
The Chi Square distribution is a mathematical distribution that is used directly or indirectly in many tests of significance. The most common use of the chi square distribution is to test differences between proportions. Although this test is by no means the only test based on the chi square distribution, it has come to be known as the chi square test. The chi square distribution has one parameter, its degrees of freedom (df). It has a positive skew; the skew is less with more degrees of freedom. The mean of a chi square distribution is its df. The mode is df - 2 and the median is approximately df -0 .7.
Formula :
The Chi-square distribution is defined by:
f(x) = {1/[2v/2 * Г(v/2)]} * [x(v/2)-1 * e-x/2]
v = 1, 2, ..., 0 < xwhere
v is the degrees of freedom
e is the base of the natural logarithm, sometimes called Euler's e (2.71...)
Г (gamma) is the Gamma function
The above animation shows the shape of the Chi-square distribution as the degrees of freedom increase (1, 2, 5, 10, 25 and 50).
Application :
One reason the normal distribution is important is that many psychological and educational variables are distributed approximately normally. Measures of reading ability, introversion, job satisfaction, and memory are among the many psychological variables approximately normally distributed. Although the distributions are only approximately normal, they are usually quite close. A second reason the normal distribution is so important is that it is easy for mathematical statisticians to work with. This means that many kinds of statistical tests can be derived for normal distributions. Almost all statistical tests discussed in this text assume normal distributions. Fortunately, these tests work very well even if the distribution is only approximately normally distributed. Some tests work well even with very wide deviations from normality. Finally, if the mean and standard deviation of a normal distribution are known, it is easy to convert back and forth from raw scores to percentiles.
Applet :
For an illustration of Chi-square distributions go to
http://stat-www.berkeley.edu/users/stark/Java/chiHiLite.htm
Performing a chi-square test
http://www.physics.csbsju.edu/stats/chi-square_form.html
Excel Function :
These functions can be accessed by clicking on Insert and then choosing Function from the drop down menu.
The Excel function for finding a Chi-square distribution for a given data set is :
Chitest(actual_range, expected_range)
Returns the test for independence: the value from the chi-squared distribution of the statistic and the appropriate degrees of freedom.