إيجاد القيم الحرجة لمربع كاي

The chi-square distribution is asymmetrical, unlike symmetric distributions. For a 95% confidence interval, the remaining 5% is split equally as 2.5% on each side. Since the distribution's shape differs on each side, the left and right critical values that separate these 2.5% areas must be calculated independently using the chi-square table.

Locate the degrees of freedom in the left column, then find the significance level (0.025 for a 95% confidence interval) across the top row. The intersection gives the right-tailed critical value. For example, with nine degrees of freedom and 0.025 significance level, the value is 19.023.

Since chi-square tables provide cumulative areas from the right, subtract the significance level from 1 to find the left-tailed area. For a 0.025 right-side area, calculate 1 minus 0.025 equals 0.975. Then locate 0.975 in the table's column to obtain the left-tailed critical value.

A 95% confidence level covers 95% of the area under the chi-square curve, with the remaining 5% distributed equally on both sides as 2.5% each. These tail areas represent unlikely values, while the central 95% represents likely results for estimating population standard deviation.

Degrees of freedom, calculated as sample size minus one (df = n - 1), determines which row to use in the chi-square table. Each row corresponds to a different degrees of freedom value, and the correct row must be selected before finding the critical value at the intersection with the appropriate significance level column.

Chi-square tables use right-cumulative values because the distribution is asymmetrical and naturally skewed. This format allows direct reading of right-tailed critical values. For left-tailed values, you must convert by subtracting the desired tail area from 1 to find the cumulative area from the right.

Critical values separate likely sample results from unlikely ones at a specified confidence level. They define the boundaries of the confidence interval, ensuring that 95% of possible sample standard deviations fall within the interval for normally distributed populations, making them essential for reliable population parameter estimation.