{"id":21929,"date":"2026-09-05T21:11:09","date_gmt":"2026-09-05T15:41:09","guid":{"rendered":"https:\/\/www.geographynotespdf.com\/?p=21929"},"modified":"2026-09-05T21:13:08","modified_gmt":"2026-09-05T15:43:08","slug":"chi-square-test","status":"publish","type":"post","link":"https:\/\/geographynotespdf.com\/cms\/chi-square-test\/","title":{"rendered":"Chi-Square Test (\u0915\u093e\u0908-\u0938\u094d\u0915\u094d\u0935\u093e\u092f\u0930 \u092a\u0930\u0940\u0915\u094d\u0937\u0923) | 100% Concept Clear | Formula, Types, Hypothesis & Numerical | Statistics for Research"},"content":{"rendered":"\n\n\n
Chi-Square Test<\/strong><\/span><\/p>\n

(\u0915\u093e\u0908-\u0938\u094d\u0915\u094d\u0935\u093e\u092f\u0930 \u092a\u0930\u0940\u0915\u094d\u0937\u0923)<\/strong><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

\u092a\u0930\u093f\u091a\u092f (Introduction):<\/strong><\/span><\/h2>\n

\u00a0 \u00a0 \u00a0 Chi-Square Test (\u03c7\u00b2 Test) \u090f\u0915 \u092a\u094d\u0930\u092e\u0941\u0916 Non-Parametric Statistical Test \u0939\u0948, \u091c\u093f\u0938\u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u092e\u0941\u0916\u094d\u092f\u0924\u0903 Categorical Data (\u0936\u094d\u0930\u0947\u0923\u0940\u092c\u0926\u094d\u0927 \u0906\u0901\u0915\u0921\u093c\u094b\u0902) \u0915\u0947 \u0935\u093f\u0936\u094d\u0932\u0947\u0937\u0923 \u0915\u0947 \u0932\u093f\u090f \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964 \u0907\u0938\u0915\u0947 \u092e\u093e\u0927\u094d\u092f\u092e \u0938\u0947 \u092f\u0939 \u091c\u093e\u0901\u091a\u093e \u091c\u093e\u0924\u093e \u0939\u0948 \u0915\u093f \u092a\u094d\u0930\u093e\u092a\u094d\u0924 Observed Frequency (\u092a\u094d\u0930\u0947\u0915\u094d\u0937\u093f\u0924 \u0906\u0935\u0943\u0924\u094d\u0924\u093f) \u0914\u0930 Expected Frequency (\u0905\u092a\u0947\u0915\u094d\u0937\u093f\u0924 \u0906\u0935\u0943\u0924\u094d\u0924\u093f) \u0915\u0947 \u092c\u0940\u091a \u0905\u0902\u0924\u0930 \u0915\u0947\u0935\u0932 \u0938\u0902\u092f\u094b\u0917 \u0915\u0947 \u0915\u093e\u0930\u0923 \u0939\u0948 \u092f\u093e \u0938\u093e\u0902\u0916\u094d\u092f\u093f\u0915\u0940\u092f \u0930\u0942\u092a \u0938\u0947 \u092e\u0939\u0924\u094d\u0935\u092a\u0942\u0930\u094d\u0923 \u0939\u0948\u0964<\/span><\/p>\n

\u00a0 \u00a0 \u00a0 \u092d\u0942\u0917\u094b\u0932, \u0938\u092e\u093e\u091c\u0936\u093e\u0938\u094d\u0924\u094d\u0930, \u091c\u0928\u0938\u0902\u0916\u094d\u092f\u093e \u0905\u0927\u094d\u092f\u092f\u0928, \u0915\u0943\u0937\u093f, \u0905\u0930\u094d\u0925\u0936\u093e\u0938\u094d\u0924\u094d\u0930 \u0924\u0925\u093e \u0905\u0928\u094d\u092f \u0938\u093e\u092e\u093e\u091c\u093f\u0915 \u0935\u093f\u091c\u094d\u091e\u093e\u0928\u094b\u0902 \u092e\u0947\u0902 \u0907\u0938\u0915\u093e \u0935\u094d\u092f\u093e\u092a\u0915 \u0909\u092a\u092f\u094b\u0917 \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964<\/span><\/p>\n

Chi-Square Test \u0915\u093e \u0905\u0930\u094d\u0925-<\/strong><\/span><\/p>\n

\u00a0 \u00a0 \u00a0 \u00a0Chi-Square Test \u092e\u0947\u0902 \u092f\u0939 \u0924\u0941\u0932\u0928\u093e \u0915\u0940 \u091c\u093e\u0924\u0940 \u0939\u0948 \u0915\u093f \u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u0930\u0942\u092a \u0938\u0947 \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0906\u0935\u0943\u0924\u094d\u0924\u093f\u092f\u093e\u0901 \u0905\u092a\u0947\u0915\u094d\u0937\u093f\u0924 \u0906\u0935\u0943\u0924\u094d\u0924\u093f\u092f\u094b\u0902 \u0938\u0947 \u0915\u093f\u0924\u0928\u0940 \u092d\u093f\u0928\u094d\u0928 \u0939\u0948\u0902\u0964<\/span><\/p>\n

\u00a0 \u00a0 \u00a0 \u00a0\u0907\u0938\u0947 \u0938\u093e\u092e\u093e\u0928\u094d\u092f\u0924\u0903 \u03c7\u00b2 (Chi-Square)<\/strong> \u0938\u0947 \u092a\u094d\u0930\u0926\u0930\u094d\u0936\u093f\u0924 \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964<\/span><\/p>\n

\u00a0“Observed Frequency (O) \u0914\u0930 Expected Frequency (E) \u0915\u0947 \u092c\u0940\u091a \u0905\u0902\u0924\u0930 \u091c\u093f\u0924\u0928\u093e \u0905\u0927\u093f\u0915 \u0939\u094b\u0917\u093e, \u03c7\u00b2 \u0915\u093e \u092e\u093e\u0928 \u0909\u0924\u0928\u093e \u0939\u0940 \u0905\u0927\u093f\u0915 \u0939\u094b \u0938\u0915\u0924\u093e \u0939\u0948\u0964”<\/strong><\/span><\/p>\n

Chi-Square \u0915\u093e \u0938\u0942\u0924\u094d\u0930-<\/strong><\/span><\/p>\n

\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u03c72<\/sup> = \u2211E(O\u2212E)2<\/sup><\/strong><\/span><\/span><\/p>\n

\u091c\u0939\u093e\u0901-<\/span><\/p>\n

\u03c7\u00b2<\/strong> = Chi-Square \u0915\u093e \u092e\u093e\u0928<\/span><\/p>\n

O<\/strong> = Observed Frequency (\u092a\u094d\u0930\u0947\u0915\u094d\u0937\u093f\u0924 \u0906\u0935\u0943\u0924\u094d\u0924\u093f)<\/span><\/p>\n

E<\/strong> = Expected Frequency (\u0905\u092a\u0947\u0915\u094d\u0937\u093f\u0924 \u0906\u0935\u0943\u0924\u094d\u0924\u093f)<\/span><\/p>\n

\u03a3<\/strong> = \u0938\u092d\u0940 \u0935\u0930\u094d\u0917\u094b\u0902 \u0915\u0947 \u0932\u093f\u090f \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u092e\u093e\u0928\u094b\u0902 \u0915\u093e \u092f\u094b\u0917<\/span><\/p>\n

Expected Frequency \u0915\u0948\u0938\u0947 \u091c\u094d\u091e\u093e\u0924 \u0915\u0930\u0947\u0902?<\/strong><\/span><\/p>\n

\u00a0 \u00a0 \u00a0\u0926\u094b-way contingency table \u092e\u0947\u0902 Expected Frequency \u0915\u093e \u0938\u093e\u092e\u093e\u0928\u094d\u092f \u0938\u0942\u0924\u094d\u0930 \u0939\u0948:<\/span><\/p>\n

E = (Row Total \u00d7 Column Total\u200b)\/Grand Total<\/strong><\/span><\/p>\n

\u0905\u0930\u094d\u0925\u093e\u0924\u094d-<\/span><\/p>\n

Expected Frequency = (\u092a\u0902\u0915\u094d\u0924\u093f \u092f\u094b\u0917 \u00d7 \u0938\u094d\u0924\u092e\u094d\u092d \u092f\u094b\u0917) \u00f7 \u0915\u0941\u0932 \u092f\u094b\u0917<\/strong><\/span><\/p>\n

\u00a0 \u00a0 \u00a0 Chi-Square Test \u0915\u0947 \u092a\u094d\u0930\u092e\u0941\u0916 \u092a\u094d\u0930\u0915\u093e\u0930<\/strong><\/span><\/p>\n

(1) Test of Goodness of Fit:<\/span><\/strong><\/span><\/p>\n

\u00a0 \u00a0 \u00a0Test of Goodness of Fit<\/strong> \u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u092f\u0939 \u091c\u093e\u0901\u091a\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948 \u0915\u093f \u092a\u094d\u0930\u093e\u092a\u094d\u0924 Observed Frequencies (\u092a\u094d\u0930\u0947\u0915\u094d\u0937\u093f\u0924 \u0906\u0935\u0943\u0924\u094d\u0924\u093f\u092f\u093e\u0901)<\/strong> \u0915\u093f\u0938\u0940 \u0928\u093f\u0930\u094d\u0927\u093e\u0930\u093f\u0924 Theoretical \u092f\u093e Expected Distribution (\u0905\u092a\u0947\u0915\u094d\u0937\u093f\u0924 \u0935\u093f\u0924\u0930\u0923)<\/strong> \u0915\u0947 \u0905\u0928\u0941\u0930\u0942\u092a \u0939\u0948\u0902 \u092f\u093e \u0928\u0939\u0940\u0902\u0964<\/span><\/p>\n

\u0907\u0938\u092e\u0947\u0902 \u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u0906\u0901\u0915\u0921\u093c\u094b\u0902 \u0915\u0940 \u0924\u0941\u0932\u0928\u093e \u092a\u0939\u0932\u0947 \u0938\u0947 \u0928\u093f\u0930\u094d\u0927\u093e\u0930\u093f\u0924 \u0905\u0928\u0941\u092a\u093e\u0924 \u092f\u093e \u0935\u093f\u0924\u0930\u0923 \u0938\u0947 \u0915\u0940 \u091c\u093e\u0924\u0940 \u0939\u0948\u0964 \u092f\u0926\u093f \u0926\u094b\u0928\u094b\u0902 \u0915\u0947 \u092c\u0940\u091a \u0905\u0902\u0924\u0930 \u0938\u093e\u0902\u0916\u094d\u092f\u093f\u0915\u0940\u092f \u0930\u0942\u092a \u0938\u0947 \u092e\u0939\u0924\u094d\u0935\u092a\u0942\u0930\u094d\u0923 \u0928\u0939\u0940\u0902 \u0939\u0948, \u0924\u094b \u092e\u093e\u0928\u093e \u091c\u093e\u0924\u093e \u0939\u0948 \u0915\u093f observed data \u0905\u092a\u0947\u0915\u094d\u0937\u093f\u0924 \u0935\u093f\u0924\u0930\u0923 \u0915\u0947 \u0905\u0928\u0941\u0915\u0942\u0932 \u0939\u0948\u0964<\/span><\/p>\n

\u0909\u0926\u093e\u0939\u0930\u0923 \u0915\u0947 \u0932\u093f\u090f, \u0915\u093f\u0938\u0940 \u0915\u094d\u0937\u0947\u0924\u094d\u0930 \u092e\u0947\u0902 \u091a\u093e\u0930 \u092a\u094d\u0930\u0915\u093e\u0930 \u0915\u0940 \u092d\u0942\u092e\u093f \u0915\u093e \u0905\u092a\u0947\u0915\u094d\u0937\u093f\u0924 \u0905\u0928\u0941\u092a\u093e\u0924 25 : 25 : 25 : 25 \u0939\u094b, \u0924\u094b Chi-Square \u0926\u094d\u0935\u093e\u0930\u093e \u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u0935\u093f\u0924\u0930\u0923 \u0915\u0940 \u0909\u092a\u092f\u0941\u0915\u094d\u0924\u0924\u093e \u091c\u093e\u0901\u091a\u0940 \u091c\u093e \u0938\u0915\u0924\u0940 \u0939\u0948\u0964<\/span><\/p>\n

(2) Test of Independence:<\/strong><\/span><\/p>\n

\u00a0 \u00a0 \u00a0Test of Independence<\/strong> \u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0926\u094b Categorical Variables (\u0936\u094d\u0930\u0947\u0923\u0940\u092c\u0926\u094d\u0927 \u091a\u0930\u094b\u0902)<\/strong> \u0915\u0947 \u092c\u0940\u091a \u0915\u093f\u0938\u0940 \u0938\u093e\u0902\u0916\u094d\u092f\u093f\u0915\u0940\u092f \u0938\u0902\u092c\u0902\u0927 \u092f\u093e Association<\/strong> \u0915\u0940 \u091c\u093e\u0901\u091a \u0915\u0930\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964 \u0907\u0938\u0915\u093e \u092e\u0941\u0916\u094d\u092f \u0909\u0926\u094d\u0926\u0947\u0936\u094d\u092f \u092f\u0939 \u0928\u093f\u0930\u094d\u0927\u093e\u0930\u093f\u0924 \u0915\u0930\u0928\u093e \u0939\u0948 \u0915\u093f \u090f\u0915 variable \u0915\u0940 categories \u0926\u0942\u0938\u0930\u0947 variable \u0915\u0940 categories \u0938\u0947 \u0938\u094d\u0935\u0924\u0902\u0924\u094d\u0930 \u0939\u0948\u0902 \u092f\u093e \u0909\u0928\u0915\u0947 \u092c\u0940\u091a \u0915\u094b\u0908 \u092e\u0939\u0924\u094d\u0935\u092a\u0942\u0930\u094d\u0923 \u0938\u0902\u092c\u0902\u0927 \u092e\u094c\u091c\u0942\u0926 \u0939\u0948\u0964<\/span><\/p>\n

\u00a0 \u00a0 \u0907\u0938\u092e\u0947\u0902 \u0938\u093e\u092e\u093e\u0928\u094d\u092f\u0924\u0903 Contingency Table<\/strong> \u0915\u093e \u092a\u094d\u0930\u092f\u094b\u0917 \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964 \u0909\u0926\u093e\u0939\u0930\u0923 \u0915\u0947 \u0932\u093f\u090f, \u0936\u093f\u0915\u094d\u0937\u093e \u0938\u094d\u0924\u0930 \u0914\u0930 \u0930\u094b\u091c\u0917\u093e\u0930 \u0915\u0940 \u0938\u094d\u0925\u093f\u0924\u093f \u0915\u0947 \u092c\u0940\u091a \u0938\u0902\u092c\u0902\u0927 \u0915\u0940 \u091c\u093e\u0901\u091a \u0915\u0940 \u091c\u093e \u0938\u0915\u0924\u0940 \u0939\u0948\u0964 \u092f\u0926\u093f \u0926\u094b\u0928\u094b\u0902 variables \u0938\u094d\u0935\u0924\u0902\u0924\u094d\u0930 \u0928\u0939\u0940\u0902 \u0939\u0948\u0902, \u0924\u094b \u0909\u0928\u0915\u0947 \u092c\u0940\u091a statistically significant association \u092e\u093e\u0928\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948\u0964<\/span><\/p>\n

(3) Test of Homogeneity:<\/span><\/p>\n

\n
\n
\n

\u00a0 \u00a0 \u00a0 Test of Homogeneity<\/strong> \u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0935\u093f\u092d\u093f\u0928\u094d\u0928 Populations \u092f\u093e Groups<\/strong> \u092e\u0947\u0902 \u0915\u093f\u0938\u0940 categorical characteristic \u0915\u0947 \u0935\u093f\u0924\u0930\u0923 \u0915\u0940 \u0938\u092e\u093e\u0928\u0924\u093e \u0915\u0940 \u091c\u093e\u0901\u091a \u0915\u0930\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964 \u0907\u0938\u0915\u093e \u0909\u0926\u094d\u0926\u0947\u0936\u094d\u092f \u092f\u0939 \u092a\u0924\u093e \u0932\u0917\u093e\u0928\u093e \u0939\u094b\u0924\u093e \u0939\u0948 \u0915\u093f \u0905\u0932\u0917-\u0905\u0932\u0917 \u0938\u092e\u0942\u0939\u094b\u0902 \u092e\u0947\u0902 \u0915\u093f\u0938\u0940 \u0935\u093f\u0936\u0947\u0937 \u0935\u093f\u0936\u0947\u0937\u0924\u093e \u0915\u093e \u0935\u093f\u0924\u0930\u0923 \u0938\u092e\u093e\u0928 \u0939\u0948 \u092f\u093e \u0928\u0939\u0940\u0902\u0964<\/span><\/p>\n

\u00a0 \u00a0 \u0907\u0938\u092e\u0947\u0902 \u0935\u093f\u092d\u093f\u0928\u094d\u0928 \u0938\u092e\u0942\u0939\u094b\u0902 \u0915\u0940 observed frequencies \u0915\u0940 \u0924\u0941\u0932\u0928\u093e \u0915\u0940 \u091c\u093e\u0924\u0940 \u0939\u0948\u0964 \u0909\u0926\u093e\u0939\u0930\u0923 \u0915\u0947 \u0932\u093f\u090f, \u092c\u093f\u0939\u093e\u0930 \u0915\u0947 \u0935\u093f\u092d\u093f\u0928\u094d\u0928 \u091c\u093f\u0932\u094b\u0902 \u092e\u0947\u0902 \u0915\u093f\u0938\u0940 \u0935\u093f\u0936\u0947\u0937 \u0915\u0943\u0937\u093f \u092b\u0938\u0932 \u0915\u0947 \u0935\u093f\u0924\u0930\u0923 \u0915\u0940 \u0938\u092e\u093e\u0928\u0924\u093e \u091c\u093e\u0901\u091a\u0940 \u091c\u093e \u0938\u0915\u0924\u0940 \u0939\u0948\u0964 \u092f\u0926\u093f \u0905\u0902\u0924\u0930 statistically significant \u0928\u0939\u0940\u0902 \u0939\u0948, \u0924\u094b \u0935\u093f\u092d\u093f\u0928\u094d\u0928 \u0938\u092e\u0942\u0939\u094b\u0902 \u092e\u0947\u0902 distribution \u0915\u094b homogeneous \u092e\u093e\u0928\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964<\/span><\/p>\n

Null Hypothesis \u0914\u0930 Alternative Hypothesis<\/span><\/h2>\n

\u00a0 \u00a0 \u00a0Chi-Square Test \u092e\u0947\u0902 \u0915\u093f\u0938\u0940 \u092d\u0940 statistical test \u0915\u0940 \u0924\u0930\u0939 \u092a\u0939\u0932\u0947 Null Hypothesis (H\u2080)<\/strong> \u0914\u0930 Alternative Hypothesis (H\u2081)<\/strong> \u0928\u093f\u0930\u094d\u0927\u093e\u0930\u093f\u0924 \u0915\u0940 \u091c\u093e\u0924\u0940 \u0939\u0948\u0902\u0964 \u0907\u0928\u0915\u0947 \u092e\u093e\u0927\u094d\u092f\u092e \u0938\u0947 \u092f\u0939 \u0924\u092f \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948 \u0915\u093f \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0906\u0901\u0915\u0921\u093c\u094b\u0902 \u092e\u0947\u0902 \u0926\u093f\u0916\u093e\u0908 \u0926\u0947\u0928\u0947 \u0935\u093e\u0932\u093e \u0905\u0902\u0924\u0930 \u092f\u093e \u0938\u0902\u092c\u0902\u0927 \u0938\u093e\u0902\u0916\u094d\u092f\u093f\u0915\u0940\u092f \u0930\u0942\u092a \u0938\u0947 \u092e\u0939\u0924\u094d\u0935\u092a\u0942\u0930\u094d\u0923 (statistically significant)<\/strong> \u0939\u0948 \u092f\u093e \u0915\u0947\u0935\u0932 \u0938\u0902\u092f\u094b\u0917 \u0915\u0947 \u0915\u093e\u0930\u0923 \u0909\u0924\u094d\u092a\u0928\u094d\u0928 \u0939\u0941\u0906 \u0939\u0948\u0964<\/span><\/p>\n

✍️ <\/span><\/span>Null Hypothesis (H\u2080)<\/span><\/h3>\n

\u00a0 \u00a0 \u00a0Null Hypothesis<\/strong> \u0915\u093e \u0905\u0930\u094d\u0925 \u0939\u0948 \u0915\u093f observed data \u092e\u0947\u0902 \u092a\u093e\u092f\u093e \u0917\u092f\u093e \u0905\u0902\u0924\u0930 \u092f\u093e variables \u0915\u0947 \u092c\u0940\u091a \u0938\u0902\u092c\u0902\u0927 \u0938\u093e\u0902\u0916\u094d\u092f\u093f\u0915\u0940\u092f \u0930\u0942\u092a \u0938\u0947 \u092e\u0939\u0924\u094d\u0935\u092a\u0942\u0930\u094d\u0923 \u0928\u0939\u0940\u0902 \u0939\u0948<\/strong>\u0964<\/span><\/p>\n

Goodness of Fit \u092e\u0947\u0902:<\/strong><\/span><\/p>\n

\u00a0 \u00a0 \u00a0 H\u2080:<\/strong> Observed \u0914\u0930 Expected frequencies \u0915\u0947 \u092c\u0940\u091a \u0905\u0902\u0924\u0930 statistically significant \u0928\u0939\u0940\u0902 \u0939\u0948\u0964<\/span><\/p>\n

\u0905\u0930\u094d\u0925\u093e\u0924\u094d \u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 (Observed) \u0935\u093f\u0924\u0930\u0923 \u0914\u0930 \u0905\u092a\u0947\u0915\u094d\u0937\u093f\u0924 (Expected) \u0935\u093f\u0924\u0930\u0923 \u092e\u0947\u0902 \u0915\u094b\u0908 \u092e\u0939\u0924\u094d\u0935\u092a\u0942\u0930\u094d\u0923 \u0905\u0902\u0924\u0930 \u0928\u0939\u0940\u0902 \u0939\u0948\u0964<\/span><\/p>\n

Test of Independence \u092e\u0947\u0902:<\/strong><\/span><\/p>\n

\u00a0 \u00a0 H\u2080:<\/strong> \u0926\u094b\u0928\u094b\u0902 categorical variables \u090f\u0915-\u0926\u0942\u0938\u0930\u0947 \u0938\u0947 independent \u0939\u0948\u0902\u0964<\/span><\/p>\n

\u0905\u0930\u094d\u0925\u093e\u0924\u094d \u0926\u094b\u0928\u094b\u0902 variables \u0915\u0947 \u092c\u0940\u091a \u0915\u094b\u0908 statistically significant association \u092f\u093e \u0938\u0902\u092c\u0902\u0927 \u0928\u0939\u0940\u0902 \u0939\u0948\u0964<\/span><\/p>\n

✍️<\/span><\/span> Alternative Hypothesis (H\u2081)<\/span><\/h3>\n

\u00a0 \u00a0 Alternative Hypothesis<\/strong> \u092f\u0939 \u092c\u0924\u093e\u0924\u0940 \u0939\u0948 \u0915\u093f observed data \u092e\u0947\u0902 \u092a\u093e\u092f\u093e \u0917\u092f\u093e \u0905\u0902\u0924\u0930 \u092f\u093e variables \u0915\u0947 \u092c\u0940\u091a \u0938\u0902\u092c\u0902\u0927 \u0938\u093e\u0902\u0916\u094d\u092f\u093f\u0915\u0940\u092f \u0930\u0942\u092a \u0938\u0947 \u092e\u0939\u0924\u094d\u0935\u092a\u0942\u0930\u094d\u0923 \u0939\u0948<\/strong>\u0964<\/span><\/p>\n

Goodness of Fit \u092e\u0947\u0902:<\/strong><\/span><\/p>\n

\u00a0 \u00a0 \u00a0 H\u2081:<\/strong> Observed \u0914\u0930 Expected frequencies \u0915\u0947 \u092c\u0940\u091a \u0905\u0902\u0924\u0930 statistically significant \u0939\u0948\u0964<\/span><\/p>\n

\u0905\u0930\u094d\u0925\u093e\u0924\u094d \u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u0935\u093f\u0924\u0930\u0923 \u0905\u092a\u0947\u0915\u094d\u0937\u093f\u0924 \u0935\u093f\u0924\u0930\u0923 \u0938\u0947 \u092e\u0939\u0924\u094d\u0935\u092a\u0942\u0930\u094d\u0923 \u0930\u0942\u092a \u0938\u0947 \u0905\u0932\u0917 \u0939\u0948\u0964<\/span><\/p>\n

Test of Independence \u092e\u0947\u0902:<\/strong><\/span><\/p>\n

\u00a0 \u00a0 \u00a0H\u2081:<\/strong> \u0926\u094b\u0928\u094b\u0902 categorical variables \u0915\u0947 \u092c\u0940\u091a statistically significant association \u0939\u0948\u0964<\/span><\/p>\n

\u0905\u0930\u094d\u0925\u093e\u0924\u094d \u0926\u094b\u0928\u094b\u0902 variables \u090f\u0915-\u0926\u0942\u0938\u0930\u0947 \u0938\u0947 \u0938\u094d\u0935\u0924\u0902\u0924\u094d\u0930 \u0928\u0939\u0940\u0902 \u0939\u0948\u0902 \u0914\u0930 \u0909\u0928\u0915\u0947 \u092c\u0940\u091a \u0915\u094b\u0908 \u092e\u0939\u0924\u094d\u0935\u092a\u0942\u0930\u094d\u0923 \u0938\u0902\u092c\u0902\u0927 \u092e\u094c\u091c\u0942\u0926 \u0939\u0948\u0964<\/span><\/p>\n

Degrees of Freedom (df)<\/span><\/strong><\/h2>\n

\u00a0 \u00a0 \u00a0Degrees of Freedom (df)<\/strong> Chi-Square Test \u0915\u093e \u090f\u0915 \u092e\u0939\u0924\u094d\u0935\u092a\u0942\u0930\u094d\u0923 \u0918\u091f\u0915 \u0939\u0948\u0964 \u0907\u0938\u0915\u093e \u0905\u0930\u094d\u0925 \u0939\u0948 \u0915\u093f \u0915\u093f\u0938\u0940 statistical calculation \u092e\u0947\u0902 \u0915\u093f\u0924\u0928\u0940 observations \u092f\u093e values \u0938\u094d\u0935\u0924\u0902\u0924\u094d\u0930 \u0930\u0942\u092a \u0938\u0947 \u092c\u0926\u0932 \u0938\u0915\u0924\u0940 \u0939\u0948\u0902, \u091c\u092c\u0915\u093f \u0906\u0935\u0936\u094d\u092f\u0915 constraints \u092f\u093e totals \u0938\u094d\u0925\u093f\u0930 \u0930\u0939\u0924\u0947 \u0939\u0948\u0902\u0964 <\/span><\/p>\n

\u00a0 \u00a0 \u00a0 Chi-Square \u0915\u0940 calculated value \u0915\u0940 \u0924\u0941\u0932\u0928\u093e \u0938\u0939\u0940 critical value<\/strong> \u0938\u0947 \u0915\u0930\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f Degrees of Freedom \u0915\u0940 \u0906\u0935\u0936\u094d\u092f\u0915\u0924\u093e \u0939\u094b\u0924\u0940 \u0939\u0948\u0964 \u0907\u0938\u0915\u093e \u0928\u093f\u0930\u094d\u0927\u093e\u0930\u0923 test \u0915\u0947 \u092a\u094d\u0930\u0915\u093e\u0930 \u0914\u0930 data \u0915\u0940 \u0938\u0902\u0930\u091a\u0928\u093e \u0915\u0947 \u0906\u0927\u093e\u0930 \u092a\u0930 \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964<\/span><\/p>\n

(1) Goodness of Fit \u0915\u0947 \u0932\u093f\u090f<\/span><\/strong><\/h3>\n

\u092f\u0926\u093f \u0915\u093f\u0938\u0940 distribution \u092e\u0947\u0902 n categories<\/strong> \u0939\u0948\u0902, \u0924\u094b Degrees of Freedom \u0915\u093e \u0938\u0942\u0924\u094d\u0930 \u0939\u0948-<\/span><\/p>\n

\u00a0 \u00a0 \u00a0 df = n-1<\/span><\/p>\n

\u091c\u0939\u093e\u0901 n = Categories \u0915\u0940 \u0938\u0902\u0916\u094d\u092f\u093e<\/strong><\/span><\/p>\n

\u0909\u0926\u093e\u0939\u0930\u0923: <\/strong>\u092f\u0926\u093f \u0915\u093f\u0938\u0940 dataset \u092e\u0947\u0902 5 categories \u0939\u0948\u0902, \u0924\u094b-<\/span><\/p>\n

\u00a0 \u00a0 \u00a0d f = 5\u22121 = 4<\/span><\/p>\n

(2) Contingency Table \u0915\u0947 \u0932\u093f\u090f<\/span><\/strong><\/span><\/h3>\n

\u00a0 \u00a0 \u092f\u0926\u093f Chi-Square Test of Independence \u0915\u0947 \u0932\u093f\u090f r rows<\/strong> \u0914\u0930 c columns<\/strong> \u0935\u093e\u0932\u0940 contingency table \u0939\u0948, \u0924\u094b Degrees of Freedom \u0915\u093e \u0938\u0942\u0924\u094d\u0930 \u0939\u0948-<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n

\u00a0 \u00a0 \u00a0 df = (r\u22121) (c\u22121)<\/span><\/p>\n

\u091c\u0939\u093e\u0901-<\/span><\/p>\n

r = Rows \u0915\u0940 \u0938\u0902\u0916\u094d\u092f\u093e<\/strong><\/span><\/p>\n

c = Columns \u0915\u0940 \u0938\u0902\u0916\u094d\u092f\u093e<\/strong><\/span><\/p>\n

\u0909\u0926\u093e\u0939\u0930\u0923: <\/strong>\u092f\u0926\u093f contingency table \u092e\u0947\u0902 4 rows \u0914\u0930 3 columns<\/strong> \u0939\u0948\u0902, \u0924\u094b-<\/span><\/p>\n

     df = (4-1)(3-1)\r\n     df = 3\u00d72 = 6\u200b<\/span><\/pre>\n
Chi-Square Test \u0915\u0940 \u092a\u094d\u0930\u0915\u094d\u0930\u093f\u092f\u093e:<\/strong><\/span><\/h3>\n
\u00a0 \u00a0 Research \u092e\u0947\u0902 Chi-Square Test \u0915\u094b \u0938\u093e\u092e\u093e\u0928\u094d\u092f\u0924\u0903 \u0928\u093f\u092e\u094d\u0928 \u091a\u0930\u0923\u094b\u0902 \u092e\u0947\u0902 \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948:<\/span><\/p>\n
Step 1:<\/strong> Research problem \u0928\u093f\u0930\u094d\u0927\u093e\u0930\u093f\u0924 \u0915\u0930\u0947\u0902\u0964<\/span><\/p>\n
Step 2:<\/strong> Null \u0914\u0930 Alternative Hypothesis \u092c\u0928\u093e\u090f\u0902\u0964<\/span><\/p>\n
Step 3:<\/strong> Observed Frequencies (O) \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0915\u0930\u0947\u0902\u0964<\/span><\/p>\n
Step 4:<\/strong> Expected Frequencies (E) calculate \u0915\u0930\u0947\u0902\u0964<\/span><\/p>\n
Step 5:<\/strong> \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 category \u0915\u0947 \u0932\u093f\u090f (O\u2212E)2\/E(O-E)^2\/E<\/span> calculate \u0915\u0930\u0947\u0902\u0964<\/span><\/p>\n
Step 6:<\/strong> \u0938\u092d\u0940 values \u0915\u094b \u091c\u094b\u0921\u093c\u0915\u0930 \u03c7\u00b2 \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0915\u0930\u0947\u0902\u0964<\/span><\/p>\n
Step 7:<\/strong> Degrees of Freedom \u0928\u093f\u0930\u094d\u0927\u093e\u0930\u093f\u0924 \u0915\u0930\u0947\u0902\u0964<\/span><\/p>\n
Step 8:<\/strong> Calculated \u03c7\u00b2 \u0915\u0940 \u0924\u0941\u0932\u0928\u093e Critical\/Table \u03c7\u00b2 \u0938\u0947 \u0915\u0930\u0947\u0902 \u0905\u0925\u0935\u093e p-value<\/strong> \u0926\u0947\u0916\u0947\u0902\u0964<\/span><\/p>\n
Step 9:<\/strong> Statistical decision \u0932\u0947\u0902\u0964<\/span><\/p>\n
Decision Rule<\/strong><\/span><\/p>\n
\u092f\u0926\u093f: \u03c72<\/sup>calculated\u200b>\u03c72<\/sup>critical<\/span><\/p>\n
\u0924\u094b H\u2080 \u0915\u094b reject<\/strong> \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964<\/span><\/p>\n
\u092f\u0926\u093f: \u03c72<\/sup>calculated\u200b\u2264\u03c72<\/sup>critical<\/span><\/p>\n
\u0924\u094b H\u2080 \u0915\u094b reject \u0928\u0939\u0940\u0902 \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e\u0964<\/strong><\/span><\/p>\n
p-value \u0915\u0947 \u0906\u0927\u093e\u0930 \u092a\u0930<\/span><\/h3>\n
\u092f\u0926\u093f:<\/span><\/p>\n
p < \u03b1<\/strong> \u2192 H\u2080 reject<\/span><\/p>\n
p \u2265 \u03b1<\/strong> \u2192 H\u2080 reject \u0928\u0939\u0940\u0902<\/span><\/p>\n
\u0938\u093e\u092e\u093e\u0928\u094d\u092f\u0924\u0903 \u03b1 = 0.05<\/strong> \u0932\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964<\/span><\/p>\n
\u0909\u0926\u093e\u0939\u0930\u0923: \u092e\u093e\u0928 \u0932\u0940\u091c\u093f\u090f \u0915\u093f\u0938\u0940 \u0915\u094d\u0937\u0947\u0924\u094d\u0930 \u092e\u0947\u0902 \u091a\u093e\u0930 \u092a\u094d\u0930\u0915\u093e\u0930 \u0915\u0940 \u092d\u0942\u092e\u093f \u0915\u093e observed distribution \u0939\u0948<\/span><\/strong><\/p>\n\n\n\n\n\n\n\n
\u092d\u0942\u092e\u093f \u092a\u094d\u0930\u0915\u093e\u0930<\/span><\/td>\nObserved (O)<\/span><\/td>\nExpected (E)<\/span><\/td>\n<\/tr>\n
\u0915\u0943\u0937\u093f<\/span><\/td>\n40<\/span><\/td>\n25<\/span><\/td>\n<\/tr>\n
\u0935\u0928<\/span><\/td>\n20<\/span><\/td>\n25<\/span><\/td>\n<\/tr>\n
\u092c\u0902\u091c\u0930<\/span><\/td>\n15<\/span><\/td>\n25<\/span><\/td>\n<\/tr>\n
\u0905\u0928\u094d\u092f<\/span><\/td>\n25<\/span><\/td>\n25<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
\u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 category \u0915\u0947 \u0932\u093f\u090f:<\/strong><\/span><\/p>\n
\u00a0 \u00a0(O\u2212E)2<\/sup>\u200b\/E<\/strong><\/span><\/span><\/p>\n
\u00a0 \u00a0 \u0928\u093f\u0915\u093e\u0932\u093e \u091c\u093e\u090f\u0917\u093e \u0914\u0930 \u0938\u092d\u0940 values \u0915\u094b \u091c\u094b\u0921\u093c\u0915\u0930 \u03c7\u00b2 calculated value<\/strong> \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0915\u0940 \u091c\u093e\u090f\u0917\u0940\u0964<\/span><\/p>\n
\u00a0 \u00a0 \u00a0\u0907\u0938\u0915\u0947 \u092c\u093e\u0926 \u0907\u0938\u0947 appropriate critical \u03c7\u00b2 value<\/strong> \u0938\u0947 compare \u0915\u093f\u092f\u093e \u091c\u093e\u090f\u0917\u093e\u0964<\/span><\/p>\n
Chi-Square Test \u0915\u0940 \u092a\u094d\u0930\u092e\u0941\u0916 \u0936\u0930\u094d\u0924\u0947\u0902 (Assumptions):<\/span><\/strong><\/span><\/p>\n
\u00a0 \u00a0 \u00a0 \u00a0 Chi-Square Test \u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0915\u0930\u0924\u0947 \u0938\u092e\u092f \u0915\u0941\u091b \u092e\u0939\u0924\u094d\u0935\u092a\u0942\u0930\u094d\u0923 conditions \u0915\u093e \u0927\u094d\u092f\u093e\u0928 \u0930\u0916\u0928\u093e \u091a\u093e\u0939\u093f\u090f:<\/span><\/p>\n
    \n
  1. Data \u0938\u093e\u092e\u093e\u0928\u094d\u092f\u0924\u0903 categorical\/frequency data<\/strong> \u0939\u094b\u0928\u093e \u091a\u093e\u0939\u093f\u090f\u0964<\/span><\/li>\n
  2. Observations \u0938\u094d\u0935\u0924\u0902\u0924\u094d\u0930 (independent) \u0939\u094b\u0928\u0940 \u091a\u093e\u0939\u093f\u090f\u0964<\/span><\/li>\n
  3. \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 observation \u0915\u094b \u0915\u0947\u0935\u0932 \u090f\u0915 category \u092e\u0947\u0902 classify \u0915\u093f\u092f\u093e \u091c\u093e\u0928\u093e \u091a\u093e\u0939\u093f\u090f\u0964<\/span><\/li>\n
  4. Expected frequencies \u092c\u0939\u0941\u0924 \u091b\u094b\u091f\u0940 \u0928\u0939\u0940\u0902 \u0939\u094b\u0928\u0940 \u091a\u093e\u0939\u093f\u090f\u0964<\/span><\/li>\n
  5. Sample \u0915\u094b research population \u0915\u093e \u0909\u091a\u093f\u0924 \u092a\u094d\u0930\u0924\u093f\u0928\u093f\u0927\u093f\u0924\u094d\u0935 \u0915\u0930\u0928\u093e \u091a\u093e\u0939\u093f\u090f\u0964<\/span><\/li>\n
  6. Test \u0915\u093e \u091a\u092f\u0928 research question \u0915\u0947 \u0905\u0928\u0941\u0930\u0942\u092a \u0939\u094b\u0928\u093e \u091a\u093e\u0939\u093f\u090f\u0964<\/span><\/li>\n<\/ol>\n
    Chi-Square Test \u0915\u0947 \u0932\u093e\u092d:<\/strong><\/span><\/p>\n
    ✍️ <\/span><\/span>\u092f\u0939 non-parametric test \u0939\u0948\u0964<\/span><\/p>\n
    ✍️ <\/span><\/span>\u0907\u0938\u0915\u093e \u0909\u092a\u092f\u094b\u0917 categorical data \u0915\u0947 \u0932\u093f\u090f \u0915\u093f\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948\u0964<\/span><\/p>\n
    ✍️ <\/span><\/span>\u0907\u0938\u0915\u093e calculation \u0905\u092a\u0947\u0915\u094d\u0937\u093e\u0915\u0943\u0924 \u0938\u0930\u0932 \u0939\u0948\u0964<\/span><\/p>\n
    ✍️ <\/span><\/span>Social Science \u0914\u0930 Geography \u092e\u0947\u0902 \u0909\u092a\u092f\u094b\u0917\u0940 \u0939\u0948\u0964<\/span><\/p>\n
    ✍️ <\/span><\/span>\u0926\u094b categorical variables \u0915\u0947 \u092c\u0940\u091a association \u0915\u0940 \u091c\u093e\u0901\u091a \u0915\u0940 \u091c\u093e \u0938\u0915\u0924\u0940 \u0939\u0948\u0964<\/span><\/p>\n
    ✍️ <\/span><\/span>\u092c\u0921\u093c\u0947 frequency datasets \u092e\u0947\u0902 \u0909\u092a\u092f\u094b\u0917\u0940 \u0939\u094b \u0938\u0915\u0924\u093e \u0939\u0948\u0964<\/span><\/p>\n
    Chi-Square Test \u0915\u0940 \u0938\u0940\u092e\u093e\u090f\u0901:<\/strong><\/span><\/p>\n
    ✍️ <\/span><\/span>\u092f\u0939 continuous numerical data \u0915\u0947 \u0932\u093f\u090f \u0938\u093e\u092e\u093e\u0928\u094d\u092f\u0924\u0903 \u0909\u092a\u092f\u0941\u0915\u094d\u0924 test \u0928\u0939\u0940\u0902 \u0939\u0948\u0964<\/span><\/p>\n
    ✍️ <\/span><\/span>\u092c\u0939\u0941\u0924 \u091b\u094b\u091f\u0940 expected frequencies \u0939\u094b\u0928\u0947 \u092a\u0930 \u092a\u0930\u093f\u0923\u093e\u092e unreliable \u0939\u094b \u0938\u0915\u0924\u0947 \u0939\u0948\u0902\u0964<\/span><\/p>\n
    ✍️ <\/span><\/span>\u092f\u0939 association \u0915\u0940 statistical significance<\/strong> \u092c\u0924\u093e \u0938\u0915\u0924\u093e \u0939\u0948, \u0932\u0947\u0915\u093f\u0928 \u0905\u092a\u0928\u0947 \u0906\u092a causation \u0938\u093f\u0926\u094d\u0927 \u0928\u0939\u0940\u0902 \u0915\u0930\u0924\u093e\u0964<\/span><\/p>\n
    ✍️ <\/span><\/span>\u0905\u0924\u094d\u092f\u0927\u093f\u0915 \u092c\u0921\u093c\u0947 sample \u092e\u0947\u0902 \u091b\u094b\u091f\u0947 differences \u092d\u0940 statistically significant \u0939\u094b \u0938\u0915\u0924\u0947 \u0939\u0948\u0902\u0964<\/span><\/p>\n
    ✍️ <\/span><\/span>\u0915\u0947\u0935\u0932 \u03c7\u00b2 significant \u0939\u094b\u0928\u0947 \u0938\u0947 relationship \u0915\u0940 strength \u0938\u094d\u092a\u0937\u094d\u091f \u0928\u0939\u0940\u0902 \u0939\u094b\u0924\u0940\u0964<\/span><\/p>\n
    Geography \u092e\u0947\u0902 Chi-Square Test \u0915\u093e \u0909\u092a\u092f\u094b\u0917:<\/strong><\/span><\/p>\n
    \u00a0 \u00a0 \u00a0 \u092d\u0942\u0917\u094b\u0932 \u092e\u0947\u0902 \u0907\u0938\u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0935\u093f\u092d\u093f\u0928\u094d\u0928 \u092a\u094d\u0930\u0915\u093e\u0930 \u0915\u0947 spatial \u0914\u0930 socio-economic categorical data \u092e\u0947\u0902 \u0915\u093f\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948, \u091c\u0948\u0938\u0947-<\/span><\/p>\n
    ✍️ <\/span><\/span>\u0917\u094d\u0930\u093e\u092e\u0940\u0923 \u090f\u0935\u0902 \u0936\u0939\u0930\u0940 \u091c\u0928\u0938\u0902\u0916\u094d\u092f\u093e \u0915\u093e \u0905\u0927\u094d\u092f\u092f\u0928<\/span><\/p>\n
    ✍️ <\/span><\/span>\u0936\u093f\u0915\u094d\u0937\u093e \u0914\u0930 \u0930\u094b\u091c\u0917\u093e\u0930 \u0915\u0947 \u092c\u0940\u091a \u0938\u0902\u092c\u0902\u0927<\/span><\/p>\n
    ✍️ <\/span><\/span>\u092d\u0942\u092e\u093f \u0909\u092a\u092f\u094b\u0917 \u0914\u0930 \u0915\u0943\u0937\u093f \u092a\u094d\u0930\u0915\u093e\u0930<\/span><\/p>\n
    ✍️ <\/span><\/span>\u092b\u0938\u0932 \u090f\u0935\u0902 \u0938\u093f\u0902\u091a\u093e\u0908 \u0915\u0947 \u092c\u0940\u091a \u0938\u0902\u092c\u0902\u0927<\/span><\/p>\n
    ✍️ <\/span><\/span>\u091c\u0928\u091c\u093e\u0924\u0940\u092f \u090f\u0935\u0902 \u0917\u0948\u0930-\u091c\u0928\u091c\u093e\u0924\u0940\u092f \u091c\u0928\u0938\u0902\u0916\u094d\u092f\u093e \u0915\u093e \u0935\u093f\u0924\u0930\u0923<\/span><\/p>\n
    ✍️ <\/span><\/span>\u0935\u093f\u092d\u093f\u0928\u094d\u0928 \u0915\u094d\u0937\u0947\u0924\u094d\u0930\u094b\u0902 \u092e\u0947\u0902 \u0938\u0941\u0935\u093f\u0927\u093e\u0913\u0902 \u0915\u093e \u0935\u093f\u0924\u0930\u0923<\/span><\/p>\n
    ✍️ <\/span><\/span>\u0906\u0935\u093e\u0938\u0940\u092f \u092a\u094d\u0930\u0915\u093e\u0930 \u0914\u0930 \u0938\u093e\u092e\u093e\u091c\u093f\u0915 \u0935\u0930\u094d\u0917<\/span><\/p>\n
    ✍️ <\/span><\/span>\u092a\u094d\u0930\u093e\u0915\u0943\u0924\u093f\u0915 \u0906\u092a\u0926\u093e \u090f\u0935\u0902 \u092a\u094d\u0930\u092d\u093e\u0935\u093f\u0924 \u0915\u094d\u0937\u0947\u0924\u094d\u0930<\/span><\/p>\n
    ✍️ <\/span><\/span>\u0935\u093f\u092d\u093f\u0928\u094d\u0928 \u091c\u093f\u0932\u094b\u0902 \u092e\u0947\u0902 \u0915\u0943\u0937\u093f \u092b\u0938\u0932\u094b\u0902 \u0915\u093e \u0935\u093f\u0924\u0930\u0923<\/span><\/p>\n
    \u0928\u093f\u0937\u094d\u0915\u0930\u094d\u0937 (Conclusion):<\/strong><\/span><\/p>\n
    \u00a0 \u00a0 \u00a0Chi-Square Test<\/strong> \u0938\u093e\u092e\u093e\u091c\u093f\u0915 \u0935\u093f\u091c\u094d\u091e\u093e\u0928 \u090f\u0935\u0902 \u092d\u0942\u0917\u094b\u0932 \u092e\u0947\u0902 \u0936\u094d\u0930\u0947\u0923\u0940\u092c\u0926\u094d\u0927 \u0906\u0901\u0915\u0921\u093c\u094b\u0902 \u0915\u0947 \u0935\u093f\u0936\u094d\u0932\u0947\u0937\u0923 \u0915\u0940 \u090f\u0915 \u092e\u0939\u0924\u094d\u0935\u092a\u0942\u0930\u094d\u0923 \u0938\u093e\u0902\u0916\u094d\u092f\u093f\u0915\u0940\u092f \u0935\u093f\u0927\u093f \u0939\u0948\u0964 \u0907\u0938\u0915\u0947 \u092e\u093e\u0927\u094d\u092f\u092e \u0938\u0947 observed \u0914\u0930 expected frequencies \u0924\u0925\u093e \u0935\u093f\u092d\u093f\u0928\u094d\u0928 categorical variables \u0915\u0947 \u092c\u0940\u091a \u0938\u0902\u092c\u0902\u0927 \u0915\u0940 \u091c\u093e\u0901\u091a \u0915\u0940 \u091c\u093e\u0924\u0940 \u0939\u0948\u0964 \u092f\u0939 \u0936\u094b\u0927\u0915\u0930\u094d\u0924\u093e \u0915\u094b \u0906\u0901\u0915\u0921\u093c\u094b\u0902 \u092e\u0947\u0902 \u092a\u093e\u090f \u0917\u090f \u0905\u0902\u0924\u0930 \u092f\u093e \u0938\u0902\u092c\u0902\u0927 \u0915\u0940 \u0938\u093e\u0902\u0916\u094d\u092f\u093f\u0915\u0940\u092f \u092e\u0939\u0924\u094d\u0924\u093e \u0928\u093f\u0930\u094d\u0927\u093e\u0930\u093f\u0924 \u0915\u0930\u0928\u0947 \u092e\u0947\u0902 \u0938\u0939\u093e\u092f\u0924\u093e \u0915\u0930\u0924\u093e \u0939\u0948\u0964<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"
    Chi-Square Test (\u0915\u093e\u0908-\u0938\u094d\u0915\u094d\u0935\u093e\u092f\u0930 \u092a\u0930\u0940\u0915\u094d\u0937\u0923) \u092a\u0930\u093f\u091a\u092f (Introduction): \u00a0 \u00a0 \u00a0 Chi-Square Test (\u03c7\u00b2 Test) \u090f\u0915 \u092a\u094d\u0930\u092e\u0941\u0916 Non-Parametric Statistical Test \u0939\u0948, \u091c\u093f\u0938\u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u092e\u0941\u0916\u094d\u092f\u0924\u0903 Categorical Data (\u0936\u094d\u0930\u0947\u0923\u0940\u092c\u0926\u094d\u0927 \u0906\u0901\u0915\u0921\u093c\u094b\u0902) \u0915\u0947 \u0935\u093f\u0936\u094d\u0932\u0947\u0937\u0923 \u0915\u0947 \u0932\u093f\u090f \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964 \u0907\u0938\u0915\u0947 \u092e\u093e\u0927\u094d\u092f\u092e \u0938\u0947 \u092f\u0939 \u091c\u093e\u0901\u091a\u093e \u091c\u093e\u0924\u093e \u0939\u0948 \u0915\u093f \u092a\u094d\u0930\u093e\u092a\u094d\u0924 Observed Frequency (\u092a\u094d\u0930\u0947\u0915\u094d\u0937\u093f\u0924 \u0906\u0935\u0943\u0924\u094d\u0924\u093f) \u0914\u0930 Expected Frequency (\u0905\u092a\u0947\u0915\u094d\u0937\u093f\u0924 \u0906\u0935\u0943\u0924\u094d\u0924\u093f) \u0915\u0947 \u092c\u0940\u091a \u0905\u0902\u0924\u0930 \u0915\u0947\u0935\u0932 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