Chi-Square Test

A chi-squared test is basically a data analysis on the basis of observations of a random set of variables. Usually, it’s a comparison of two statistical data sets. It is also represented as χ2 test. This test was introduced by Karl Pearson in 1900 for categorical data analysis and distribution. So it was mentioned as Pearson’s chi-squared test. With the assumption of the null hypothesis as true, this test is used to estimate how likely the observations that are made would be.

A hypothesis is a consideration, that a given condition or statement might be true, which we can test afterwards. Chi-squared tests are usually created from a sum of squared falsities or errors else via the sample variance.

Chi-Square Distribution

When we consider, the null speculation as true, the sampling distribution of the test statistic is called as chi-squared distribution. The chi-squared test helps to determine whether there is a notable difference between the normal frequencies and the observed frequencies in one or more classes or categories. It gives the probability of independent variables.

Probability is all about chance or risk or uncertainty. It is the possibility of the outcome of the sample or the occurrence of an event. But when we talk about statistics, it is more about how we handle various data using different techniques. It helps to represent complicated data or bulk data in a very easy and understandable way. It describes the collection, analysis, interpretation, presentation, and organization of data. The concept of both probability and statistics is related to the chi-squared test.

Chi-Square Test Formula

The chi-squared test is done to check if there is any difference between the observed value and expected value. The formula for chi-square can be written as;

Chi-square Test Formula

Chi-Square Table

The chi-square distribution table with three probability level is provided here. The statistic here is used to examine whether distributions of certain variables vary from one another. The categorical variable will produce data in the categories and numerical variables will produce data in numerical form. The distribution of χ2 with (r-1)(c-1) degrees of freedom(DF), is represented in the table given below. Here, r represents the number of rows in the two-way table and c represents the number of columns.

DF

Value of P

0.05 0.01 0.001
1 3.84 6.64 10.83
2 5.99 9.21 13.82
3 7.82 11.35 16.27
4 9.49 13.28 18.47
5 11.07 15.09 20.52
6 12.59 16.81 22.46
7 14.07 18.48 24.32
8 15.51 20.09 26.13
9 16.92 21.67 27.88
10 18.31 23.21 29.59
11 19.68 24.73 31.26
12 21.03 26.22 32.91
13 22.36 27.69 34.53
14 23.69 29.14 36.12
15 25.00 30.58 37.70
16 26.30 32.00 39.25
17 27.59 33.41 40.79
18 28.87 34.81 42.31
19 30.14 36.19 43.82
20 31.41 37.57 45.32
21 32.67 38.93 46.80
22 33.92 40.29 48.27
23 35.17 41.64 49.73
24 36.42 42.98 51.18
25 37.65 44.31 52.62
26 38.89 45.64 54.05
27 40.11 46.96 55.48
28 41.34 48.28 56.89
29 42.56 49.59 58.30
30 43.77 50.89 59.70
31 44.99 52.19 61.10
32 46.19 53.49 62.49
33 47.40 54.78 63.87
34 48.60 56.06 65.25
35 49.80 57.34 66.62
36 51.00 58.62 67.99
37 52.19 59.89 69.35
38 53.38 61.16 70.71
39 54.57 62.43 72.06
40 55.76 63.69 73.41
41 56.94 64.95 74.75
42 58.12 66.21 76.09
43 59.30 67.46 77.42
44 60.48 68.71 78.75
45 61.66 69.96 80.08
46 62.83 71.20 81.40
47 64.00 72.44 82.72
48 65.17 73.68 84.03
49 66.34 74.92 85.35
50 67.51 76.15 86.66
51 68.67 77.39 87.97
52 69.83 78.62 89.27
53 70.99 79.84 90.57
54 72.15 81.07 91.88
55 73.31 82.29 93.17
56 74.47 83.52 94.47
57 75.62 84.73 95.75
58 76.78 85.95 97.03
59 77.93 87.17 98.34
60 79.08 88.38 99.62
61 80.23 89.59 100.88
62 81.38 90.80 102.15
63 82.53 92.01 103.46
64 83.68 93.22 104.72
65 84.82 94.42 105.97
66 85.97 95.63 107.26
67 87.11 96.83 108.54
68 88.25 98.03 109.79
69 89.39 99.23 111.06
70 90.53 100.42 112.31
71 91.67 101.62 113.56
72 92.81 102.82 114.84
73 93.95 104.01 116.08
74 95.08 105.20 117.35
75 96.22 106.39 118.60
76 97.35 107.58 119.85
77 98.49 108.77 121.11
78 99.62 109.96 122.36
79 100.75 111.15 123.60
80 101.88 112.33 124.84
81 103.01 113.51 126.09
82 104.14 114.70 127.33
83 105.27 115.88 128.57
84 106.40 117.06 129.80
85 107.52 118.24 131.04
86 108.65 119.41 132.28
87 109.77 120.59 133.51
88 110.90 121.77 134.74
89 112.02 122.94 135.96
90 113.15 124.12 137.19
91 114.27 125.29 138.45
92 115.39 126.46 139.66
93 116.51 127.63 140.90
94 117.63 128.80 142.12
95 118.75 129.97 143.32
96 119.87 131.14 144.55
97 120.99 132.31 145.78
98 122.11 133.47 146.99
99 123.23 134.64 148.21
100 124.34 135.81 149.48

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