Polaris’ Help Center:How to Use the Chi-Square Test for Survey Data Analysis

Note: This section is fairly technical in nature and
we do not provide a Chi-Square Calculator. The Chi-Square test is used when cross-tabulating survey results and all popular cross-tab software packages include this as a test option. Please use our handy Chi-Square Significance Reference Table for help interpreting your cross tabulation results.

Chi-Square Reference Table

df = (columns-1) x (rows-1)

                        Significance Level
        df                90%                 95%                99%
         1             2.7055              3.8415             6.6349
         2             4.6052              5.9915             9.2104
         3             6.2514              7.8147            11.3449
         4             7.7794              9.4877            13.2767
         5             9.2363             11.0705            15.0863
         6            10.6446             12.5916            16.8119
         7             12.017             14.0671            18.4753
         8            13.3616             15.5073            20.0902
         9            14.6837              16.919             21.666
        10            15.9872              18.307            23.2093
        11             17.275             19.6752             24.725
        12            18.5493             21.0261             26.217
        13            19.8119              22.362            27.6882
        14            21.0641             23.6848            29.1412
        15            22.3071             24.9958             30.578
        16            23.5418             26.2962            31.9999
        17             24.769             27.5871            33.4087
        18            25.9894             28.8693            34.8052
        19            27.2036             30.1435            36.1908
        20             28.412             31.4104            37.5663
        21            29.6151             32.6706            38.9322
        22            30.8133             33.9245            40.2894
        23            32.0069             35.1725            41.6383
        24            33.1962              36.415            42.9798
        25            34.3816             37.6525             44.314
        26            35.5632             38.8851            45.6416      
        27            36.7412             40.1133            46.9628
        28            37.9159             41.3372            48.2782
        29            39.0875             42.5569            49.5878
        30             40.256              43.773            50.8922
        31            41.4217             44.9853            52.1914
        32            42.5847             46.1942            53.4857
        33            43.7452             47.3999            54.7754
        34            44.9032             48.6024            56.0609
        35            46.0588             49.8018             57.342
        36            47.2122             50.9985            58.6192
        37            48.3634             52.1923            59.8926
        38            49.5126             53.3835             61.162
        39            50.6598             54.5722            62.4281
        40             51.805             55.7585            63.6908
        41            52.9485             56.9424              64.95
        42            54.0902              58.124            66.2063
        43            55.2302             59.3035            67.4593
        44            56.3685             60.4809            68.7096
        45            57.5053             61.6562            69.9569
        46            58.6405             62.8296            71.2015
        47            59.7743             64.0011            72.4432
        48            60.9066             65.1708            73.6826
        49            62.0375             66.3387            74.9194
        50            63.1671             67.5048            76.1538
        51            64.2954             68.6693             77.386
        52            65.4224             69.8322            78.6156
        53            66.5482             70.9934            79.8434
        54            67.6728             72.1532            81.0688
        55            68.7962             73.3115             82.292
        56            69.9185             74.4683            83.5136
        57            71.0397             75.6237            84.7327
        58            72.1598             76.7778            85.9501
        59            73.2789             77.9305            87.1658
        60             74.397              79.082            88.3794
        61            75.5141             80.2321            89.5912
        62            76.6302              81.381            90.8015
        63            77.7454             82.5287            92.0099
        64            78.8597             83.6752            93.2167
        65             79.973             84.8206             94.422
        66            81.0855             85.9649            95.6256
        67            82.1971              87.108            96.8277
        68            83.3079             88.2502            98.0283
        69            84.4179             89.3912            99.2274
        70             85.527             90.5313           100.4251
        71            86.6354             91.6703           101.6214
        72            87.7431             92.8083           102.8163
        73            88.8499             93.9453           104.0098
        74            89.9561             95.0815           105.2019
        75            91.0615             96.2167           106.3929
        76            92.1662              97.351           107.5824
        77            93.2702             98.4844           108.7709
        78            94.3735              99.617           109.9582
        79            95.4762            100.7486            111.144
        80            96.5782            101.8795           112.3288
        81            97.6796            103.0095           113.5123
        82            98.7803            104.1387           114.6948
        83            99.8805            105.2672           115.8762
        84             100.98            106.3949           117.0566
        85           102.0789            107.5217           118.2356
        86           103.1773            108.6479           119.4137
        87            104.275            109.7733           120.5909
        88           105.3723             110.898           121.7672
        89           106.4689             112.022           122.9422
        90            107.565            113.1452           124.1162
        91           108.6606            114.2679           125.2893
        92           109.7556            115.3898           126.4616
        93           110.8501             116.511            127.633
        94           111.9442            117.6317           128.8032
        95           113.0377            118.7516           129.9725
        96           114.1307            119.8709           131.1411
        97           115.2232            120.9897           132.3089
        98           116.3153            122.1077           133.4756
        99           117.4069            123.2252           134.6415
       100            118.498            124.3421           135.8069

The Chi-Square is a statistical test used to examine
differences with categorical variables. The Chi-Square test
is used in two circumstances: 1) for estimating how closely
an observed distribution matches an expected distribution
(a “goodness-of-fit” test), or 2) for estimating whether two
random variables are independent.

For survey results, the Chi-Square statistical test comes in
most handy when analyzing cross tabulations of the survey
data. Since crosstabs show the frequency and percentage
of responses to questions by different segments or categories of respondents (gender, income, profession, etc.), the Chi-Square test can tell us whether there is a statistical difference between the segments/categories in how they answered the question.

Note #1: The Chi-Square statistic only tests whether two
variables are independent in a “yes” or “no” format. It does
not indicate the degree of difference between the respondent categories in terms of which is greater or less.

Note #2: The Chi-Square test requires that you use
numerical values (frequency counts), not percentages or
ratios.

Chi-Square Calculation Formula

chi-sq = sum[(Ei,j – Ai,j) / Ei,j ]

where Ei,j represents the expected value for cell i, j

E i,j = (Ti x Tj) / N Ti = sum of values in columni, Tj = sum of values in row j, N = total of values in table

A i,j represents the actual value for cell i, j

For significance, test chi-sq against critical values for the Chi Square Distribution

df = (columns-1) x (rows-1)