How to Read Pearson's Goodness of Fit Output

Chi-Square Goodness-of-Fit Test in SPSS Statistics

Introduction

The chi-square goodness-of-fit exam is a single-sample nonparametric test, also referred to as the one-sample goodness-of-fit test or Pearson'southward chi-square goodness-of-fit examination. It is used to decide whether the distribution of cases (e.g., participants) in a single categorical variable (e.g., "gender", consisting of two groups: "males" and "females") follows a known or hypothesised distribution (e.g., a distribution that is "known", such as the proportion of males and females in a country; or a distribution that is "hypothesised", such as the proportion of males versus females that nosotros conceptualize voting for a particular political party in the next elections). The proportion of cases expected in each grouping of the categorical variable tin be equal or unequal (e.one thousand., we may anticipate an "equal" proportion of males and females voting for the Republican Political party, or an "unequal" proportion, with 70% of those voting for the Republican Political party existence male and simply 30% female person).

When you comport out a chi-square goodness-of-fit examination, "hypothesising" whether you lot await the proportion of cases in each group of your chiselled variable to be "equal" or "unequal" is critical. Not only is it an important aspect of your research design, only from a practical perspective, information technology volition determine how you behave out the chi-square goodness-of-fit test in SPSS Statistics, equally well equally how yous interpret and write upwardly your results.

In this "quick get-go" guide, we prove you how to carry out a chi-foursquare goodness-of-fit exam using SPSS Statistics when y'all have "equal" expected proportions (e.g., you anticipated an "equal" proportion of males and females voting for the Republican Party). In addition, we explain how to interpret the results from this test. However, if you lot have "unequal" expected proportions (e.yard., y'all anticipated 70% of those voting for the Republican Political party being male person and merely 30% female), we show you how to do this in our enhanced chi-square goodness-of-fit guide (N.B., you can learn almost our enhanced content on our Features: Overview folio). Therefore, assuming that you lot would like to know the SPSS Statistics procedure and estimation of the chi-square goodness-of-fit test when y'all have equal expected proportions, you lot first need to understand the different assumptions that your data must see in order for a chi-square goodness-of-fit to give y'all a valid result. We discuss these assumptions next.

SPSS Statistics

Assumptions

When you lot cull to analyse your data using the chi-foursquare goodness-of-fit test, office of the process involves checking to make sure that the data you want to analyse can actually exist analysed using a chi-square goodness-of-fit test. Yous demand to do this because information technology is only appropriate to utilise a chi-square goodness-of-fit examination if your data meets four assumptions that are required for a chi-square goodness-of-fit exam to give you a valid event. In practice, checking for these assumptions is a relatively simple process, only requiring you to utilize SPSS Statistics. Permit'southward have a look at these four assumptions:

• Assumption #1: One categorical variable (i.e., the variable tin can be dichotomous, nominal or ordinal). Examples of dichotomous variables include gender (two groups: male person or female), treatment type (two groups: medication or no medication), educational level (two groups: undergraduate or postgraduate) and religious (2 groups: yes or no). Examples of nominal variables include ethnicity (e.g., 3 groups: Caucasian, African American and Hispanic), and profession (eastward.g., 5 groups: surgeon, doctor, nurse, dentist, therapist). Examples of ordinal variables include Likert scales (e.grand., a 7-point calibration from "strongly agree" through to "strongly disagree"), amid other ways of ranking categories (due east.g., a 5-point scale for measuring job satisfaction, ranging from "most satisfied" to "least satisfied"; a 4-point scale determining how piece of cake information technology was to navigate a new website, ranging from "very easy" to "very difficult; or a 3-point scale explaining how much a customer liked a product, ranging from "Not very much", to "It is OK", to "Yes, a lot"), and physical activeness level (due east.m., 4 groups: sedentary, depression, moderate and high).
• Assumption #2: You should have independence of observations, which means that there is no human relationship betwixt any of the cases (e.g., participants).
• Supposition #3: The groups of the categorical variable must be mutually exclusive. For instance, if the four groups of a categorical variable, Physical Activity Level, were "Sedentary", "Low", "Moderate" and "High", a case (due east.g., a participant in an practice study) could just be in one of these four groups (east.g., a participant could not exist classified as having a "High" action level and a "Low" action level, but only one or the other).
• Assumption #4: There must be at least 5 expected frequencies in each group of your categorical variable. This is an assumption of the chi-square goodness-of-fit test and volition be shown in your SPSS Statistics output when you run the test.

Therefore, before proceeding, bank check that your report design meets assumptions #1, #2 and #iii. Bold they do, you will now demand to check that your information meets assumption #iv, which you tin can practice using SPSS Statistics. We explicate how to test for assumption #4 and how to interpret the SPSS Statistics output in our enhanced chi-square goodness-of-fit guide to assistance you get this right. You can detect out near our enhanced content as a whole on our Features: Overview page, or more specifically, learn how we aid with testing assumptions on our Features: Assumptions page.

In the section, Procedure, we illustrate the SPSS Statistics procedure required to perform a chi-square goodness-of-fit examination bold that no assumptions have been violated and when you take equal expected proportions. First, we prepare out the example nosotros use to explain the chi-square goodness-of-fit procedure in SPSS Statistics.

SPSS Statistics

Example

A website owner, Christopher, wants to offering a free gift to people that purchase a subscription to his website. New subscribers can choose one of three gifts of equal value: a souvenir voucher, a cuddly toy or free cinema tickets. After 1000 people have signed up, Christopher wants to review the figures to see if the three gifts offered were equally pop.

In this instance, the 3 gifts – a gift voucher, a cuddly toy or free movie house tickets – reflect the iii groups of the chiselled variable, gift_type. The 1000 people that have signed up reflect the "cases" (i.east., cases can be anything from "people", to "animals", "objects", "organisations", and and so forth).

SPSS Statistics

Data Setup in SPSS Statistics

In that location are two methods of inbound data into SPSS Statistics in club to run a chi-square goodness-of-fit examination in SPSS Statistics. Common to both methods is a column in the SPSS Statistics data file for the chiselled variable, which in this instance, we shall name gift_type. We have assigned codes of "1" for the souvenir certificate, which we labelled "Gift Certificate", "2" for the cuddly toy, which we labelled "Cuddly Toy", and "3" for the complimentary cinema tickets, which we labelled "Cinema Tickets". If the frequency data has already been summated for the diverse categories, we need to create a second column that contains the respective frequency counts; we take called this variable frequency. This type of information entry is shown below:

Published with written permission from SPSS Statistics, IBM Corporation.

Note: If you have entered your information in this way, y'all cannot run the chi-square goodness-of-fit examination without first "weighting" your cases. This is a procedure that tells SPSS Statistics that you have summated your categories. It is required because information technology changes the fashion that SPSS Statistics deals with your data in order to run the chi-foursquare goodness-of-fit examination. If you are unsure how to weight your cases, we prove you how to practise this in our enhanced chi-square goodness-of-fit guide.

Alternatively, you may take the data in raw form (i.east., you have not summated the frequencies). In this case, you lot exercise not demand a second column as SPSS Statistics tin calculate the frequencies of occurrence of each category for yous. This would mean that, in this example, in that location are yard rows of information, of which the beginning of said information is shown beneath:

Published with written permission from SPSS Statistics, IBM Corporation.

If you are however unsure how to enter your data accurately into the Data View and Variable View of SPSS Statistics, we should y'all how to practice this in our enhanced chi-square goodness-of-fit test guide. You tin learn almost our enhanced data setup content on our Features: Data Setup page.

SPSS Statistics

Test Procedure in SPSS Statistics

The four steps below show you how to analyse your information using a chi-square goodness-of-fit test in SPSS Statistics when y'all have hypothesised that you take equal expected proportions (Due north.B., if yous are unclear well-nigh the differences betwixt equal and unequal expected proportions, see the Introduction). Also, it is important to annotation that this procedure will simply give y'all the correct results if you lot have gear up your data correctly in SPSS Statistics (N.B., if yous have entered the summated frequencies for each group of your categorical variable, this process volition only piece of work if you lot have already "weighted" your cases, as we explained in the Data Setup department earlier, but if you have entered all of your information into SPSS Statistics in raw form, this process will not give the correct results). In our enhanced chi-square goodness-of-fit examination guide, nosotros show all the SPSS Statistics procedures for when you take equal and unequal expected proportions, also as when yous have to weight your cases or have non summated your data. If you lot only need to follow this "quick start" guide for equal expected proportions (without the weighting of cases), the 4 steps you lot need are shown below. At the end of these four steps, we evidence you how to interpret the results from this exam.

1. Click Analyze > Nonparametric Tests > Legacy Dialogs > Chi-foursquare... on the pinnacle menu as shown beneath:

   Note: The procedure that follows is identical for SPSS Statistics versions 18 to 28, as well as the subscription version of SPSS Statistics, with version 28 and the subscription version being the latest versions of SPSS Statistics. Yet, in version 27 and the subscription version, SPSS Statistics introduced a new look to their interface called "SPSS Light", replacing the previous wait for versions 26 and earlier versions, which was called "SPSS Standard". Therefore, if you accept SPSS Statistics versions 27 or 28 (or the subscription version of SPSS Statistics), the images that follow will be low-cal grey rather than blueish. Nonetheless, the procedure is identical.

   

   Published with written permission from SPSS Statistics, IBM Corporation.

2. You lot volition be presented with the Chi-foursquare Examination dialogue box, as shown below:

   

   Published with written permission from SPSS Statistics, IBM Corporation.

3. Transfer the gift_type variable into the Test Variable List: box past using the button, every bit shown below:

   

   Published with written permission from SPSS Statistics, IBM Corporation.

   
   Keep the All categories equal pick selected in the –Expected Values– area as we are assuming equal proportions for each category.
   
4. Click on the button to generate the output.

SPSS Statistics

SPSS Statistics Output for Chi-Square Goodness-of-Fit Test

The SPSS Statistics output that is generated for the chi-foursquare goodness-of-fit test will depend on whether you have hypothesised that the proportion of cases expected in each group of the chiselled variable is equal or unequal. For a consummate caption of the output you have to interpret for the chi-square goodness-of-fit test for both scenarios (as well every bit the testing of Assumption #four), you can access our enhanced chi-square goodness-of-fit guide, every bit well equally all of our other SPSS Statistics guides by becoming a member of Laerd Statistics. Beneath nosotros evidence you the SPSS Statistics output when y'all accept hypothesised that the proportion of cases expected in each group of your categorical variable is equal.

The table beneath, gift_type, provides the observed frequencies (Observed N) for each gift, too as the expected frequencies (Expected Due north), which are the frequencies expected if the goose egg hypothesis is truthful. The difference betwixt the observed and expected frequencies is provided in the Residual column.

Published with written permission from SPSS Statistics, IBM Corporation.

The tabular array below, Test Statistics, provides the actual result of the chi-square goodness-of-fit test. We tin can see from this table that our examination statistic is statistically meaning: χⁱⁱ(2) = 49.4, p < .0005. Therefore, we can reject the nada hypothesis and conclude that at that place are statistically significant differences in the preference of the type of sign-up gift, with less people preferring the "Cuddly Toy" (N = 230) compared to either the "Gift Certificate" (N = 370) or the "Cinema Tickets" (North = 400).

Published with written permission from SPSS Statistics, IBM Corporation.

In our enhanced chi-square goodness-of-fit test guide, we evidence you how to write up the results if you lot need to report this in a dissertation/thesis, assignment or research written report. We do this using the Harvard and APA styles. You tin learn more about our enhanced content on our Features: Overview page.

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