As you learned in your statistics course, statistics play an essential role in scientific research. Since it is nearly impossible to study large populations, researchers rely on samples and use inferential statistical techniques to determine whether or not their data supports their hypothesis. For the purposes of our assignment, we conducted a Pearson’s r Correlation Coefficient or a Two-Way Chi-Square. What follows are instructions on how to read, interpret, and report out the results of your statistical test for the Results section of your research paper. Remember, the purpose of the results section is to communicate information about your data, allow replication of your study, for your statistical methodology to be scrutinized by the scientific community (or just looked at kindly by me), and it is a space for you to show off how smart you are.

Feel free to skip ahead to the calculator that you used…formatting guidelines (and an extra credit opportunity) are at the bottom of the page.

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Pearson’s r Correlation Coefficient

If both of your variables were measured quantitatively, you used Pearson’s r Correlation Coefficient calculator. At this point, you should have a clear idea of what your X variable is and what your Y variable is, and you should have conducted the analysis. We are meeting on the results page. Here we see our data’s scatterplot, some computational tables, and the very important Result Details & Calculations table. Below is an example of some random numbers that I typed into the calculator. I was thinking about snacks when I did it, so let’s pretend my X variable is Handfuls of Goldfish Crackers Consumed per Day and my Y variable is Cans of Seltzer Water Consumed per Day, over 13 days.

We can skip down to the Result Details & Calculations table and check out the results of our analysis.

This shows the Result Details & Calculation table from the Pearson’s R correlation calculator.

In the Result Details & Calculations table, I can see some summary information about my X and Y variables (yellow and red boxes), including the sum of scores (indicated by sigma), and the mean for each variable. Looks like over 13 days, I consumed about 5 handfuls of Goldfish (X) per day (M = 4.62) and had about 6 cans of Seltzer Water (Y) per day (M = 5.39). Sponsor me, Bubly. Next, we find our value for N in under the ‘X and Y Combined’ heading (blue square; N = 13). Finally, we have our value for r, and we can ignore that they’ve capitalized it (green square) and R2, which is the coefficient of determination (black square; which determines how much variation in one variable is explained by the variation in the other variable). It looks like we’ve found a weak positive relationship between Goldfish and Seltzer Water consumption.

At this point, I jot the important details I will need to report my results into my notes:

N = 13 (to calculate degrees of freedom)

r = 0.42

R2= 0.18 (which I multiply by 100 to get a percentage of 18%)

Just like in our statistics courses, now that we have a value for r we need to determine statistical significance. To do this, click ‘Click here to calculate the p-value (Links to an external site.)’ (purple square) on this page. We will use this new calculator to determine statistical significance. Essentially, it asks for our value for r, the number of participants in our study (N), and determines the critical value at the .05 level of significance. Plug in your values, select the 0.05 Significance Level, and click Calculate. After we click calculate, the results will populate on the same page (pink square).

This shows the p-value significance calculator for the Pearson’s R correlation.Here we can see the results of my “study.” We are looking for a value for p below 0.05, because that is our threshold for significance. With a p-value of 0.15, I do not have statistical significance. Shh, don’t tell Bubly. Unfortunately, my results were not statistically significant, so I cannot reasonably say there is a relationship between Goldfish and Seltzer Water consumption, but it is still important that I report out my results. I jot down my value for p, because I will need that later (p= 0.15).

Here is how I would report the results in APA format:

Here is how I would report the results in APA format:

Results

A Pearson’s correlation coefficient was conducted to assess the relationship between the number of handfuls of Goldfish Crackers consumed and cans of Seltzer water consumed per day for one adult female. Thirteen days were analyzed (N = 13). There was no statistically significant correlation between handfuls of Goldfish Crackers consumed and cans of Seltzer Water consumed per day, r(11) = 0.42, p = 0.15), with the number of handfuls of Goldfish Crackers explaining 18% of the variation in Seltzer Water consumption.

(Yours will not have the box around it, I just do this to make it stand out on the page)

Here are the general formatting guidelines of this paragraph:

Opening sentence: Introduces the statistical test used and the variables being compared, as well as the population studied (for us, it’s college students enrolled in a psychology course)

Second sentence: Tells the reader how many participants are in the study with the value for N italicized and in parentheses at the end of the sentence.

Third sentence: A description of the results of the significant test and the results of the Pearson’s r. I have indicated that there was not a statistically significant relationship, reiterated the relationship I was assessing, provided the results of the statistical analysis, and used my value for R2 to write about my coefficient of determination.

Note: If you do have statistical significance, you also need to comment on the strength and direction of the relationship. The first part of this sentence might look something like this: “There was a statistically significant, weak positive correlation between handfuls of Goldfish Crackers consumed and cans of Seltzer Water consumed per day, r(11) = 0.42, p < 0.05…” Here you will notice that we indicate either the actual value for p, or put “p < .05.”

Format of results: r(degrees of freedom) = [value for r], p = [value for p]

The degrees of freedom calculation for r: df = N – 2

Plug in your values into the “equation” above for this part of the sentence

Reporting coefficient of determination (after you’ve converted it into a percentage): “with [Variable X] explaining [value for R2 converted to a percentage] of the variation in [Variable Y].”

That’s it! In more advanced statistical analysis, we would expect the results to be much more in-depth, but for the purposes of this class, as long as you follow the appropriate formatting and content guidelines, a single paragraph will suffice.

Two-Way Chi-Square

If your variables were measured qualitatively, you used a Two-Way Chi-Square to analyze your data. At this point, you should have put your values into the calculator and conducted the analysis. We are meeting after the results have been generated into the calculator. For the purposes of this example, I used numbers from an example in my statistics course, where we compared Personality (introversion and extroversion) and Color Preference (Red, Green, Yellow, and Blue). To the right, I have provided a (non-APA) example of a chi-square table. I put these values into the calculator and clicked “Calculate.”

Here are my results:

This image shows the results from the chi-square calculator.

Lucky for me, everything that I need to report the results of my study are listed here. I have Chi-Square value, degrees of freedom, and p-value.

I jot down the following into my notes:

X2= 35.60

degrees of freedom (df) = 3

p-value = …uh oh.

So, my p-value is so low that it is listed in scientific format. Here’s a handy calculator that you can use to convert a value that looks like this to a decimal. I plugged in my values to the second form because I have a negative number in my notation (-8), and it converts it to 0.00000009098. That’s a big number, but I know that it is WAY less than my threshold for statistical significance, which is a p-value of 0.05 or below. I decide to write the following in my notes: p < .05.

Since I do have statistical significance (my value for p is well below the .05 threshold), I have to figure out what the strength of the relationship is – if you do not have statistical significance (your value for p is above .05), then you can skip this step. I need to determine Cramer’s V, as this tells us about the strength of the relationship in the same way that the value for r tells us about the strength and direction of a relationship for Pearson’s r. To do so, please go to this calculator (not included in the video). (Links to an external site.)You can use the results from the previous calculator (above) to determine the number of rows and columns (this is an essential step.) After the number of rows and columns you have populated, fill in your values again and click calculate. You will notice that you get the same value for the chi-square, but now you also have a value for Cramer’s V. We interpret Cramer’s V the same way that we interpret Pearson’s r. The closer the value is to 1, the stronger the relationship. This looks to me as though we have a moderately weak association (V = 0.42). Additionally, at the intersection of both the “Totals” rows/columns, I have my value for N, which is 200. I will add these to my notes and get ready to write up the results of my study comparing personality type and color preference.

Here is how I would report the results in APA format:

Results

A two-way chi-square test was conducted to determine the association between personality type and color preference (N = 200). There was a statistically significant association between personality type and color preference, X2(3)= 35.60, p < .05. The association was moderately weak, Cramer’s V = 0.42.

(Yours will not have the box around it, I just do this to make it stand out on the page)

Here are the general formatting guidelines of this paragraph:

Opening sentence: Introduces the statistical test used and the variables being compared, as well as the number of participants in the study italicized and in parentheses.

Second sentence: A description of the results of the significant test and the results of the two-way chi-square are provided. I have indicated that there was a statistically significant relationship, reiterated the relationship I was assessing, provided the results of the statistical analysis.

Note: If you do not have a statistically significant relationship, it is still important to report out the results. You would simply say “There was NOT a significantly significant relationship… and continue on as normal (but you would omit the last sentence).

Format of results: X2(degrees of freedom) = [value for the Chi-Square], p = [value for p]

Plug in your values into the “equation” above for this part of the sentence

If your value for p is very small, p < .05 will suffice.

Third sentence: The final sentence reports on the strength of the relationship following the format I did above.

That’s it! In more advanced statistical analysis, we would expect the results to be much more in-depth, but for the purposes of this class, as long as you follow the appropriate formatting and content guidelines, a single paragraph will suffice.

All Students: Assignment Expectations

EXTRA CREDIT: For an additional 10 points, you may provide a graph or table that you create on your own (do not use the scatterplot, for example, from the calculator) and follows APA formatting guidelines.

For Pearson’s r, the appropriate graph would be a scatterplot.

Here’s information on APA formatting of figures (graphs). (Links to an external site.)

Here’s a video on how to make an XY bivariate scatterplot using Google Sheets. (Links to an external site.)

For a two-way chi-square, it would be most appropriate to include a table.

Here’s an example of a chi-square table.

Here’s an example of what a chi-square table would look like. It provides the reader with the observed frequencies.

Here’s information on APA formatting of tables. (Links to an external site.)

Here’s a video on how to format tables using Google Docs. (Links to an external site.)

General Formatting Guidelines:

In this assignment, you will write your Results section following APA formatting. There will be no references required for the Results section and it is expected that your results will be less than one page.

Font options (name & size): Calibri 12, Arial 11, Lucida Sans Unicode 10, Times New Roman 12, or Georgia 11

Double-spaced, 1” margins

Citation format: APA

All formatting should follow APA formatting guidelines

Submission format: .docx, doc, .pdf, .rtf ONLY (It is your responsibility to check your submissions and make sure they are in the appropriate format that I can access. I cannot access google docs links or PAGES. You need to save the file in one of these formats.)

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