• Vertical (y-axis, dv)

• Horizontal (x-axis, IV)

Bar-graphs are good to compare categories. Histograms are good for a continuous IV.

F . Results: Inferential Statistics

Even if you suffer from calculatophobia (fear of statistics), you must include inferential statistical analysis of your results in this section.

You probably have some idea of the results that you wanted from your experiment. Do not try to be a cheerleader rooting for your specific results. In research, you want to be a non-partisan (not taking sides) bystander. Don’t be so in love with your outcome that you lose your professional objectivity and become biased in your evaluation of the data. You must interpret the data regardless of the outcome—even if you failed to show a significant change between rock music and classical music (for example).

Inferential statistics ask if your random sample really represents the whole population. Can we infer from the small group how the large group would behave?

• Example: Could I ask 10% of the class to place a vote on something, and then apply that vote to the entire class?

• Could 1 handful of beans tell me the health and taste of the entire bin of beans?

• If 10 states voted for one presidential candidate, could we infer that the rest of the country will likely vote the same way?

• Is it likely that the math anxiety experienced by economics majors could be applied to the entire population of people majoring in economics that could have been sampled?

The Null Hypothesis (N₀)

The Null Hypothesis (N₀) assumes that there is no significant difference in your data. In statistics, The Null is assumed to be true…your experiment is either proving the Null to be right or wrong. Think of this like a court of law where you are assumed innocent until proven guilty.

So, if the starting assumption is the NULL hypothesis (N₀), than this means the results of your experiment were caused by probability or chance (like flipping a coin). This also means that your IV (Independent Variable) did not really have an effect. To test this theory, we need to conduct an inferential statistics test.

For example: I flip a coin 20 times while listening to heavy metal rock music. My results are 11 heads and 9 tails. A statistical test would show no real effect from the rock music. The results were probably just chance.

The further apart your observed results are from your expected results, the more likely it is that your IV (rock music) really did make a difference (example: if my results were 3 tails to 17 heads).

Do I Accept or Reject the Null Hypothesis?

The goal of research is to either accept or reject the null hypothesis. We want to establish that there actually is a relationship between the IV and DV and that any results we have obtained were not just due to chance.

Again, working off of your null hypothesis, if the null is true…then there is no significant difference caused by your Independent Variable (IV). This means you could have played rock music, classical music, or no music and the results would have been the same.

• This statistics test will establish that the results are not due to chance.

• This means that the data are significant (95% or higher, or p<0.05).

• When the data are significant (95% or higher, or p<0.05) we ”reject the null hypothesis.” This simply means that the IV did cause significant changes in the DV.

• When our hypothesis is not proven to be statistically significant, we must ”accept the null hypothesis.” (You can score just as high in the IA process even if you have to accept the null. The fact that you understand why and how is what is important).

After analyzing your data with inferential statistics, you need to make a decision on the significance of your findings. Were your results due to a real difference between the groups, or were they the result of chance or error?

Here are some examples of how to properly state the null:

• “The results were found to be significant (p<0.05), which means that the probability of the results occurring by chance if the null hypothesis were true is less than 5%. Therefore the null hypothesis was rejected and the research hypothesis was accepted. There does appear to be a real difference in the number of words recalled by males and females.”

• “A chi-squared (X²) test was carried out because the data were nominal and two independent samples were being compared. A significant difference was found (X²=4.78, p<0.05) and we can therefore reject the null hypothesis.”

• “There was a significant difference between the number of letters recalled by the experimental group and the control group. (X²=4.2, (degree of freedom) d.f.=1, p<0.05)”

Finding the Best Statistical Test

There are 3 statistical tests that are generally used for the IA:

• Chi-Squared Test

• Mann-Whitney U Test

• Wilcoxon Signed-Ranks Test

You should refer to the design of your experiment and the level of measurement of your data to justify your choice. You will need to explain why you chose the test that you did. The charts below should help you make a solid decision in selecting the right test for the job.

• Parametric: if the data is ratio or interval, evenly distributed (homogenous, bell curve), often best with a large sample. A T-Test is an example (not a recommended choice).

• Nonparametric: if the data is ordinal or nominal, this test gets rid of outliers if your data is skewed (important because of our small sample size).

Next, figure out if you need a paired or unpaired test (also based on the design method used).

• Paired: when you use the same people (repeated measures), the values are matched (before & after with the same group).

• Unpaired: when you use 2 groups of people (independent sample), when the values are not matched (control & experimental group, smokers & non-smokers).

Chi-Squared Test: used for nominal data in an independent samples design in an experiment testing the difference between two conditions (or categories).

• Examples: dog people or cat people; smokers & non-smokers; wear glasses & don’t wear glasses; Republicans & Democrats; male & female.

Mann Whitney U Test: used for ordinal data (or higher data) in an independent samples design in an experiment testing the difference between two conditions.

• Examples: The control group solved a puzzle with no noise, the experimental group solved the puzzle with loud music playing.

• The control group performed a task alone, the experimental group performed a task in front of others (social facilitation).

• The control group memorized a word list. The experimental group smelled vanilla while memorizing a word list.

Wilcoxon Signed Ranks Test: used for at least ordinal data (or higher data) in a repeated measures design in an experiment testing the difference between two conditions.

• Examples: Before and after; same group of 10 participants.

• A group of 10 first tried to learn new words; then the same group tried to learn new words with pictures next to them.

• A group of 10 first tried to solve a puzzle with no noise; then the same group tried to solve a puzzle with white noise playing in their ears.

For video explanations of these tests, visit: www.statisticslectures.com

You can double check your calculations at: www.graphpad.com/quickcalcs/index.cfm

The Chi-Squared (X²) Test

When you are interested in the relationship between 2 categories, you can use a chi-squared test.

Example: Students created flyers on different kinds of color paper—white paper, pink paper, and yellow paper. They watch the table to see if participants would take a white flyer, a pink flyer, or a yellow flyer.

STEP 1: Draw a chart with boxes.

STEP 2: Fill in the boxes with the results that you observed (O=observed results).

STEP 3: Add up the observed numbers to be sure your math is correct (columns and rows).

STEP 4: Find the expected results (E) for each cell. Take the column total X the row total then divide by the total number of participants. Columns go up and down; rows go across.

Example: 90 x 40 / 150 = 24

• O: Observed: the data that actually happened.

• E: Expected: The expected amount that there should be (based on percentages).

• The further apart the observed is from the expected, the more likely that there is a significant difference.

STEP 5: Double check that the expected values also add up to the column and row total.

STEP 6: (Almost Done!) Now you have to plug each cell into the formula.

• Cell 1: (O-E)² divided by E = (32-24)²/24= (8)²/24 = 64/24 = 2.6

• Cell 2: (O-E)² divided by E = (38-36)²/36 = (2)²/36 = 4/36 = 0.11

• Cell 3: (O-E)² divided by E = (20-30)²/30 = (-10)²/30 = 100/30 = 3.33

• Cell 4: (O-E)² divided by E = (8-16)²/16 = (-8)²/16 = 64/16 = 4

• Cell 5: (O-E)² divided by E = (22-24)²/24 = (-2)²/24 = 4/24 = 0.166

• Cell 6: (O-E)² divided by E = (30-20)²/20 = (10)²/20 = 100/20 = 5

STEP 7: Add the total number of cells. Total: 15.206

STEP 8: Now you check a Chi-Squared Chart/Table to see if your results show significance.

• For Chi-Squared tests, there is a number called the “degrees of freedom” (df) which has to do with how many columns and rows you used. To calculate the df you use the following formula: the # of rows in your table-1; multiply that by the number of columns -1= df.

Here is the easy version:

1 df (4 cells) your answer must be greater than (>) 3.84 in order to be significant (p<0.05). Anything less than (<) 3.84 means you must accept the NULL.

2 df (6 cells) must be greater than (>) 5.99. Anything less means you must accept the NULL.

Chi-Square Chart

Mann-Whitney U

(also called the Mann–Whitney–Wilcoxon (MWW) or Wilcoxon rank-sum test)

STEP 1: Rank all of your scores in order from lowest (Rank 1) to highest (Rank 15). Do this together for both groups—Control Group and Experimental Group.

*My example only goes to 15, but if you used 20 participants, then Rank all the way to 20.

Rank: 1, 2, 3, 4, 5, 6, 7, 8, 9. . .20

STEP 2: Add the sum of each group.

Experimental Group Control Group

Time (min) Rank Time (min) Rank

140 4 130 1

147 6 135 2

153 8 138 3

160 10 144 5

165 11 148 7

170 13 155 9

171 14 168 12

193 15

________________________________________________

(N₁=8; R₁=81) (N₂=7; R₂=39)

N= Number of participants in each group

R= the total when we add all of the ranks together

STEP 3: Now use the following formula to find U (U= Hypothetical Data).

R=Rank; N=Total Number of Observations

U=N₁*N₂+ N₁(N₁+1)-R₁

Example: U = (8) x (7) + 8*(9) - 81

U= 56 + (8 x 9) - 81

U = 56 + 72 – 81

U= 128 – 81 = 47

U = 47

STEP 4: Go to the Mann-Whitney Chart/Table (on the next page).

• U = 47.

• Is 47 between the N₁ to N₂ range of numbers on the chart?

• If YES, then reject the NULL Hypothesis.

• If NO, then accept the NULL Hypothesis.

• 47 is NOT between the N₁ to N₂ range of numbers on the chart so we must accept the Null hypothesis.

• This means that our independent variable (IV) did not make a significant difference in the results (Dependent Variable, DV).

Wilcoxon Signed-Ranks Test

• This test is used for the repeated measures design to see if there is a significant difference between the before and after trials of your experiment.

• The significance for this test must also be 0.05 (called the Alpha Level for this test).

• This test involves finding the “critical Z-score” from a Z-table which is 1.96.

• If your Z-score is less than -1.96 or greater than 1.96 then reject the null and accept your alternate hypothesis (H₁).

• This will mean that your Independent Variable (IV) did make a significant difference.

Step 1: Make a chart showing the scores before and after.

Before After

28 12

17 31

36 19

35 14

32 20

33 19

Step 2: Add a 3^rd column showing the difference between the before and after scores. You find this difference by subtracting.

Before After Difference

28 -(minus) 12 16

17 31 -14

36 19 17

35 14 21

32 20 12

33 19 14

Step 3: From now on, only this “Difference” column matters. Rank the “Difference” scores.

• The highest number ranks #6 (there are only 6 numbers in this sample. If there were 10 or 20 then your highest rank would go to 10 or 20).

• The lowest number ranks #1.

• Our ranks for this sample would be:

16 Rank # 4

-14 # 1

17 # 5

21 # 6

12 # 2

14 # 3

Step 4: add up your positive ranks (not the scores, but the rank #s). R=Rank

∑ R₊ = 4 + 5 + 6 + 2 + 3 = 20

Step 5: add up your negative ranks (not the scores, but the rank #s). R=Rank

∑ R_- = 1

Step 6: T= the smallest of these numbers

T= 1

Step 7: Solve the following equation to find the z score.

T = 1 N (n) = 6 (we had 6 participants in this data set)

If Z is less than -1.96, or greater than 1.96, you have to reject the Null Hypothesis.

Z = -1.99 (Reject the Null). The IV did make a difference.

T-Test

Not a recommended test to use because our sample size is so small (and it’s considered a parametric test). If you decide to use the t-test be sure to justify its use (for example: saying that although the criteria for parametric tests are not met these tests are very robust). Use this test only if no other statistical test will work.

• Example: Use when you have 1 variable in 2 situations. You want to find the mean (average) between 2 conditions.

STEP 1: Find the mean from your data set.

• Data: 4, 2, 3, 4, 1, 3, 4 = the sum of these is 21 there are 7 scores (N=7)

• Mean = total (21) divided by 7 = 3

• Mean = 3

STEP 2: Subtract the mean from each score.

• 4-3= 1 2-3=-1 3-3=0 4-3=1 1-3=-2

• 3-3=0 4-3=1

STEP 3: Rank the items in order (largest to smallest).

• 1, 1, 1, 1, 0, 0, -1, -2

STEP 4: Add the sum of the Positive Ranks.

• 1 + 1 + 1 + 1 = 4

STEP 5: Add the sum of your Negative Ranks.

• -1 + -2 = -3

STEP 6: Take the smallest score (t) from Steps 4 & 5.

• t=-3

STEP 7: If t>1.96 or <-1.96 then p<.05 (significant or valid). This means that the independent variable (IV) did cause the change in the Dependent Variable (DV).

• -3 is less than <1.96 so the test is NOT valid. You must accept the NULL.

• The results are caused by chance, not the IV.

Section G. Discussion Approx. 600 words

The discussion is the final part of the paper. This is an important part of your report, so make sure that you do everything that is set out in the assessment criteria. In this section you:

• Interpret your own results in the light of previous research.

• You must relate your findings to each of the theories or studies referred to in the introduction, and say how your results differed and where they were in line with the study replicated. No new studies or citation should be introduced.

• Then you need to say why you think you achieved the results you did.

• You should analyze and evaluate your methodology. You are not expected to conduct a ‘perfect’ experiment.

• Be sure to discuss the limitations that may have affected the outcome of the experiment.

Scoring G. Discussion: (8 marks)

• Analysis of results is well-developed and complete.

• Descriptive and inferential statistics are discussed.

• The findings of the student’s experimental study are discussed with reference to relevant background theories or studies.

• Limitations of the design and procedure are highly relevant and have been rigorously analyzed.

• Modifications are suggested and ideas for further research may be mentioned.

• The conclusion is appropriate.

WORD COUNT ENDS HERE

• The word limit is 1500-2000 words.

• The word count does not include supplementary information such as the abstract, title page, references, and appendices

H. Citation of Sources

This section is often called a Bibliography or References Section. For your IA, you must use APA style (from the American Psychological Association). I have included many sources to help guide you through this process.

• Primary source documents from respected academic journals are best (use WFU library)

• Secondary sources may also be used but are not as strong academically.

• You should use more books than Internet sources.

• Many journal articles are stored on-line (or in on-line databases). If these sources were originally published, list the source by its original published format and date. You can put the URL where the article is now archived (stored).

• Make sure you list all sources in alphabetical order.

• Every source used in your Introduction and Discussion must be listed here.

Your APA format will use the following information:

• Book: author(s), book title, publisher, date of publication, and page number(s) if appropriate.

• Journal: author(s), article title, journal title, date of publication, and page number(s).

• Newspaper: author(s), article title, name of newspaper, section title and page number(s) if desired, date of publication.

• Web site: author(s), article and publication title where appropriate, as well as a URL, and a date when the site was accessed.

Your study must contain proper APA citation format. Every time you refer to your research there should be a citation (reference). The full reference goes here in this section.

Here are several examples of how to cite your source within the text:

• Smith (1998) found that psychology students were motivated by grades.

• In 1998, Smith found that psychology students were motivated by grades.

• Research shows that psychology students are highly motivated by grades (Smith, 1998).

• For in-text citations without page numbers (such as websites), help the reader pinpoint the quote by listing the section name or paragraph number instead.

Plagiarism (copying someone’s words or work without giving credit) is an automatic zero (F) in IB and can cost you your certificate or diploma.

Here are some tutorials to help you learn the APA style:

How to Write an APA Style Paper | eHow.com, http://www.ehow.com/how_2002020_apa_style_paper.html#ixzz1untWl5FB

http://www.apastyle.org/learn/index.aspx

http://owl.english.purdue.edu/owl/resource/560/01/

http://psychology.about.com/od/apastyle/a/apageneral.htm

http://www.sciencebuddies.org/science-fair-projects/project_apa_format_examples.shtml

http://psychology.vanguard.edu/wp-content/uploads/2011/07/apastyleessentials.pdf

Appendix (Appendices, plural)

In this section include blank copies of any supplementary information (anything too bulky for the main text). Number in order with lower case Roman numerals (i, ii, iii, iv, v, vi, etc).

• One copy of standardized instructions

• One copy of any briefing/debriefing notes

• One copy of any materials used (like a word list)

• One copy of informed consent letters

• Any calculations. Tables of raw data must be included here, but it is not necessary to include all participant responses.

• Make sure that each appendix is numbered and has an appropriate title (example: Appendix for Calculation of the Mann-Whitney U).

Special Thanks:

Dr. David W. Martin, NC State, Department of Psychology for his book: Doing Psychology Experiments, 7^th Edition, Thomson Publishing, 2008.

John Crane, IB Psychology.

Numerous other IB Psychology teachers who have shared their experience in the IA process.