Group Paper
The Introduction Section ( at least 1 page )
- Must contain an introduction on reflexes and reaction and how they are important for human survival. You may want to explain the differences.
- In another paragraph explain what the purpose of your specific experiment is and eventually formulate your hypothesis.
- Remember that you should never include your data at this point, because a hypothesis is an educated guess of the outcome, provided you have some previous experiences in this field.
- For this exercise, the best hypothesis is the null hypothesis. In other words, you do NOT expect to see any differences among the different groups. A null hypothesis is always the easiest starting point and makes it easier to discuss the actual data with your proposed hypothesis.
The Materials and Methods Section ( Not more than 1 page)
You can already start with this as well and include the way the experimental data was collected.
For example, you need to write about the set up of the experiment, how many groups, how many participants per group, what the participants were asked to do , what instructions were given, and what data were collected.
Your writing style should be in the passive voice.
Example :Instead of writing “We told the participants that…..”
You write “The participants were told to….”
Instead of writing “ We divided the participants in four groups…”
Your write “ The participants were divided into four groups ….”
To finish up, you should mention that data were analyzed by means of a statistical ANOVA test.
The Results and Discussion Section( at least 2 pages, including the tables).
The following should appear in your results and discussion.
1) A table with your groups, the 10 average data points per group and the average of each group at the bottom of each column. The table should have a legend under it, as shown in this table below. To be discussed in a later handout but start by putting the average reaction time of each individual into an excel spread sheet. (example below – need 10 data per column)
Exp Group 1 / Exp Group 2 / Exp. Group 3 / Exp. Group 40.222 / 0.285 / 0.356 / 0.333
0.212 / 0.275 / 0.311 / 0.354
0.234 / 0.255 / 0.341 / 0.378
Average 1 / Average 2 / Average 3 / Average 4
Table 1 :This table represent ……
2) There should also be a second table that has all the statistical elements shown in the ANOVA discussion below. The # represents your calculated data
Source / SS / Df / MS / FAmong / ## / # / ### / #
Within / ## / # / ###
Total / ## / ##
Table 2 : ANOVA test results of the experiment.
3) You should discuss your data. I like to see a discussion that logically explains your data. It is OK if they are not in agreement with your hypothesis, but you have to provide possible explanations as to what may have influenced the data, etc...
Addendum: You do not need to provide references, but I like to see all your collected data attached to your paper. I also like to see your calculations that results in the data going into table 2.( see below : every member of your team can do the calculation for each included experimental group…. See below)
Anova step by step
Introduction
The one-way analysis of variance (ANOVA) is used to determine whether there are any significant differences between the means of three or more independent (unrelated) groups.
The one-way ANOVA compares the means between the groups you are interested in and determines whether any of those means are significantly different from each other. Specifically, it tests the null hypothesis:
where µ = group mean and k = number of groups. In other words, it tests the null hypothesis that there are no differences between the means of each group. If, however, the one-way ANOVA returns a significant result, we accept the alternative hypothesis (HA), which is that there are at least 2 groups where the means are significantly different from each other.
It is important to realize that the one-way ANOVA is a statistical that that can only indicate if there is a difference between at least two groups; it cannot tell you which specific groups were significantly different from each other. To determine which specific groups differed from each other, one will need to use a post hoc test.
Example
The best way to see how we use this method is by showing a simple problem example.
A curious researcher predicts that students will learn most effectively with a constant background sound, as opposed to an unpredictable sound or no sound at all. She randomly divides twenty-four students into three groups of eight. ( Remember : your experiment has 4 groups of ten … or 3 groups of ten). All students study a passage of text for 30 minutes. Those in group 1 study with background sound at a constant volume in the background. Those in group 2 study with noise that changes volume periodically. Those in group 3 study with no sound at all. After studying, all students take a 10 point multiple choice test over the material. The scores of the participants were as follow:
Sound type / Constant / Random / None7 / 5 / 2
4 / 5 / 4
6 / 3 / 7
8 / 4 / 1
6 / 4 / 2
6 / 7 / 1
2 / 2 / 5
9 / 2 / 5
Question 1 : how many groups do we have ? k=3
Question 2 : how many data are there per groupn= 8
Question 2 : how many total data are there ?N = k x n = 8 x 3 = 24
( So, if your team has 4 groups, k= 4, n = 10 and N= 40)
( If your team has 3 groups, k = 3 , n = 10 and N = 30)
First steps of ANOVA execution. Recreate the table and add a column next to each column and 3 or 4 rows below( can easily be done in EXCEL).
Sound type / Constant / A / Random / B / None / C7 / 72 = 49 / 5 / 52 = 25 / 2 / 4
4 / 42 = 16 / 5 / 52 = 25 / 4 / 16
6 / 36 / 3 / 9 / 7 / 49
8 / 64 / 4 / 16 / 1 / 1
6 / 36 / 4 / 16 / 2 / 4
6 / 36 / 7 / 49 / 1 / 1
2 / 4 / 2 / 4 / 5 / 25
9 / 81 / 2 / 4 / 5 / 25
Step 2 / 48 / 322 / 32 / 148 / 27 / 125
Step 3 / 482=2304 / 32^2=1024 / 272=729
Step 4 / 48/8=6 / 32/8=4 / 27/8=3.38
Step 1 : for each row, column, enter square the data and enter in the next column ( see columns A, B and C)
Step 2 : Create the sum from each column
Step 3 : Square only the sums of the original data
Step 4 : Calculate the average of each original column
Now that we have the basic information, we calculate what is called the sum of squares for the total group. W will do this in parts
Step 5 : (Sum of Column A) + (Sum of Column B) + (Sum of Column C)
Step 6 : (Sum of Group 1) + (Sum of Group 2) + (Sum of Group 3)
Step 7 : [(result from step 6)2]/N
Step 8 : (Result from step 5) – step 7
Lets do this
Step 5 = 322 + 148 + 125 = 595
Step 6 = 48 + 32 + 27 = 107
Step 7 = (1072)/24 = 477.04
Step 8 = 595 – 477.04 = 117.96
Thus, SStotal = 117.96
Next we calculate the sym of square among groups
Step 9 : [(sum of squares of Group1)/n ] + [(sum of squares of Group2)/n ]
+ [(sum of squares of Group3)/n ]
Step 10 = Result from step 9 – result from step 7
Calculating :
Step 9 = ( 2304/8 + 1024/8 + 729/8) = 507.13
Step 10 = 507.13 – 477.04 = 30.08
Hence, SSamong = 30.08
Finally, SS within = SStotal - SSamong = 117.96 – 30.08 = 87.88
Before proceeding with the final calculations we need some extra info, provided by our experiment.
The degrees of freedom (df) are approximately equal to the number of items that went into computing the corresponding SS minus 1. So if n things went in, then df = n - 1.
Because SSamongis really based on the deviations of each treatment mean from the grand mean, the number of items in this SS is the number of treatment means = k (the number of groups). So the dfamong= k – 1 (for this exercise 3-1 = 2)( for your experiment, if you had 4 groups, dfamong= k – 1 = 4-1=3 )
Because SStotal was computed using the entire set of N scores, the associated degrees of freedom for SStotal are dftotal = N - 1 (for this exercise N-1 = 24-1 =23)( for your experiment, if you had 4 groups, dftotal = N - 1 = 40-1 = 39 ;if you had 3 groups, dftotal = N - 1 = 30-1 = 29)
To get the degrees of freedom for SSwithin we first computed the SS for each level and then added them up. This is the same for dfwithin in a sense. For each level we have "n - 1" degrees of freedom. Then we sum those n - 1 degrees of freedom across the levels: (n - 1) + (n - 1) + (n - 1) + ... If you simplify this, you get for dfwithin= N - k or , for this exercise, 24-3 = 21 )( ( for your experiment, if you had 4 groups,dfwithin= N - k = 40-4 = 36 ; if you had 3 groups, dfwithin= N - k = 30-3 = 27)
Note 1: You should Always check to see if: dftotal = dfbetween + dfwithin
If this check does not come out right - you have made a mistake in your calculations.
A variance is SS/df. In Anova the variances we compute are called Mean Squares, symbolized "MS" (Because they are essentially mean squared deviations)
So, MSamong = SSamong /dfamong and MSwithin =SSwithin / dfwithin
Note 2: In general we do not compute MStotal. Also, it is NOT TRUE that MStotalmust be =MSamong + MSwithin.
Finally, because the F test is the variance among divided by the variance within, we get our F-ratio:
F = MSamong / MSwithin
If we put this in a table we get
Source / SS / Df / MS / FAmong / 30.08 / 2 / 30.08/2=15.04 / =15.04/4.18 = 3.59
Within / 87.88 / 21 / 87.88/21=4.18
Total / 117.96 / 23
The F value becomes important now. We have to use The table below to explain what it means. We first need to look for the critical F value relating to this experiment. We have an experiment with (2,21) degrees of freedom. ( In your experiment, it will be either an experiment with (2, 27) or (3, 36) degrees of freedom). If we look at the table, the critical F value for (2,21) is 3.467 !If the Calculated F value falls below this number, the differences observed between the means have a probability (of occurring by chance) of greater than 5 % . With most biological data, this is considered too much and hence, these differences are not considered to be significant at a P>0.05 level. However, if the calculated F value is greater than 3.467, the chance (probability=P) that the observed differences would occur would be less than 5 % ( <0.05) and these differences are then thus considered to be significant at a P<0.05 level.
With respect to this experiment, the calculated F = 3.59 > 3.467. Thus, the researcher can conclude that her hypothesis may be supported. The means are as she predicted, in that the constant music group has the highest score. However, the significant F only indicates that at least two means are significantly different from one another, but she can't know which specific mean pairs significantly differ until she conducts a post-hoc analysis.
We will not do a post hoc analysis, but you should be able to provide statistical evidence if your data have a P>0.05 ( meaning that the means are not significantly different from each other, provides evidence in favor of a null hypothesis) or that they are significantly different with a P <0.05. ( evidence in favor of rejecting the null hypothesis).
Upper critical values of the F distribution
for ν1 numerator degrees of freedom and ν2 denominator degrees of freedom
5% significance level
F.05(ν1,ν2)
\ ν1 1 2 3 4 5 6 7 8 9 10
ν2
1 161.448 199.500 215.707 224.583 230.162 233.986 236.768 238.882 240.543 241.882
2 18.513 19.000 19.164 19.247 19.296 19.330 19.353 19.371 19.385 19.396
3 10.128 9.552 9.277 9.117 9.013 8.941 8.887 8.845 8.812 8.786
4 7.709 6.944 6.591 6.388 6.256 6.163 6.094 6.041 5.999 5.964
5 6.608 5.786 5.409 5.192 5.050 4.950 4.876 4.818 4.772 4.735
6 5.987 5.143 4.757 4.534 4.387 4.284 4.207 4.147 4.099 4.060
7 5.591 4.737 4.347 4.120 3.972 3.866 3.787 3.726 3.677 3.637
8 5.318 4.459 4.066 3.838 3.687 3.581 3.500 3.438 3.388 3.347
9 5.117 4.256 3.863 3.633 3.482 3.374 3.293 3.230 3.179 3.137
10 4.965 4.103 3.708 3.478 3.326 3.217 3.135 3.072 3.020 2.978
11 4.844 3.982 3.587 3.357 3.204 3.095 3.012 2.948 2.896 2.854
12 4.747 3.885 3.490 3.259 3.106 2.996 2.913 2.849 2.796 2.753
13 4.667 3.806 3.411 3.179 3.025 2.915 2.832 2.767 2.714 2.671
14 4.600 3.739 3.344 3.112 2.958 2.848 2.764 2.699 2.646 2.602
15 4.543 3.682 3.287 3.056 2.901 2.790 2.707 2.641 2.588 2.544
16 4.494 3.634 3.239 3.007 2.852 2.741 2.657 2.591 2.538 2.494
17 4.451 3.592 3.197 2.965 2.810 2.699 2.614 2.548 2.494 2.450
18 4.414 3.555 3.160 2.928 2.773 2.661 2.577 2.510 2.456 2.412
19 4.381 3.522 3.127 2.895 2.740 2.628 2.544 2.477 2.423 2.378
20 4.351 3.493 3.098 2.866 2.711 2.599 2.514 2.447 2.393 2.348
21 4.325 3.467 3.072 2.840 2.685 2.573 2.488 2.420 2.366 2.321
22 4.301 3.443 3.049 2.817 2.661 2.549 2.464 2.397 2.342 2.297
23 4.279 3.422 3.028 2.796 2.640 2.528 2.442 2.375 2.320 2.275
24 4.260 3.403 3.009 2.776 2.621 2.508 2.423 2.355 2.300 2.255
25 4.242 3.385 2.991 2.759 2.603 2.490 2.405 2.337 2.282 2.236
26 4.225 3.369 2.975 2.743 2.587 2.474 2.388 2.321 2.265 2.220
27 4.210 3.354 2.960 2.728 2.572 2.459 2.373 2.305 2.250 2.204
28 4.196 3.340 2.947 2.714 2.558 2.445 2.359 2.291 2.236 2.190
29 4.183 3.328 2.934 2.701 2.545 2.432 2.346 2.278 2.223 2.177
30 4.171 3.316 2.922 2.690 2.534 2.421 2.334 2.266 2.211 2.165
31 4.160 3.305 2.911 2.679 2.523 2.409 2.323 2.255 2.199 2.153
32 4.149 3.295 2.901 2.668 2.512 2.399 2.313 2.244 2.189 2.142
33 4.139 3.285 2.892 2.659 2.503 2.389 2.303 2.235 2.179 2.133
34 4.130 3.276 2.883 2.650 2.494 2.380 2.294 2.225 2.170 2.123
35 4.121 3.267 2.874 2.641 2.485 2.372 2.285 2.217 2.161 2.114
36 4.113 3.259 2.866 2.634 2.477 2.364 2.277 2.209 2.153 2.106
37 4.105 3.252 2.859 2.626 2.470 2.356 2.270 2.201 2.145 2.098
38 4.098 3.245 2.852 2.619 2.463 2.349 2.262 2.194 2.138 2.091
39 4.091 3.238 2.845 2.612 2.456 2.342 2.255 2.187 2.131 2.084
40 4.085 3.232 2.839 2.606 2.449 2.336 2.249 2.180 2.124 2.077