Is the Dart Throwing Data a Gaussian Distribution?

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SUMMARY

The discussion centers on analyzing dart throw data to determine if it follows a Gaussian distribution. The data, ranging from -20 to 20, was graphed, and initial observations suggested a fit to the Gaussian model. However, issues arose during the Chi-squared test in Excel, where the CHITEST function returned 0 and the CHIINV function produced unrecognized values. Participants noted that improper binning and adherence to Chi-squared test assumptions could lead to misleading results.

PREREQUISITES
  • Understanding of Gaussian distribution principles
  • Familiarity with Excel functions, specifically CHITEST and CHIINV
  • Knowledge of statistical binning techniques
  • Basic grasp of Chi-squared test assumptions and limitations
NEXT STEPS
  • Review the principles of Gaussian distribution and its applications in data analysis
  • Learn how to properly use Excel's CHITEST and CHIINV functions for statistical testing
  • Study effective binning strategies for Chi-squared tests to avoid misinterpretation
  • Explore common pitfalls in Chi-squared test applications and how to avoid them
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Statisticians, data analysts, students in statistics courses, and anyone involved in data analysis using Excel who seeks to understand Gaussian distributions and Chi-squared testing.

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Homework Statement



We threw a bunch of darts at a dart board to find the statistics of the positions. They ranged from -20 to 20 and the data is right here. ( each column corresponds to the dart number, it is just a syntax thing) The frequencies are below.

Dart 1 Dart 2 Dart 3 Dart 4 Dart 5
1 -7 -4 4 4 5
2 -19 -6 -5 -2 0
3 -10 -6 -5 6 7
4 -13 -8 -3 8 11
5 -8 -4 -2 0 7
6 -7 -1 0 1 2
7 -13 -11 -2 -2 0
8 -8 -4 -1 8 9
9 -13 -9 -6 -3 5
10 -6 7 12 12 14
11 9 -8 -2 -2 0
12 -12 -4 -4 3 10
13 -8 -7 -4 -3 4
14 -7 -4 2 4 4
15 -5 -3 0 6 11
16 -16 -5 -3 3 8
17 -11 -3 3 1 20
18 -6 -3 3 5 7
19 -4 -2 3 5 12
20 -4 0 3 4 6
21 -11 -9 -4 -5 3
22 0 1 2 7 11
23 -8 -7 -4 -4 0
24 -2 0 6 11 13
25 -10 -8 0 0 4
26 -2 0 3 5 9
27 -5 -1 -1 0 7
28 -13 -6 -5 5 12
29 -4 -8 -2 0 3
30 -15 -5 0 1 4
31 -10 -2 -2 0 4
32 -4 1 5 7 8
33 -10 1 2 6 -1.5
34 -6 -2.5 -2 -7 6
35 -9 -3 -2 5.5 6
36 -15 -7 1 0 3
37 -6 -1 -1 3 2
38 2 1 1 -7 -8
39 -4 -3 1 4 9
40 -3 -4 -2 1 -1
41 1 -2 -2 3 4
42 -4 -3 -1 3 3
43 -7 -4 -1 3 7
44 0 -1 -1 1 2
45 0 -7 -1 -3 7
46 -7 1 -1 -1 0
47 -6 2 -3 -2 6
48 -11 -8 -1 -3 2
49 -6 -3 -2 2 5
50 -7 -4 -3 1 1
51 -2 -2 4 4 11
52 0 -1 -4 -6 7
53 -3 -3 -1 1 2
54 -1 -1 -2 2 6
55 1 -3 4 4 2
56 -8 -8 -9 8 -3
57 2 6 -7 -3 -1
58 -5 -1 -1 1 2
59 3 1 2 -1 -3
60 2 0 -3 -4 -5



-20 0
-19 1
-18 0
-17 0
-16 1
-15 2
-14 0
-13 4
-12 1
-11 4
-10 4
-9 4
-8 12
-7 13
-6 11
-5 10
-4 21
-3 23
-2 23
-1 24
0 23
1 20
2 17
3 16
4 15
5 8
6 11
7 10
8 5
9 4
10 1
11 5
12 4
13 1
14 1
15 0
16 0
17 0
18 0
19 0
20 1

I graphed it, and I did a gaussian distribution and all the data fits fine and dandy. But when I perform the Chi squared test in excel, what ends up happening is that I will get CHITEST = 0
and then when I do CHIINV I get that the number isn't recognized by excel. Or when I mess around with bins I end up getting a number that's roughly around 300...


I honestly have no idea why I am not getting that this data is a gaussian distribution.





The Attempt at a Solution

 
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I'm not intending to address your question, but putting code tags around your data makes it much more readable.

Dgray101 said:

Homework Statement



We threw a bunch of darts at a dart board to find the statistics of the positions. They ranged from -20 to 20 and the data is right here. ( each column corresponds to the dart number, it is just a syntax thing) The frequencies are below.
Code: 
	Dart 1	Dart 2	Dart 3	Dart 4	Dart 5
1	-7	-4	4	4	5
2	-19	-6	-5	-2	0
3	-10	-6	-5	6	7
4	-13	-8	-3	8	11
5	-8	-4	-2	0	7
6	-7	-1	0	1	2
7	-13	-11	-2	-2	0
8	-8	-4	-1	8	9
9	-13	-9	-6	-3	5
10	-6	7	12	12	14
11	9	-8	-2	-2	0
12	-12	-4	-4	3	10
13	-8	-7	-4	-3	4
14	-7	-4	2	4	4
15	-5	-3	0	6	11
16	-16	-5	-3	3	8
17	-11	-3	3	1	20
18	-6	-3	3	5	7
19	-4	-2	3	5	12
20	-4	0	3	4	6
21	-11	-9	-4	-5	3
22	0	1	2	7	11
23	-8	-7	-4	-4	0
24	-2	0	6	11	13
25	-10	-8	0	0	4
26	-2	0	3	5	9
27	-5	-1	-1	0	7
28	-13	-6	-5	5	12
29	-4	-8	-2	0	3
30	-15	-5	0	1	4
31	-10	-2	-2	0	4
32	-4	1	5	7	8
33	-10	1	2	6	-1.5
34	-6	-2.5	-2	-7	6
35	-9	-3	-2	5.5	6
36	-15	-7	1	0	3
37	-6	-1	-1	3	2
38	2	1	1	-7	-8
39	-4	-3	1	4	9
40	-3	-4	-2	1	-1
41	1	-2	-2	3	4
42	-4	-3	-1	3	3
43	-7	-4	-1	3	7
44	0	-1	-1	1	2
45	0	-7	-1	-3	7
46	-7	1	-1	-1	0
47	-6	2	-3	-2	6
48	-11	-8	-1	-3	2
49	-6	-3	-2	2	5
50	-7	-4	-3	1	1
51	-2	-2	4	4	11
52	0	-1	-4	-6	7
53	-3	-3	-1	1	2
54	-1	-1	-2	2	6
55	1	-3	4	4	2
56	-8	-8	-9	8	-3
57	2	6	-7	-3	-1
58	-5	-1	-1	1	2
59	3	1	2	-1	-3
60	2	0	-3	-4	-5

Code: 
-20	0
-19	1
-18	0
-17	0
-16	1
-15	2
-14	0
-13	4
-12	1
-11	4
-10	4
-9	4
-8	12
-7	13
-6	11
-5	10
-4	21
-3	23
-2	23
-1	24
0	23
1	20
2	17
3	16
4	15
5	8
6	11
7	10
8	5
9	4
10	1
11	5
12	4
13	1
14	1
15	0
16	0
17	0
18	0
19	0
20	1

I graphed it, and I did a gaussian distribution and all the data fits fine and dandy. But when I perform the Chi squared test in excel, what ends up happening is that I will get CHITEST = 0
and then when I do CHIINV I get that the number isn't recognized by excel. Or when I mess around with bins I end up getting a number that's roughly around 300...


I honestly have no idea why I am not getting that this data is a gaussian distribution.





The Attempt at a Solution

 
Dgray101 said:

Homework Statement



We threw a bunch of darts at a dart board to find the statistics of the positions. They ranged from -20 to 20 and the data is right here. ( each column corresponds to the dart number, it is just a syntax thing) The frequencies are below.

Dart 1 Dart 2 Dart 3 Dart 4 Dart 5
1 -7 -4 4 4 5
2 -19 -6 -5 -2 0
3 -10 -6 -5 6 7
4 -13 -8 -3 8 11
5 -8 -4 -2 0 7
6 -7 -1 0 1 2
7 -13 -11 -2 -2 0
8 -8 -4 -1 8 9
9 -13 -9 -6 -3 5
10 -6 7 12 12 14
11 9 -8 -2 -2 0
12 -12 -4 -4 3 10
13 -8 -7 -4 -3 4
14 -7 -4 2 4 4
15 -5 -3 0 6 11
16 -16 -5 -3 3 8
17 -11 -3 3 1 20
18 -6 -3 3 5 7
19 -4 -2 3 5 12
20 -4 0 3 4 6
21 -11 -9 -4 -5 3
22 0 1 2 7 11
23 -8 -7 -4 -4 0
24 -2 0 6 11 13
25 -10 -8 0 0 4
26 -2 0 3 5 9
27 -5 -1 -1 0 7
28 -13 -6 -5 5 12
29 -4 -8 -2 0 3
30 -15 -5 0 1 4
31 -10 -2 -2 0 4
32 -4 1 5 7 8
33 -10 1 2 6 -1.5
34 -6 -2.5 -2 -7 6
35 -9 -3 -2 5.5 6
36 -15 -7 1 0 3
37 -6 -1 -1 3 2
38 2 1 1 -7 -8
39 -4 -3 1 4 9
40 -3 -4 -2 1 -1
41 1 -2 -2 3 4
42 -4 -3 -1 3 3
43 -7 -4 -1 3 7
44 0 -1 -1 1 2
45 0 -7 -1 -3 7
46 -7 1 -1 -1 0
47 -6 2 -3 -2 6
48 -11 -8 -1 -3 2
49 -6 -3 -2 2 5
50 -7 -4 -3 1 1
51 -2 -2 4 4 11
52 0 -1 -4 -6 7
53 -3 -3 -1 1 2
54 -1 -1 -2 2 6
55 1 -3 4 4 2
56 -8 -8 -9 8 -3
57 2 6 -7 -3 -1
58 -5 -1 -1 1 2
59 3 1 2 -1 -3
60 2 0 -3 -4 -5



-20 0
-19 1
-18 0
-17 0
-16 1
-15 2
-14 0
-13 4
-12 1
-11 4
-10 4
-9 4
-8 12
-7 13
-6 11
-5 10
-4 21
-3 23
-2 23
-1 24
0 23
1 20
2 17
3 16
4 15
5 8
6 11
7 10
8 5
9 4
10 1
11 5
12 4
13 1
14 1
15 0
16 0
17 0
18 0
19 0
20 1

I graphed it, and I did a gaussian distribution and all the data fits fine and dandy. But when I perform the Chi squared test in excel, what ends up happening is that I will get CHITEST = 0
and then when I do CHIINV I get that the number isn't recognized by excel. Or when I mess around with bins I end up getting a number that's roughly around 300...


I honestly have no idea why I am not getting that this data is a gaussian distribution.





The Attempt at a Solution


You need to include more details. How did you obtain the Chi-squared values; that is, what are the individual "bins" that you used in submitting the data to the Chi-squared package?

I looked at your data in maple, and found that in some ways of presenting the data the normality hypothesis was rejected (at some reasonable level of significance), while in other ways of presenting the data the normality hypothesis was accepted at the same level of significance. The point is that it is easy to "misuse" the Chi-squared test if you don't really know what you are doing. There are certain rule of thumb restrictions that typically accompany descriptions of chi-squared usage, and we have no way of knowing whether or not you paid attention to these restrictions. If you violate the restrictions the chi-squared test simply does not apply; that is, the use of a Chi-squared table is inaccurate and ill-advised.
 

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