Frequency Distribution Width & 'Rounding Up'

[dotsero]

Homework Statement

I'm having trouble understanding setting up a frequency distribution. I am confident I am doing it right, but the book I'm using differs when calculating width.

The problem gives a bunch of numbers representing the number of counties, divisions, or parishes for each of the 50 states. It then asks to setup a grouped frequency distribution with 6 classes, a histogram, a frequency polygon, and an ogive with the data. The data is as follows:

67 27 15 75 58 64 8 67 159 5
102 44 92 99 105 120 64 16 23 14
83 87 82 114 56 93 16 10 21 33
62 100 53 88 77 36 67 5 46 66
95 254 29 14 95 39 55 72 23 3

Homework Equations

Width = ceil(Range / number of Classes)
**width can also be rounded up if it's value is a whole number after calculating it. Or one could increase the number of classes by one. This is an important point for me because my book failed to mention it and I was setting it up wrong as a result. Even after correcting though, I wasn't always getting an identical setup.

Range = MaxVal - MinVal

The Attempt at a Solution

MaxVal = 254
MinVal = 3
Range = 254 - 3 = 251
Width = ceil(251 / 6) ≈ ceil(41.8) = 42
Starting point = MinVal = 3

Class Limits Boundaries Tally Freq
3 - 44 2.5-44.5 didn't even finish these since CL & boundaries
45 - 86 44.5-86.5 were different from book
87 - 128 86.5-128.5
129 - 170 128.5-170.5
171 - 212 170.5-212.5
213 - 254 212.5-254.5

Here's my problem. The book has class limits:

3-45
46-88
89-131
132-174
175-217
218-260

In other words, they've increased the width by rounding up from its original approximate value (41.8) like I did, but then they rounded that up too! From 41.8 → 42 → 43. I was under the impression that you only round up if the width is an approximate value (it has decimal points). The exception, from what I gather, is that one only rounds up from one whole value to the next, is when the width is a whole number after calculating its value. So if I'd calculated W = 252 / 6 = 42, if 252 were the MaxVal. Then, and only then, would I choose to either increase the number of classes from 6 to 7, or increase the width from 42 to 43. But the author of this book seems to want it both ways: round up the value after calculating it and getting an approximate number, then round up again to the next whole number. This is my current plight. Any help is much appreciated.

 

[RUber]

I can't see any good reason for the book to want it that way.
Your low end of the lowest class is your min value...so the high end of your highest class should be about your max value...not max + 6.
If you were to use a width of 43, I think you would be better off starting at 0 and going to 257, min - 3 to max + 3.

Of course, I am not reading your text, and the author may give good justification for the method...but I would go with your method for now until some rationale becomes evident to the contrary.

 

[dotsero]

I can't see any good reason for the book to want it that way.
Your low end of the lowest class is your min value...so the high end of your highest class should be about your max value...not max + 6.
If you were to use a width of 43, I think you would be better off starting at 0 and going to 257, min - 3 to max + 3.

Of course, I am not reading your text, and the author may give good justification for the method...but I would go with your method for now until some rationale becomes evident to the contrary.

Thanks for your answer. It definitely cleared things up. My remaining question is, did I do it correctly? Setting up a grouped frequency distribution that is. I realize one could use different classes and widths and so on, for the same data set, so they need not always look identical. But is my way of going about it correct?

 

[RUber]

There is no one way to get the job done. The important thing is that you use equally-sized intervals, which you have done.

 

[dotsero]

There is no one way to get the job done. The important thing is that you use equally-sized intervals, which you have done.

Alright. Thank you so much for taking time out to answer my question. I really appreciate it.