Measures of central tendency-Histogram

[Zelda22]

The mode it is not concerned with which bar of the histogram is the tallest. The bars of the histogram mask all information about whether or not there are any repeated individual values, so I think I could not estimate those values. Is there any way I could estimate the mode? Thank you.

 

[Dr.Peterson]

The mode it is not concerned with which bar of the histogram is the tallest. The bars of the histogram mask all information about whether or not there are any repeated individual values, so I think I could not estimate those values. Is there any way I could estimate the mode? Thank you.

That's an interesting question, because some students are actually taught a formula for the "mode" of such a grouped distribution. What that formula actually gives is a reasonable guess, assuming the underlying distribution is smooth. So it is definitely possible to make an estimate; but there is no guarantee that that estimated interval will be correct.

So the answer depends on just what they mean by "rough range".

 

[Zelda22]

This is very confusing for me.

The possible data points for each bar 5
5-9 (3 data points) Possible repeated values 3,2
10-14 (4 data points) Possible repeated values 4,3,2
15-19 (5 data points) Possible repeated values 5,4,3,2
20-24 ( 2 data points) Possible repeated values 2
25-29 ( 1 data point) Possible repeated values 0


I found this way to find the mode graphically, but I don't think I could get the mode like this.

-Identify the highest rectangle.
-Join the top corners with the immediately next corners of the adjacent rectangles. The two lines must be cutting each other.
-Let the point where the joining lines cut each other be 'A'. Draw a perpendicular line from point A onto the x-axis. The point 'P' where the perpendicular will meet the x-axis will give the mode.

 

[BigBeachBanana]

This is very confusing for me.

The possible data points for each bar 4
5-9 (3 data points) Possible repeated values 3,2
15-19 (5 data points) Possible repeated values 4,3,2
20-24 ( 2 data points) Possible repeated values 2
25-29 ( 1 data point) Possible repeated values 0


I found this way to find the mode graphically, but I don't think I could get the mode like this.

-Identify the highest rectangle.
-Join the top corners with the immediately next corners of the adjacent rectangles. The two lines must be cutting each other.
-Let the point where the joining lines cut each other be 'A'. Draw a perpendicular line from point A onto the x-axis. The point 'P' where the perpendicular will meet the x-axis will give the mode.

Here's my take on this question. Let's assume there's only one mode. Which range would the mode belong to?
Now, let's assume all the modes in the said range are repeated. Which range would the mode belong to?

 

[Zelda22]

Possible data point for each bar 5
5-9 (3 data points) Possible repeated values (3+0), (2+1)
10-14 (4 data points) Possible repeated values (4+0), (3+1),(2+2)
15-19 (5 data points) Possible repeated values (5+0), (4+1),(3+2), (2+1+1+1), (2+2+1)
20-24 ( 2 data points) Possible repeated values (2+0)
25-29 ( 1 data point) Possible repeated values 0

5-9, 10-14, 15-19, and 20-24 could all have values repeated twice, and (10-14) and (15-19) could have two values repeated twice.
I don't have an option from 5 to 24. Plus could be any other value. So I don't think is possible here.

 

[Dr.Peterson]

I found this way to find the mode graphically, but I don't think I could get the mode like this.

-Identify the highest rectangle.
-Join the top corners with the immediately next corners of the adjacent rectangles. The two lines must be cutting each other.
-Let the point where the joining lines cut each other be 'A'. Draw a perpendicular line from point A onto the x-axis. The point 'P' where the perpendicular will meet the x-axis will give the mode.

Yes, that's the basis of the formula I was referring to which can be found here:

​

This would give [math]15+\frac{5-4}{(5-4)+(5-2)}5=15+\frac{5}{4}=16.25[/math]
This answer is a reasonable guess, but can't be the mode of the actual underlying data, since it isn't a whole number.

It is most likely that the actual mode is in or near the modal class; but it's conceivable that the actual data are

7,8,9, 11,12,13,14, 15,16,17,18,19, 23,23, 26​

in which case the mode is 23, though that is in the class with second-lowest frequency!

So I'll say again, we can estimate that the mode is likely to be in the modal class, but have no certainty except that it can't be in the class with frequency 1. And this reinforces the fact that the mode of a small dataset is not very significant.

 

[Zelda22]

Yes, that's the basis of the formula I was referring to which can be found here:

View attachment 31980​

This would give [math]15+\frac{5-4}{(5-4)+(5-2)}5=15+\frac{5}{4}=16.25[/math]
This answer is a reasonable guess, but can't be the mode of the actual underlying data, since it isn't a whole number.

It is most likely that the actual mode is in or near the modal class; but it's conceivable that the actual data are

7,8,9, 11,12,13,14, 15,16,17,18,19, 23,23, 26​

in which case the mode is 23, though that is in the class with second-lowest frequency!

So I'll say again, we can estimate that the mode is likely to be in the modal class, but have no certainty except that it can't be in the class with frequency 1. And this reinforces the fact that the mode of a small dataset is not very significant.

Thank you. So No would be the right answer to his question?

 

[Dr.Peterson]

Thank you. So No would be the right answer to his question?

I'll repeat:

So the answer depends on just what they mean by "rough range".

If you take it to mean "an interval in which the true mode must lie" (that is, the "roughness" means being an interval rather than a precise number), the answer is no; if you take it to mean "an interval in which the true mode is likely to lie" (that is, the "roughness" means not being certain), the answer may be 14 to 21, which extends the modal class slightly to make the likelihood greater. I prefer the former, because of the wording "the mode is between ...".

Ask your teacher, or the author of the problem!

 

[Zelda22]

Thank you