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The following table shows a frequency distribution of the scores obtained in a test.

Punctuation           (3, 4] (4, 5] (5, 6] (6, 7] (7, 8] (8, 9] (9, 10]
Number of participants  2      4      10     20     40     35      9

(a) The highest score reached by the bottom 20% of the participants and the score lowest obtained by the top 25% of the participants

(b) The quartiles of the distribution

For section a) to be continuous variables that can have any value within a range what I do is calculate the average for that 20%, that is to say who is in position 24. I calculate the relative frequency (that for this I need to know the absolute frequency) and multiplied it by the midpoint of the interval. I am applying according to the definition and the formula, but I am stuck.

It is a note that is between (6,7] and the average I get 0.1, so the note would be 6.1.

For b) of the quartiles I get that 1 is in (6,7], 2 is in (7,8] and 3 is in (8,9), but I do not know how to get the exact note

I've done it that way, but I do not know if it's okay since it's the first time I've done something similar, can someone verify it?

a) I have the table

Class | partici | Ni
3-4   |     2   | 2
4-5   |     4   | 6
5-6   |    10   | 16
6-7   |    20   | 36
7-8   |    40   | 76
8-9   |    35   | 111
9-10  |     9   | 120

For The highest score reached by the bottom 20% of the participants means the 20th percentile so

P20 = 6+ ((0,20*120)-16)*1 /20 = 6,4 score

The lowest score obtained by the top 25% of the participants means the 75th percentile, so it is calculated in the same way

P75 = 8 + ((0,75*120)-76)*1 /35 = 8,4 score

b)

For the quartiles, it is the same as calculating the percentiles for 25% 50% and 75%. Q2 is equal to the median

Q1 = 6 + ((0,25*120)-16)*1 /20 = 6,7 score

Q2 = 7 + ((0,5*120)-36)*1 /40 = 7,6 score

Q3 = 8 + ((0,75*120)-76)*1 /35 = 8,4 score =P75 (From the section a-))
Fernando
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  • You seem to have the right idea for both (a) and (b). It is impossible to know exactly where within your intervals the various percentiles lie. Different texts have different ways of estimating a value. // In general, several different ways of finding percentiles are in common use. Perhaps see this Q&A. – BruceET Sep 14 '18 at 07:19
  • @BruceETI edit the question and complete it better with the information you gave me and what I found. Can you take a look? – Fernando Sep 14 '18 at 14:43
  • Took a quick look. Seems on the right track for one method of estimation. // If you're using a comma for $0,1 = 1/10,$ then you have to put spaces in interval notation: (6, 7], not (6,7]. In JaX, you can use $(6,\; 7]$ to get $(6,; 7].$ – BruceET Sep 14 '18 at 17:34

1 Answers1

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You have identified correctly the intervals the $20\%$ point and the quartiles fall in. That is all you can really say. If you assume the scores in the $(6,7]$ interval are evenly distributed the $20\%$ score will be $6.4$ as you say because you want the $8^{th}$ score of $20$ in that interval. It could be that all the $20$ scores in the interval are $6.1$ or it could be that they are $6.9$. You don't have the data to distinguish between these possibilities.

You dealt with the quartiles the same way and the same comment applies to them.

Ross Millikan
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