Question 1190560: Marks of 75 students are summarised in the following frequency distribution Marks Frequency 40 - 44 7 45 - 49 10 50 - 54 22 55 - 59 F1 60 - 64 F2 65 - 69 6 70 - 74 3 If 20% of the students have marks b/n 55 and 59 find The 5th deciles, The 60th percentile, 1 quartiles
Answer by math_tutor2020(3817)   (Show Source):
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Given data table
| Marks | Frequency | | 40 - 44 | 7 | | 45 - 49 | 10 | | 50 - 54 | 22 | | 55 - 59 | F1 | | 60 - 64 | F2 | | 65 - 69 | 6 | | 70 - 74 | 3 |
We're told that "20% of the students have marks b/n 55 and 59" which I'm assuming the b/n means "between".
If 20% of the marks are between 55 and 59, then
20% of 75 = 0.20*75 = 15
This means 15 students are in the interval 55 - 59
We'll replace the F1 with 15
| Marks | Frequency | | 40 - 44 | 7 | | 45 - 49 | 10 | | 50 - 54 | 22 | | 55 - 59 | 15 | | 60 - 64 | F2 | | 65 - 69 | 6 | | 70 - 74 | 3 |
To find F2, we'll add up each of the frequencies. The total sum is 75
7+10+22+15+F2+6+3 = 75
63+F2 = 75
F2 = 75-63
F2 = 12
Here's the full completed grouped frequency table
| Marks | Frequency | | 40 - 44 | 7 | | 45 - 49 | 10 | | 50 - 54 | 22 | | 55 - 59 | 15 | | 60 - 64 | 12 | | 65 - 69 | 6 | | 70 - 74 | 3 |
You should find that the frequencies sum to 75.
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Deciles split the data set into 10 equal pieces
The 5th decile is the median point because we're 5/10 = 1/2 of the way there.
1/2 of 75 = 75/2 = 37.5
Start at the first frequency (7) and add down until we reach 37.5 or go over.
7 is too short
7+10 = 17 is too short also
7+10+22 = 39 we've gone over the mark of 37.5
The median is somewhere in the interval from 50 to 54 (when we added on that last frequency of 22)
Unfortunately we don't have enough info to pinpoint exactly where the 5th decile is. I think an acceptable answer is to state the interval 50 - 54, or to state "a score between 50 and 54".
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The 60th percentile is the marker where 60% of the scores are below this value.
60% of 75 = 0.60*75 = 45
We see that 45 students are below the 60th percentile person.
Like before, we'll add up the frequencies until we hit 45 or go over.
7 is too short
7+10 = 17 is too short
7+10+22 = 39 is too short
7+10+22+15 = 54 goes overboard
The 60th percentile scorer is somewhere in the interval of 55 to 59.
Like the previous section, we don't have enough info to pin down the exact score.
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The 1st quartile is equivalent to finding the 25th percentile
We want to know what marker has 25% of the data below this value.
25% of 75 = 0.25*75 = 18.75
We round 18.75 to 19
About 19 students are below the 25th percentile marker
We'll use the same trick as the previous sections to try to get to 19 or higher
7 is too short
7+10 = 17 is too short
7+10+22 = 39 goes overboard
The 25th percentile is in the same group as the 5th decile (aka median).
That group being the "between 50 and 54" scores.
Like the other sections, we don't have enough info to pin down the exact score.
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