The data has 13 elements or 15 elements. Both are the possible odd numbers for the given condition about the upper quartile.
What is an Upper quartile:
- The upper quartile is the median of the upper half of the data set.
- Represented by [tex]Q_3[/tex].
- The formula for the upper quartile is [tex]Q_3[/tex] = [tex]\frac{3}{4}[/tex]×(n+1)th term.
- Where n is the number of data elements in the data set.
- This is used in finding outliers for a set of data elements.
Given data:
It is given that, a data set has an odd number of elements and there are 3 elements above the upper quartile.
Calculating the number of elements in the data set:
Since we know that the upper quartile is the [tex]\frac{3}{4}[/tex]×(n+1)th term, considering the n values given in the options.
Option A:
Here n=11
So,
[tex]Q_3[/tex] = [tex]\frac{3}{4}[/tex] × (11+1)
= [tex]\frac{3}{4}[/tex] × 12
= 9th term
Since the 9th term is the upper quartile, more than 2 elements (10th and 11th terms) are there. But it is given that there should be 3 elements after the upper quartile.
So, this is not the required number for the set of data elements.
Option B:
Here n=13
So,
[tex]Q_3[/tex] = [tex]\frac{3}{4}[/tex] × (13+1)
= [tex]\frac{3}{4}[/tex] × 14
= 10.5th term
This means the median lies between the 10th and 11th terms. Thus from this, we can say that the elements at 11th, 12th, and 13th are above the upper quartile. So, there are 3 elements above the upper quartile. Satisfies the given condition.
Hence, there are 13 elements in the data set.
Option C:
Here n=15
So,
[tex]Q_3[/tex] = [tex]\frac{3}{4}[/tex] × (15+1)
= [tex]\frac{3}{4}[/tex] × 16
= 12th term
Thus from this, we can say that the elements at 13th, 14th, and 15th are above the upper quartile. So, there are 3 elements above the upper quartile. Satisfies the given condition.
Hence, there are 15 elements in the data set.
Therefore, option B) 13 and option C) 15 are the required number of elements the data set has.
Learn more about outliers and quartiles here:
https://brainly.com/question/3514929
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