Answer :
To find the 49th-percentile of a data set with 117 elements, follow these steps:
1. Understand Percentile: The 49th percentile is the value below which 49% of the data falls.
2. Calculate the Index:
- First, determine the index where this percentile lies in the sorted data set.
- The formula for finding the percentile index is:
[tex]\[
\text{Index} = \left(\frac{p}{100}\right) \times (n - 1)
\][/tex]
where [tex]\( p \)[/tex] is the percentile you want, and [tex]\( n \)[/tex] is the total number of data points.
3. Apply the Formula:
- Here, [tex]\( p = 49 \)[/tex] and [tex]\( n = 117 \)[/tex].
- Thus, the index calculation becomes:
[tex]\[
\text{Index} = \left(\frac{49}{100}\right) \times (117 - 1) = 0.49 \times 116 = 56.84
\][/tex]
4. Locate the Position:
- The index 56.84 indicates that the 49th percentile lies between the 57th and 58th data points in the ordered data set (remember: index counting starts from 0).
- The lower index is 56 and the upper index is 57.
5. Interpolate Between Two Values:
- To find the exact percentile value, interpolate between the 57th and 58th values.
- The 57th value (at index 56) is 64.5, and the 58th value (at index 57) is 64.7.
- Calculate the fractional part of the index, which is [tex]\( 0.84 \)[/tex].
6. Perform the Calculation:
- Use linear interpolation to estimate the 49th percentile:
[tex]\[
P_{49} = \text{Lower Value} + \text{Fraction} \times (\text{Upper Value} - \text{Lower Value})
\][/tex]
[tex]\[
P_{49} = 64.5 + 0.84 \times (64.7 - 64.5) = 64.5 + 0.84 \times 0.2 = 64.684
\][/tex]
Therefore, the 49th-percentile of the given data set is approximately [tex]\( \boxed{64.684} \)[/tex].
1. Understand Percentile: The 49th percentile is the value below which 49% of the data falls.
2. Calculate the Index:
- First, determine the index where this percentile lies in the sorted data set.
- The formula for finding the percentile index is:
[tex]\[
\text{Index} = \left(\frac{p}{100}\right) \times (n - 1)
\][/tex]
where [tex]\( p \)[/tex] is the percentile you want, and [tex]\( n \)[/tex] is the total number of data points.
3. Apply the Formula:
- Here, [tex]\( p = 49 \)[/tex] and [tex]\( n = 117 \)[/tex].
- Thus, the index calculation becomes:
[tex]\[
\text{Index} = \left(\frac{49}{100}\right) \times (117 - 1) = 0.49 \times 116 = 56.84
\][/tex]
4. Locate the Position:
- The index 56.84 indicates that the 49th percentile lies between the 57th and 58th data points in the ordered data set (remember: index counting starts from 0).
- The lower index is 56 and the upper index is 57.
5. Interpolate Between Two Values:
- To find the exact percentile value, interpolate between the 57th and 58th values.
- The 57th value (at index 56) is 64.5, and the 58th value (at index 57) is 64.7.
- Calculate the fractional part of the index, which is [tex]\( 0.84 \)[/tex].
6. Perform the Calculation:
- Use linear interpolation to estimate the 49th percentile:
[tex]\[
P_{49} = \text{Lower Value} + \text{Fraction} \times (\text{Upper Value} - \text{Lower Value})
\][/tex]
[tex]\[
P_{49} = 64.5 + 0.84 \times (64.7 - 64.5) = 64.5 + 0.84 \times 0.2 = 64.684
\][/tex]
Therefore, the 49th-percentile of the given data set is approximately [tex]\( \boxed{64.684} \)[/tex].
