Data Science and A.I. : Numerical Related to Quartile

Numerical Example to Compute Quartile

Problem Statement:

Given the data set:
2, 4, 6, 8, 10, 12, 14, 16, 18, 20, calculate the first quartile, second quartile, and third quartile using the quartile formulas.

Understanding Quartiles:

Quartiles divide the data set into four equal parts. There are three quartiles, denoted by \( Q_1 \), \( Q_2 \), and \( Q_3 \).

First Quartile \( (Q_1) \)

The formula to compute the first quartile is:
\[
Q_1 = \left( \frac{n+1}{4} \right)^{th} \text{ observation}
\]
Where \( n \) is the number of observations.

For the given data set, \( n = 10 \).
\[
Q_1 = \left( \frac{10+1}{4} \right) = 2.75
\]

Now, we interpolate between the 2nd and 3rd observations (4 and 6 respectively):
\[
Q_1 = 4 + 0.75 \times (6 – 4) = 5.50
\]

Second Quartile \( (Q_2) \)

The second quartile (also the median) is calculated using:
\[
Q_2 = \left( \frac{n+1}{2} \right)^{th} \text{ observation}
\]

For an even number of observations, the second quartile is the arithmetic mean of the \( \frac{n}{2} \)-th and \( \frac{n}{2} + 1 \)-th observations.

\[
Q_2 = \frac{10}{2} + \frac{10}{2} + 1 = \frac{10 + 12}{2} = 11
\]

Third Quartile \( (Q_3) \)

The formula for the third quartile is:
\[
Q_3 = \left( \frac{3(n+1)}{4} \right)^{th} \text{ observation}
\]

For \( n = 10 \):
\[
Q_3 = \left( \frac{3(10+1)}{4} \right) = 8.25
\]

Interpolating between the 8th and 9th observations (16 and 18):
\[
Q_3 = 16 + 0.25 \times (18 – 16) = 16.50
\]

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Conclusion:

The quartiles for the given data set are:
– First Quartile \( Q_1 = 5.50 \)
– Second Quartile \( Q_2 = 11 \)
– Third Quartile \( Q_3 = 16.50 \)