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In a data set, the first quartile is 175 and the quartile deviation is 20.

Find the third quartile and the coefficient of quartile deviation.

Sagot :

GinnyAnsweridn
Absolutely, let's solve this problem step by step.

Given:
- The first quartile ([tex]\( Q1 \)[/tex]) is 175.
- The quartile deviation ([tex]\( QD \)[/tex]) is 20.

### Step 1: Calculate the Third Quartile ([tex]\( Q3 \)[/tex])

We know that the quartile deviation (QD) is defined as half the difference between the first and third quartiles. Mathematically, it is represented as:
[tex]\[ QD = \frac{Q3 - Q1}{2} \][/tex]

Rearranging this formula to solve for [tex]\( Q3 \)[/tex]:
[tex]\[ Q3 = Q1 + 2 \cdot QD \][/tex]

Substitute the given values:
[tex]\[ Q3 = 175 + 2 \cdot 20 = 175 + 40 = 215 \][/tex]

So, the third quartile ([tex]\( Q3 \)[/tex]) is 215.

### Step 2: Calculate the Coefficient of Quartile Deviation ([tex]\( CQD \)[/tex])

The coefficient of quartile deviation is defined as the ratio of the quartile deviation to the sum of the first and third quartiles. Mathematically, it is given by:
[tex]\[ CQD = \frac{QD}{Q3 + Q1} \][/tex]

Substitute the given values along with the calculated [tex]\( Q3 \)[/tex]:
[tex]\[ CQD = \frac{20}{215 + 175} = \frac{20}{390} \approx 0.05128205128205128 \][/tex]

So, the coefficient of quartile deviation ([tex]\( CQD \)[/tex]) is approximately 0.0513.

### Summary

The third quartile ([tex]\( Q3 \)[/tex]) is 215, and the coefficient of quartile deviation ([tex]\( CQD \)[/tex]) is approximately 0.0513.
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