Question

A sociologist records the annual household income (in thousands of dollars) among a sample of families living in a high-crime neighborhood. Locate the lower, median, and upper quartiles for the times listed below. Hint: First arrange the data in numerical order. lower quartile ____ thousand dollarsmedian ______ thousand dollars upper quartile ______ thousand dollars33 55 39 44 34 48 25 39 23 45

Answer 1

Answer:

The lower quartile is $Q_1=33$ thousand dollars.

The median quartile is $Q_2=39$ thousand dollars.

The upper quartile is $Q_3=45$ thousand dollars.

Step-by-step explanation:

The lower quartile is the median value of the lower half of a data set at the 25th percentile of a distribution.

The median quartile is the median value of a data set at the 50th percentile of a distribution.

The upper quartile is the median value of the upper half of a data set at the 75th percentile of a distribution.

To locate each quartile in a data set, we follow four steps:

Step 1: Put the numbers in order: 23, 25, 33, 34, 39, 39, 44, 45, 48, 55

Step 2: The median is given by $\frac{n+1}{2}$ where n is all scores in the data set.  

Because n = 10, the median position is $\frac{10+1}{2}=5.5$

The median is the average of the fifth and sixth positioned scores

$Q_2=\frac{39+39}{2} =39$

Step 3:  Compute $\frac{n+1}{2}$ where n is all scores below $Q_2$.

For scores below $Q_2$, use only 23, 25, 33, 34, 39.

Because n = 5, the median position is $\frac{5+1}{2}=3$

The median is the third positioned score: $Q_1=33$

Step 4: Compute $\frac{n+1}{2}$ where n is all scores above $Q_2$.

For scores above $Q_2$, use only  39, 44, 45, 48, 55

Because n = 5, the median position is $\frac{5+1}{2}=3$

The median is the third positioned score: $Q_3=45$