Problem 31 - Basic College Mathematics

Question

Find the quartiles for each set of numbers. $$ 4,8,10,16,12,25,30,32,14,28 $$

Step-by-Step Solution

Answer

Q1 = 10, Q2 = 15, Q3 = 28.

Step 1: Arrange the Data in Ascending Order

Firstly, arrange the given data set in ascending order: 4, 8, 10, 12, 14, 16, 25, 28, 30, 32.

Step 2: Find the Median (Q2)

For a data set with 10 numbers, the median is the average of the 5th and 6th numbers. Here, these numbers are 14 and 16. So, the median (Q2) is: \[ Q2 = \frac{14 + 16}{2} = 15 \]

Step 3: Find the Lower Quartile (Q1)

The lower quartile (Q1) is the median of the first half of the data (excluding Q2). For the numbers 4, 8, 10, 12, 14, the median is 10. Hence, Q1 = 10.

Step 4: Find the Upper Quartile (Q3)

The upper quartile (Q3) is the median of the second half of the data (excluding Q2). For the numbers 16, 25, 28, 30, 32, the median is 28. Hence, Q3 = 28.

Key Concepts

Median Calculation

The median is a key measure in statistics. It represents the middle value of a data set. To find the median, first arrange the data in ascending order.
The median is particularly useful in data sets because it is not affected by extremely high or low values.
For an even number of values, like in our case with 10 numbers, the median is calculated by taking the average of the 5th and 6th values.
This way you get an accurate middle point.
Following these steps:
Arrange the data: 4, 8, 10, 12, 14, 16, 25, 28, 30, 32.
Find the middle values, which are 14 and 16 in this case.
The median (Q2) is calculated as:
Q2 = \[\frac{14 + 16}{2} = 15\]
This means the median for our given data set is 15.

Data Set Arrangement

Arranging your data set properly is crucial for accurate statistical analysis.
The data should be sorted in ascending order from the smallest value to the largest value. This allows for a clear, organized view of your data, making it easier to find quartiles and other statistics.
Let's see how we did this for our example:
Original data set: 4, 8, 10, 16, 12, 25, 30, 32, 14, 28.
Sorted in ascending order: 4, 8, 10, 12, 14, 16, 25, 28, 30, 32.
Sorting the values like this helps in identifying the median and quartiles systematically. So always start your analysis by arranging the data in ascending order.

Finding Q1 and Q3

Quartiles divide a data set into four parts, helping to understand its spread.
Q1 (lower quartile) represents the 25th percentile and Q3 (upper quartile) represents the 75th percentile.
To find the quartiles, first identify the median (Q2) which we've found as 15.
Next, split the data into two halves, excluding the median:
First half: 4, 8, 10, 12, 14.
Second half: 16, 25, 28, 30, 32.
Q1 is the median of the first half. For 4, 8, 10, 12, 14, the median is 10. Thus, Q1 = 10.
Q3 is the median of the second half. For 16, 25, 28, 30, 32, the median is 28. Thus, Q3 = 28.
So, the lower quartile (Q1) is 10 and the upper quartile (Q3) is 28.
This division helps in understanding the distribution of the data more thoroughly.