How to find Q1 and Q3 with even numbers?
Understand the Problem
The question is asking how to calculate the first quartile (Q1) and the third quartile (Q3) from a dataset that contains an even number of elements. This typically involves sorting the data and finding the median values of the lower and upper halves of the dataset.
Answer
Q1 = 2, Q3 = 6
Answer for screen readers
The first quartile (Q1) is 2 and the third quartile (Q3) is 6.
Steps to Solve
- Sort the Dataset
Begin by arranging the numbers in the dataset in ascending order. For example, if the dataset is: $$ 3, 7, 1, 2, 6, 4 $$ After sorting, it will look like: $$ 1, 2, 3, 4, 6, 7 $$
- Find the Median
Since the dataset contains an even number of elements (6 in this case), calculate the median by averaging the two central numbers. The central numbers are the third and fourth numbers in the sorted dataset. $$ \text{Median} = \frac{3 + 4}{2} = 3.5 $$
This step helps to separate the dataset into two halves.
- Identify Quartiles
Now, identify the two halves of the dataset:
- Lower half: $$ 1, 2, 3 $$
- Upper half: $$ 4, 6, 7 $$
- Calculate Q1
Q1 is the median of the lower half of the dataset. For the lower half $$ 1, 2, 3 $$, the median is the second number: $$ Q1 = 2 $$
- Calculate Q3
Q3 is the median of the upper half of the dataset. For the upper half $$ 4, 6, 7 $$, the median is the second number: $$ Q3 = 6 $$
The first quartile (Q1) is 2 and the third quartile (Q3) is 6.
More Information
The first quartile (Q1) represents the 25th percentile, meaning 25% of the data falls below this value. The third quartile (Q3) represents the 75th percentile, meaning 75% of the data falls below this value. These statistics are commonly used in box plots to summarize data.
Tips
- Not sorting the dataset before finding the quartiles. Always ensure the data is in ascending order.
- Confusing the median calculation for odd and even datasets. Remember that for even datasets, the median is the average of the two middle numbers.
