In statistics,
Quartiles
are key values that divide a dataset into four equal parts. When dealing with extensive numerical data in statistical analysis, various concepts and formulas come into play, making them highly applicable in research and surveys. One of the most practical applications of quartiles is in creating box and whisker plots.
Quartiles play a crucial role in dividing a dataset into three distinct sections. The middle section, which is one of the three quartiles, represents the central point of the data distribution and includes data points close to this central value. The lower section comprises roughly half of the dataset and encompasses the values that fall below the median, while the upper section represents the remaining half that falls above the median. In essence, quartiles provide valuable insights into the distribution and dispersion of a dataset, making them a fundamental tool in statistical analysis.
Quartiles Definition
Quartiles are values that divide an entire dataset into four equal parts, resulting in three quartiles: Q1, Q2, and Q3. Q2 is equivalent to the median, as it indicates the position of an item in the ordered list and serves as a positional average. To calculate quartiles for a dataset, it's necessary to first arrange the data in ascending order.
While the median provides insight into the central tendency of the data, we can further assess the distribution by considering the lower and upper quartiles. Beyond quartiles, statistics offer other measures that divide data into specific equal parts, including:
Quartiles:
As previously described, quartiles divide the data into four equal parts, focusing on the median (Q2) and the values below (Q1) and above (Q3) the median.
Deciles:
Deciles divide the data into ten equal parts. Decile values, such as D1, D2, D3, and so on, help assess the distribution and spread of data with more granularity.
Percentiles:
Percentiles divide the data into one hundred equal parts. They are often used to compare data points to a broader population and are valuable for assessing how specific data points rank relative to the entire dataset.
These different measures, including quartiles, deciles, and percentiles, allow statisticians and data analysts to gain a deeper understanding of data distributions and make more informed decisions in various fields such as finance, healthcare, and social sciences.
Quartiles Formula
Suppose, Q3 is the upper quartile is the median of the upper half of the data set. Whereas, Q1 is the lower quartile and median of the lower half of the data set. Q2 is the median. Consider, we have n number of items in a data set. Then the quartiles are given by:
Q1 = [(n+1)/4]
th
item
Q2 = [(n+1)/2]
th
item
Q3 = [3(n+1)/4]
th
item
Hence, the formula for quartile can be given by;
Where Qr is the r
th
quartile
l1 is the lower limit
l2 is the upper limit
f is the frequency
c is the cumulative frequency of the class preceding the quartile class.
Quartiles in Statistics
An excellent explanation of how quartiles divide a dataset into four equal parts, each representing a specific percentage of the data. This breakdown is particularly valuable for understanding the distribution of data and identifying key points within that distribution. Here's a summary of the quartiles and their associated percentages:
First Quartile (Q1):
The first quartile represents the 25% mark of the data when it is sorted from smallest to largest. This means that 25% of the data points are less than or equal to Q1, while 75% of the data points are greater.
Second Quartile (Q2 - Median):
The second quartile corresponds to the median, which divides the data into two equal halves. 50% of the data points are less than or equal to Q2, and the remaining 50% are greater.
Third Quartile (Q3):
The third quartile represents the 75% mark of the data. This indicates that 75% of the data points are less than or equal to Q3, while 25% are greater.
Fourth Quartile:
While not explicitly mentioned in your description, the fourth quartile would encompass the remaining 25% of the largest data points, meaning 25% of the data falls within this quartile.
Understanding these quartiles is crucial in statistics for summarizing data distributions and identifying specific data points that correspond to given percentages within the dataset.
Quartile Deviation
Quartile deviation is a measure of the spread or dispersion of data within a dataset, and it is calculated as half of the distance between the third quartile (Q3) and the first quartile (Q1). The formula for quartile deviation is as follows:
Quartile deviation = (Q
3
-Q
1
)/2
This measure provides a sense of the variability of data within the interquartile range, which contains the middle 50% of the data points. Quartile deviation is a useful statistic for understanding the spread of data and identifying the middle range of values in a dataset.
Interquartile Range
The interquartile range (IQR) and its formula. The interquartile range is a valuable statistical measure that helps us understand the spread or dispersion of data within a dataset, particularly focusing on the middle 50% of the data points. It is defined as the difference between the third quartile (Q3) and the first quartile (Q1).
IQR = Q3 – Q1
Quartiles Formula Examples
Example 1:
Quartiles Calculation
Suppose you have the following dataset of exam scores (in ascending order):
Dataset: 55, 60, 65, 70, 75, 80, 85, 90, 95, 100
To find the quartiles:
Find the Median (Q2 - Second Quartile):
The dataset has an even number of values, so the median is the average of the two middle values.
Q2 = (70 + 75) / 2 = 72.5
Find the Lower Quartile (Q1 - First Quartile):
To calculate Q1, focus on the lower half of the data (values below the median).
Q1 = (60 + 65) / 2 = 62.5
Find the Upper Quartile (Q3 - Third Quartile):
To calculate Q3, focus on the upper half of the data (values above the median).
Q3 = (90 + 95) / 2 = 92.5
So, for this dataset, the quartiles are:
First Quartile (Q1): 62.5
Second Quartile (Median - Q2): 72.5
Third Quartile (Q3): 92.5
Example 2:
Quartiles Calculation with a Larger Dataset
Consider the following dataset representing the ages of a group of individuals (in ascending order):
Dataset: 18, 20, 22, 25, 30, 35, 40, 45, 50, 60, 70, 80, 90, 100
To find the quartiles:
Find the Median (Q2 - Second Quartile):
As the dataset has an odd number of values, the median is the middle value.
Q2 = 40
Find the Lower Quartile (Q1 - First Quartile):
To calculate Q1, focus on the lower half of the data (values below the median).
Q1 = (22 + 25) / 2 = 23.5
Find the Upper Quartile (Q3 - Third Quartile):
To calculate Q3, focus on the upper half of the data (values above the median).
Q3 = (70 + 80) / 2 = 75
So, for this dataset, the quartiles are:
First Quartile (Q1): 23.5
Second Quartile (Median - Q2): 40
Third Quartile (Q3): 75
These quartiles provide valuable insights into the distribution of ages within the dataset.
