Quartiles in statistics are used to divide a data set into four equal parts, or quarters. The first quartile, also known as Q1, is the data point that separates the lowest 25% of the data from the highest 75%. The second quartile, or Q2, is the median of the data set, which is the middle value when the data is ordered from lowest to highest. The third quartile, or Q3, is the data point that separates the lowest 75% of the data from the highest 25%.
Quartiles are useful for understanding the distribution of data within a set. For example, if a data set has a high Q1 and Q3, this indicates that the data is skewed towards higher values, while if the Q1 and Q3 are low, the data is skewed towards lower values. Quartiles can also be used to identify potential outliers in the data, as values that fall outside of the range between Q1 and Q3 are considered to be potentially unusual or extreme compared to the rest of the data.
To calculate quartiles, you will need to follow these steps:
- Order the data from lowest to highest.
- Find the median of the data set by finding the middle value. If there is an odd number of data points, the median is the middle value. If there is an even number of data points, the median is the average of the two middle values.
- Divide the data set into two halves: one below the median and one above the median.
- Calculate the first quartile (Q1) by finding the median of the lower half of the data set.
- Calculate the third quartile (Q3) by finding the median of the upper half of the data set.
The formula for calculating quartiles depends on whether the data set has an odd or even number of data points.
For an odd number of data points:
Q1 = (n + 1) / 4
Q2 = (n + 1) / 2
Q3 = 3(n + 1) / 4
Where n is the number of data points in the data set.
For an even number of data points:
Q1 = (n / 2) / 2 + 1
Q2 = (n / 2) + (n / 2 + 1) / 2
Q3 = (3n / 2) / 2 + 1
Where n is the number of data points in the data set.
For example, let’s say we have the following data set with 11 data points: 4, 8, 9, 10, 12, 15, 18, 20, 22, 24, 25
Q1 = (11 + 1) / 4 = 3
Q2 = (11 + 1) / 2 = 6
Q3 = 3(11 + 1) / 4 = 9
So the quartiles for this data set are Q1 = 9, Q2 = 12, and Q3 = 20.
There are several properties of quartiles in statistics:
- The median, or Q2, is the second quartile and is always the middle value of the data set when it is ordered from lowest to highest.
- The first quartile, or Q1, is the data point that separates the lowest 25% of the data from the highest 75%.
- The third quartile, or Q3, is the data point that separates the lowest 75% of the data from the highest 25%.
- The range between Q1 and Q3 is known as the interquartile range (IQR) and is a measure of the dispersion or spread of the data.
- Outliers in the data are values that fall outside of the range between Q1 and Q3.
- Quartiles are useful for understanding the distribution of data within a set and for identifying potential outliers.
- Quartiles can be calculated using a formula that depends on whether the data set has an odd or even number of data points.