What are Quartiles?
In statistics, quartiles are the values that divide a set of numerical values into three quarters. The three quartiles are denoted by \( Q_1,~Q_2 \) and \( Q_3 \), as the first, second, and third quartiles. The second quartile, \( Q_2 \) is the value of the median for the given data.
What is an Interquartile Range?
The interquartile range (IQR) can be defined as the difference between the first quartile and the third quartile. The formula is given by, \( IQR = Q_3-Q_1 \).
Before determining the interquartile range, we first need to know the values of the first quartile and the third quartile. From the image, we can notice the occurrence of the median and the interquartile range for a particular data set.
What is the Box and Whisker Plot?
A box and whisker plot is defined as the graphical representation of displaying a variety of data. The visual display of the box plots is shown through their quartiles.
How to Calculate the Interquartile Range?
The interquartile range can be calculated in the following steps:
- The given set of numbers will first be arranged from increasing to decreasing order.
- The number of values are then counted. If the value is odd, the center value is median. If the value is even, the mean value for the two center values is obtained and denoted as \( Q_2 \). For even values, the median is the average of the middle two values.
- Median is used for giving two equal parts. These equal parts are known as \( Q_1 \) and \( Q_3 \).
- The median of data values below the median is given by \( Q_1 \).
- The median of data values above the median value is given by \( Q_3 \).
- The median values of \( Q_1 \) and \( Q_3 \) are subtracted from each other.
- The resulting value that we get is the interquartile range.
Solved Interquartile Range Examples
Example 1: Find out the interquartile range of the first 10 prime numbers.
Solution:
The first 10 prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29.
These numbers are already arranged in the ascending order.
Now, the number of values = 10
10 is an even number, so the median is the mean of 11 and 13.
Therefore, \( Q_2 = \frac{(11+13)}{2}=\frac{24}{2}=12 \).
Now, we have to find the other two parts, \( Q_1 \) and \( Q_3 \).
\( Q_1 \) part comprises 2, 3, 5, 7, 11 and, \( Q_3 \) part comprises 13, 17, 19, 23, 29.
As we can see, for both \( Q_1 \) and \( Q_3 \), the number of values is equal to 5.
Now, for \( Q_1 \), the center value is 5, that is \( Q_1 \) is equal to 5, and for \( Q_3 \), the center value is 19, that is \( Q_3 \) is equal to 19.
Now, in order to find the interquartile range, we will subtract \( Q_1 \) from \( Q_3 \).
Interquartile range \( = Q_3 – Q_1 = 19 – 5 = 14 \)
Hence, 14 is the interquartile range value in this case.
Example 2: The data given represents the number of vehicles sold from 15 manufacturers from their respective showrooms. Find the interquartile range for the data.
6, 4, 4, 8, 12, 14, 18, 10, 15, 5, 8, 15, 10, 6, 12
Solution:
Let us arrange the numbers of the data set in the ascending order. Hence, the data set will look as such, 4, 4, 5, 6, 6, 8, 8, 10, 10, 12, 12, 14, 15, 15, 18
From the above arranged data, we can find the median. As we know, median is the middle number of the data set. Since we have 15 numbers in the data set which is an odd number, the median will be one number in the middle of the set.
There will be 7 numbers to the left of the middle number and 7 numbers on the right. So, by counting the numbers from the data set, the median will be 10.
The interquartile range is found by the difference between the middle of the first half, and the middle of the second half. As this is a measure of spread, we can understand how far apart are the data points from each other.
So, now we will find the median of the first 7 numbers. Since the number of values is 7, an odd number, thus we will find the median in a similar method. The median of the first half is 6 (taking the middlemost value from 7 number values on the left) and we will find the median for the second half in a similar manner. The median of the second half will be 14 (taking the middlemost value from 7 number values on the right).
Now, the interquartile range will be the difference between the two median values that we derived from the data set.
Hence, interquartile range = 14 – 6 = 8
In simple terms, the median is the middle value in an arranged list of numbers that are arranged from the lowest to the highest order.
An outlier is a point from a data set that differs immensely from the rest of the values of the data set.
In statistics, the mean represents the average value of a data set. The mean of any data set is found by creating the sum of all the data points and then dividing the total by the number of the data points itself.
The mode of any data set can be defined as the most frequent value in the data set.