In this post, you’ll find an explanation of some important topics in statistics such as quartile and interquartile. Both of these topics could turn out to be a bit complicated if you are working on your school assignments. However, perdisco assignment help could be a great source of assistance to complete your assignments. Also with this, you can make your content and assignments related to statistics and mathematics without taking or learning any content writing course.
These concepts (Quartile & Interquartile) allow the graphic depiction of various probability distributions as well as the plotting of boxes and whiskers that describe and compare data effectively.
What are Quartiles?
The quartile in statistics is quite a basic notion. Quartiles are the three values in a ranking data collection, which divide the data into four equal portions. Each of the four segments includes 25% of the data.
- The smallest is the 1st 75% of the data set is above the first quartile, while 25% is below it.
- The median is also known as the 2nd It’s the center of the data set. Half of the data set is above the median and half is below the median.
- The biggest quartile is 3rd 25% of the data is above the third quartile while the remaining 75% is below it.
If the number of points in the data set is odd, the middle value of the second quartile or median is equal to the number of points on either side of the median. The mean is the median of the two values in the middle if the number of data points is even.
How to find Quartiles?
Take the following data set into account: 5, 6, 8, 6, 7, 9, 1, 2, 4, 3.
Steps to calculate Quartiles
- Arrange the data in the Use a least to greatest calculator to arrange your data set quickly.
1, 2, 3, 4, 5, 6, 7, 8, 9
- There is a total of 10 numbers in the data set. First, find the median (second quartile). The median is the center of the data set, ensuring that 50% of the observations are on the right and 50% on the left. The average of the two mid-range values 5 and 6 = 5.5 is the 2nd quartile or median. The 5.5 divides the set of data into the same portions essentially.
- The first quartile is 3. Median divides the set of data into two parts: the right and the left. Take the left and discover the mid-value. The center of the left section is the first quartile. This guarantees that 25% of observations will be on the left and 75% will be on the right.
- The third quartile is 7 in this case. Median divides the set of data into two parts: the right and the left. Take the right section and find its median. The third quartile will be the median of the right portion. This guarantees that 75% of the observations are on the left side and 25% on the right side.
Find the quartiles for the data set 4,6,1,7,9,2,3
- Arrange the data in the least to greatest order as we did in the previous example. 1, 2, 3, 4, 6, 7 and 9
- There is a total of 7 numbers in the data set. For odd numbers, the calculation will be a bit different. Find the median, which is 4 in this case. The data set is divided into two equal portions. When you notice, the left side of the median has 3 data points and the right side of the median has also 3 data points.
The first quartile is 2 and the third quartile is 7.
First quartile 1 is the median of the left side and the third quartile is the median of the right part.
You have learned how to locate disconnected data quartiles. The principle of quartiles remains the same for continuous data. Consider ongoing distributions of probabilities. The total area of any distribution probability curve is 1.
- 1st quartile shall be the point so that it has 0.25 area to the left and 0.75 area to the right.
- The point of the median is the 0.5 area on the left and the 0.5 area on the right.
- The third quartile is the point where it is 0.75 on the left and 0.25 on the right.
Find the Quartiles for the following.
Consider a continuous consistent distribution that takes 0 for X from 0.25 to 0.75 and 0 as indicated below.
Area of Rectangle = Height x Width
Width = 0.75 – (-0.25) = 1
Height = 1
Area of rectangle = 1
If this rectangle were divided into 4 identical parts, the area of each component would be,
Area on the left side of x = 0 area: 1 x (0 – (0.25)) = 0.25
Area on the left side of x = 0.25 area left: 1 x (0.25 – (-0.25)) = 0.5
Area on the left side of x = 0.5 area:1 Area on the left side of x (0.5 – (-0.25)) = 0.75
1st quartile: x=0 (25% of the rectangular area is on the left hand side of x = 0)
2nd quartile: x=0.25 (50% of the area on the left side x = 0.25)
3rd quartile: x=0.5 (75% of the rectangular area is x = 0.5)
Online Quartile calculator can assist you in find quartiles and interquartile without much effort.
How to find Interquartile Range?
50 percent of the data is represented by the interquartile range. The difference between the 3rd quartile and the 1st one is represented by the interquartile range. It is connected to the quartile. The interquartile range can be calculated using the following formula:
Interquartile Range = Quartile 3 − Quartile 1
After calculating quartiles, it’s rather easy to calculate an interquartile range. Simply remove the first quartile from the third one. Let’s find out we can calculate it using the above formula.
We will use the previous examples to calculate the interquartile range. The quartiles calculated in those examples can be used find IQR.
IQR = Q3 – Q1 = 4 – 3 = 1
IQR = Q3 – Q1 = 5 – 2 = 3
IQR = Q3 – Q1 = 0.9 – 0 = 0.9
The interquartile range comprises 50% of the middle data. 1st Quartile and 3rd Quartile are respectively the lower and upper positions in this mid-50 percent range. The IQR is therefore the difference between the two.
In conclusion, to compute the interquartile range, following these steps:
- Arrange the data in ascending order (least to greatest).
- Divide the data into four equal portions by making three cuts.
- The first quartile (lowest), median (middle), and the third quartile are the biggest of all of them.
- IQR = third quartile – first quartile.
There are different ways to interpret these notions now that you understand the quartile and interquartile range.
The data’s median is the median value, which divides the data in the first half and in the second part into two equal parts. First Quartile or Q1 can be read as the first part median and the third Quartile as the second part median.
Quartiles can assist you to measure the distribution of data on both sides of the median. The data are essentially symmetrical, for example, if the first and third quartiles are almost as far apart from the median. If the 3rd quartile is distant from the median, while the first quartile is nearer to the median, then greater data points than the median are far removed, whilst data points are tightly packaged less than the median.
Whilst you have known the use of quartiles, boxes and whiskers will explain how you may visually depict, analyze and compare data with Q1, Q2, Q3, and IQR.
Using Whisker Plot for Visuals
Quartiles and interquartile are some of the most important applications for drawing boxes and whiskers. Boxing and whisker plots show the data well and visualize the data distribution, outliers, data symmetry, etc.
- Build a box and whisker plot. Arrange the data as we did in the above examples i.e., ascending order.
- Use the examples given above to calculate the 1st quartile, 3rd quartile, and median. Maximum and minimum values can also be found.
The above lines are representing the minimum, first quartile, median, third quartile, and a maximum of a data set from left to right. These points are known as a five-point summary for the production of boxes and whiskers.
- Complete the box and the whisker plot by drawing a box around the middle three lines and drawing whiskers at the minimum and greatest value from the central box.
Note that the first quartile and the third quartile are the endpoints of the box. The box includes 50% of the data. The difference between the endpoints of the box, therefore, provides the interquartile range.
What are Outliers?
The finding of outlines in the data is also an important industry usage of quartiles and interquartile. Outliers are extraordinary values that probably do not properly represent the population. Data gathering failures or other problems may be caused by outliers.
Imagine collecting statistics in a school for pupils’ weights, for instance. All student weights have been found between 40 kg and 120 kg. However, one pupil is said to have a weight of 610 kilograms. A student cannot weigh 610 kg. It is impossible. This anomaly could have occurred due to an error or other reasons. Therefore, constantly consider the removal of outliers when evaluating data. The second question is: how can we find out about outliers?
Quartiles and interquartile offer an easy approach to find out about outliers. Find the lower and the higher range following after you found first quartile, third quartile and the interquartile range:
Bottom range = First Quartile – 1.5 x QRI
Upper range = Third Quartile + 1.5 x IQR
Any data point below or above the bottom range can be considered an outliner. These are the lowest and the highest data limits. They can be used to identify externally.
Further usage of IQR is the search for another statistic, which is called the half-in-quartile range or the half-in-quartile deviation.
Take the following set of data: 10, 16, 11, 7, 4, 4, 4, 12, 17, 25, 14, 0
Find first quartile, third quartile, median, interquartile, minimum value, high value, outliers, and quartile deviation, if applicable. Make a summary of the data by five points and also design a box and a whisker plot.
- Arrange the given data set in the least to greatest order.
0, 4, 4, 4, 7, 10, 11, 12, 14, 16, 17, 25
- Calculate median of this data set.
Total numbers in the data set = 12
So, the total numbers in this data set are even.
Median = Mean value 6 and 7th value, = (10+11)/2 = 10.5
- By making cuts, calculate the first quartile and third quartile. The median of 2nd step separates the data points into two equal portions. Make cuts in the center of the two portions so the right and the left parts are further separated into two equal parts.
First Quartile = (4+4)/2 = 4. Make a cut between the 3rd and 4th data points and take the average of those two points.
Third Quartile = (14+16)/2 = 15. Make a cut between the 9th and 10th data points and take the average of those two points.
- Inter-quartile range, lowest and highest value quartile deviation, etc.
Minimum = 0
Maximum = 25
IQR = 3rd quartile – 1st quartile = 15 – 4 = 11.
Deviation = IQR/2 = 5.5.
- Now we will find the lower and upper fences.
Lower fence = First Quartile – 1.5 x IQR = 4 – 1.5 x 11 = -1.5
The upper fence is an outlier in the database, as it is greater than the lower fence. Third Quartile + 1.5 x IQR = 15 + 1.5 x 11 = 20.5
25 is an outlier in the data. The lower fence does not contain any data points that are less than the bottom one.
- Create a summary of five points of data and design the whiskers and boxes.
The summary of the data in five points is 0, 4, 10.5, 15, and 25.
Below you can find the box and the whisker plot.
Note that quartiles can help you measure how the data is spread over the median on both sides. The first and the third quartiles are not as close to the median here, so it is obvious that the data are not symmetrical. The first quartile is far from the midpoint here, but the third quartile is close to the midpoint. This means that the data points that are smaller than the median are more widely spread, while the data points larger than the median are packed more closely.
In statistics, quartiles and interquartile are very important concepts. You can effectively contribute to the visual representation and interpretation of data. It is also useful for computing other statistics such as quartile deviation. It gives us an idea of how a data set is distributed. Sometimes when you look at these values, it tells us if the data is symmetric or skewed. Thus, they are simple yet useful notions and you must always remember them.