discuss the skewness for the distribution in which : mean< median < mode

Question

discuss the skewness for the distribution in which :  

 mean< median < mode 

Answer 1

Good morning, Himanshu,

So, in any distribution of data, where the mean is greater than the median and mode, your data will be skewed positively. 

Why?

A picture is worth a thousand words applies here. To explain why; it is better to provide you three examples of data sets.

What I recommend all students do when presented with a question such as this is to sketch a histogram, especially if the data set is smaller. Doing so allows you to actually see the reason why those of us in statistics provide such a characterization of the data. 

Remember: The mean, median and mode are all measures of central tendency and describe the data set in question. 


Here are the three examples: 

Data Set One: 4; 5; 6; 6; 6; 7; 7; 7; 7; 7; 7; 8; 8; 8; 9; 10 

The histogram created for this data set will show you that you have a symmetrical distribution of data. A data set is considered symmetrical if and only if you can draw a vertical line down the center of the histogram created and you have a mirror image of the histogram on each side. In perfectly symmetrical distributions you will also note that the mean and the median are both the same. You can have more than one mode also in a data set (bi-modal or even tri-modal). 

In the above data set however, both your mean and median are all 7. 

Data Set Two: 

4; 5; 6; 6; 6; 7; 7; 7; 7; 8; 

You will note that this is not a symmetrical distribution of data. Sketching a histogram will show you that the right hand seems chopped off compared to the left hand side, and we say this data set is skewed to the left because the data is pulled to the left when sketching the histogram. Also note in this example, that the mean is less than the median and they are both less than the mode. 

In the above set, the mean is 6.3, the median 6.5 and the mode is 7. 


Data Set Three: 

6; 7; 7; 7; 7; 8; 8; 8; 9; 10

In this particular data set, the mean is 7.7, the median 7.5 and the mode 7. This example is precisely descriptive of the problem you present because the mean is larger than the median, which is larger than the mode. 


To create a histogram, using a piece of paper place all values at equal distances along your x-axis. Identify the midpoint of each class interval (the mid point in each class. Remember: Class is referencing the particular data point such as 6, 7, 8, 9, 10. 

Draw a bar or column around each midpoint which represents the entire class interval. 

Continue drawing bars or columns until each of the frequencies for each of the class intervals is represented. 

Your y-axis should record the frequency of each value in those classes. In other words, how many times does that value appear in your data set. 


Solving problems in this manner, always provides you with a picture which can then be easily understood allowing you to describe that particular data set. 


I hope this information assists you, and I wish you good luck on your preparation for and the taking of your Final Exam. Please feel free to ask for clarification or make further points in the comment section below.