[Maths Class Notes] on Difference Between Mean and Median Pdf for Exam

Mean and Median hold a pivotal point in mathematics. One can’t think of any aspect of mathematics without these two terms. It becomes pertinent to understand the concept of Mean and Median and the differences. 

 

One can’t measure and compare data if it gets bigger ie., a comparison between the average heights of two cities. This data needs to be categorized and the use of Mean and Median comes in handy here.

 

An average of the data can be called a certain value that represents the whole data which signifies its characteristics.

 

There are three types of averages useful for the analysis of data.

 

They are:

1)Mean

2)Median

3)Mode

We have outlined major differences between Mean and Median for you to understand them easily with examples.

 

Mean and Median

Before we learn about the mean and median difference we must know what is mean and median.

 

What is Mean?

Mean can simply be defined as the result of the sum total of all the data values divided by the number of data values taken. There are several kinds of means and we have different methods and formulas for their calculations. The arithmetic mean is the most common type of mean.

Arithmetic mean (x̅) =sum of all individual data

 

The number of individual data x̅ is used to denote the arithmetic mean.

 

For example, let us consider a sample set having n  items x1,x2,……,xn then the arithmetic mean or simply mean can be calculated by the formula,

      

                   x̅=  (x1+x2+………..+xn) ÷ n

 

where n is the items in the sample.

 

Example:

Find the mean of the following sample: 3,7,2,18,21,9

 

Solution:

 

We add all the values and divide them by the number of values I.e. 6

 

x̅ = 3+7+2+18+21+9

 

6

3+7+2+18+21+96

 

x̅=10

 

Therefore,mean is 10.

 

Arithmetic Mean:

Mean= ∑x

 

N

 

∑xn

 

where Ʃ = Greek letter sigma, denotes ‘sum of..’

 

n = number of values

 

For Discrete Series:

Mean= ∑fx

 

N

 

∑fxN

 

meanwhile, f = frequency

 

For Continuous Series: 

Continuous series means where d =A+(Ʃfd/N+C)

 

where d= (X-A)/C

 

A =  Assumed Mean

 

C =  Common divisor

 

Different Kinds of Mean are:

1) Arithmetic Mean

2) Geometric Mean

3) Harmonic Mean

4) Quadratic Mean etc.

 

What is meant by Median?

The middlemost number in the set is the median of the set or one can also say that the number halfway into the set is the median. Median divides the set into two half sets namely-upper half set and a lower half set. To find the median, the data should first be arranged in order from least to greatest I.e. in ascending order, and then we find the middle value from the center of the distribution. This condition is suitable when we have an odd number of items, we can simply pick the middlemost item, but with an even number of items, things are slightly different.

 

In this case, we find the middle pair of numbers or items and then find the value that is halfway between them. It’s simple. Add them together and divide by two.

 

Finding out Median

Case1: When the number of values is odd.

 

Example:

 

Find the median of 3,8,1.

 

Solution: First arrange them in ascending order

 

1,3,8

 

Now pick the middle number i.e. 3

 

Hence, the median is 3.

 

Case 2:When the number of values is even

 

Example: Find the median of 4,73,45,83,2,3,9,65

 

Solution:Arranging the values in ascending order

 

2,3,4,9,45,65,73,83

 

Now there are eight numbers and so we don’t have just one middle number, we have a pair of middle numbers i.e. 9 & 45

 

To find the value halfway between them, add them and divide by 2:

 

9+45=54

 

 then 54÷2 =27

 

So the median is 27.

 

If the number of observations is odd:  

 

Median={(n+1)/2}th term

 

where n= number of observations

 

If the number of observations is even:

 

Median=

 

(n/2)th term+(n+1)/2th term

(n/2)th term+(n+1)/2th term/2

 

For continuous series:  

 

Median=l+

 

(N/2)−c/f

 

(N/2)−c/f×h

 

where l = lower limit of the median class

 

c = cumulative frequency of the preceding median class

 

f = frequency of the median class

 

h = class width

 

Fun Fact:

A quick way to find the middlemost number or the median term-count is to add 1 to the total number of data values then divide by 2. The term obtained will be the median term of that set.

 

Example: There are 47 numbers

 

47 + 1 is 48, then divide by 2 and we get 24

 

So the median is the 24th number in the sorted list.

 

So now let us understand what is the difference between mean and median. The major differences between mean and median are given below:

 

Difference Between Mean and Median

 

So, these were the main differences between mean and median.

Practical Example of Finding the Mean and Median

To better understand the concept of the Mean, let’s take a look at a practical, but hypothetical, situation.

Governments generally collect data about various things for their census data. This could include numbers related to things like the age, genders, languages, religions, income level, etc. of people. However, with such a large amount of data, it is impossible to make sense of it without arranging it into a coherent form. This is where statistical analysis methods like finding the Mean and Median help.

Take a look at the following (hypothetical) data. In this example, we will look at income levels.

The income levels of one particular area in a city have been revealed to be as follows. A total of ten people living in the neighborhood have been surveyed to gather this data. All of the figures given are for the income earned per month in rupees.

Person A: 10,000

Person B: 25,000

Person C: 1,00,000

Person D: 25,000

Person E: 15,000

Person F: 2,00,000

Person G: 50,000

Person H: 30,000

Person I: 20,000

Person J: 5,00,000

To find the Mean of the above data, we need to add up the values and divide the result by the total number of values, which in this case is 10.

(10,000+25,000+1,00,000+25,000+15,000+2,00,000+50,000+30,000+20,000+5,00,000) ÷ 10

(9,75,000) ÷ 10 = 97,500

Therefore, the Mean Income of this neighborhood is Rs. 97,500.

Now, sometimes, the mean data might be a bit skewed if the values lean towards one side. For example, in the above example, you can see that most of the people earn less than Rs. 50,000 a month, yet the mean income levels show Rs. 97,500 because of a few people who earn much more. So in this case, it would also help to find out the median income because that might give more insight.

To find the median income, arrange the data in ascending order and then find the middle value.

10,000  15,000  20,000  25,000  25,000  30,000  50,000  1,00,000  2,00,000  5,00,000

The middle number here would be between the fifth and sixth number since there are an even amount of data values (10). The fifth number is 25,000 and the sixth number is 30,000. Therefore, the median number would be in the middle of these two, which is 27,500. Therefore, the median income in this neighborhood is Rs. 27,500.

Conclusion

Arithmetic mean or Mean is considered as the best measure of central tendency as it contains all the features of an ideal measure factoring in one drawback that the sampling fluctuations influence the mean.

Similarly, the median is also unambiguously defined and easy to understand and calculate, and the best thing about this measure is that it is not affected by sampling fluctuations, but the only disadvantage of the median is that it is not based on all observations.