Difference Between Mean and Median

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 handy here.


An average of the data can be called as 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 arithmetic mean.


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

      

                   x̅= \[\frac{x1+x2+...........+xn}{n}\]


where n is the items in the sample.

 

Example- 

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


Solution:

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


x̅ = \[\frac{3+7+2+18+21+9}{6}\]

x̅=10

Therefore,mean is 10.


Arithmetic Mean:

Mean=\[\frac{\sum x}{n}\]


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

n = number of values


For Discrete Series:

Mean=\[\frac{\sum fx}{N}\]

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 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 number of values are 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)/2}th term]/2


For continuous series:  

Median=l+[{(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


MEAN

MEDIAN

  • The average arithmetic of a number is called mean.

  • Median is the middlemost number which separates the upper half sample and lower half sample usually of a probability distribution.

  • Mean is applicable for normal distribution.

  • For a skewed distribution the concept of median is applied.

  • Mean is calculated by adding up the individual entities and then dividing the sum by the number of entities.

  • Median is calculated by arranging the given sample in ascending or descending order and then finding the middlemost entity.

  • This type of average is the arithmetic average.

  • This type of average is a positional average.

  • It is sensitive to outliers.

  • It is insensitive to outliers.

  • It represents the center of gravity of a data set.

  • It represents the center of gravity of the midpoint of the data set. 

  • It takes into account every value of the data set.

  • It does not take into account every value of the data set.

  • It is affected by extreme scores.

  • It is not affected by extreme scores.

  • Its application is limited by many external factors.

  • It is much more robust and reliable to measure even for uneven data.


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


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 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.

FAQ (Frequently Asked Questions)

1) Which is More Accurate: Mean or Median?

Answer- Mean and median both are two types of averages. While mean gives the centralized tendency of a given sample, the median gives the middlemost term or value in a distribution. Both have different ways of calculation but they both might lead to the same answer. Both mean and median are accurate to a level but when we try to find which is more accurate and reliable we need to see the type of data.  This depends on the type of data we are working on or the shape of the underlying distribution.

2) What are the Uses of the Median?

Answer-The median is a good measure of the average value when the data include exceptionally high or low values because these have little influence on the outcome. 


It is actively used in reporting incomes. The median income in an area tells you more what the "average" person earns. The median is simply the point where 50% of the data is above and 50% is below. It's a good, intuitive metric of centrality that is good at representing a "typical" or "middle" value. It also is the point that minimizes the distance from it to any of the other points in the dataset.