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 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.
FAQs on Difference Between Mean and Median for JEE Main 2024
1. Which is More Accurate: Mean or Median?
Mean and median both are two types of averages. While the 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?
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.
3. What exactly is Mean?
The Mean is a statistical term used to describe the average of a given set of data. There are different kinds of means, however, the most common is the arithmetic mean. The arithmetic mean is obtained by first taking all of the values in a given data set and adding them up. Then you divide that number by the total number of values.
For example, if you have census data of the ages of people in a locality and you need to find the average age, this is how you do it.
Let's say that the ages of the people in the locality are as follows:
11, 16, 45, 23, 34, 12, 33, 44, 21, 27.
First, you add those numbers up. That will give you 11+16+45+23+34+12+33+44+21+27 = 266. Now divide 266 by the total number of values, which is 10 (there are ten people surveyed so ten values).
266÷10 = 26.6
So the mean age is 26.6
4. What exactly is Median?
The median is simply the middle number of a given set of data. It is the number right in the middle that gives a vague idea of the numbers present in the data set.
To find the median, you first need to arrange the data in ascending order, that is, from the lowest number to the highest. This is done to accurately find the middle number without scattered data interfering with the result.
Let's say that you have a government survey that has collected the ages of 11 people from one neighborhood and you need to find the median age. The ages of the people are:
12, 45, 34, 33, 56, 25, 15, 66, 23, 25, 22
First arrange these numbers in ascending order, which will give you:
12, 15, 22, 23, 25, 25, 33, 34, 45, 56, 66
Since there are 11 values in total, you need to find the middle one which is the sixth value. In this example, the sixth value is 25. Therefore, the median age is 25.
5. Where can I find out more about Mean and Median?
To find out more about Median and Mean, you can browse through the Vedantu website or app. Vedantu has a large number of resources available for free for students to refer to while they study. These resources can be downloaded for free in PDF format as long as you have a Vedantu account.
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