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# Mean, Median and Mode

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Last updated date: 11th Aug 2024
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## Introduction to Mean, Median, Mode

Mean, median and mode are some of the measures of central tendency. These are three different properties of data sets that can give us useful, easy to understand information about a data set to see the big picture and understand what the data means about the world in which we live.

### Mean

“Mean” and “average” are just two different terms for the same property of a data set. It is also known as the arithmetic mean. The mean or average is beneficial to property and one of the most significant, easy and most used calculations out of all the three central tendencies. The mean is basically the summation of all the values in the set of data after it is divided by the total number of values in the set of the data.

There are three methods of taking out averages – or mean in this case – and they are: direct method, assumed mean approach and step deviation method.

The above definition is of Arithmetic Mean, one of the many types of Mean. In detail, the types of mean are explained although most of them are out of scope for elementary Statistics

1. Arithmetic Mean

Arithmetic Mean is the average of all the observations. Generally, if the mean is mentioned without any adjective, it is assumed to be Arithmetic Mean.

Example- We have a set of observations-x=1,3,5,7,91,3,5,7,91,3,5,7,9. The Arithmetic Mean is computed as (x/n) where n is the number of observations which is equal to 5 in this case. Thus x=25 in this case and n=5  so the mean comes out to be 5

1. Weighted Mean

Weighted mean is almost the same as Arithmetic Mean, the difference being that in weighted Mean, some values contribute more than the others. 2 Cases arise while calculating Weighted Mean. The weighted mean is useful in situations when one observation is more important than others.

Case 1- When the sum of weights is 1- Simply multiply each weight by its corresponding value and sum it all up.

Example- In the previous example, let us assume that w=0.2 for all the observations, then the weighted mean is- W_mean= (0.2*1)+(0.2*3)+(0.2*5)+(0.2*7)+(0.2*9)=5 which is the same as Arithmetic Mean but if we change the weights then the mean also changes.

Case 2- When the sum of weights is not equal to 1- In this case it is beneficial to make a table that shows each weight against each observation. Then calculate the product of each observation and its corresponding weight.

1. Harmonic Mean

Harmonic Mean is calculated by dividing the total number of observations by the reciprocal of each observation. It is quite useful in Physics and has many other applications

(example- average speed when the duration of several trips is known).

It is given by the formula- $H.M= \frac{ n}{(1/x1)+(1/x2)+(1/x3)+.....(1/xn)}$

1. Geometric Mean

The Geometric Mean indicates the central tendency using the product of the observations rather than their sum(which is used in calculating Arithmetic Mean). It is used in the field of finance and social sciences. In finance, it is used to calculate the average growth rates. The Geometric Mean is most useful when the observations are dependent on each other or they have large fluctuations. It is given by(INSERT EQUATION)

### Solved Example of Mean

1. Find the mean for the following frequency table:

 x f 1 5 20 9 25 8 30 1 40 10 50 7

Solution :

Arithmetic mean  =  $\Sigma \frac{fx}{N}$

 x f fx 1 5 5 20 9 180 25 8 200 30 1 30 40 10 400 50 7 350 N = 40 $\Sigma fx = 1165$

Arithmetic mean  =  $\Sigma \frac{fx}{N}$  =  1165 / 40

=  29.125

Hence the required arithmetic mean for the given data is 29.125.

### Median

As the name suggests, the median is nothing but the middle – or “mid” – of all the values presented in the data set. This shows what the middle of the data is. For example: in a data set of 5, 10, 15, 20, 25, 15 is the median.

There are two different methods of finding out the mean. They are the odd number of values and even numbers of values.

### Solved Example of Median

1. Find the median for the following frequency table:

 x f 1 5 20 9 25 8 30 1 40 10 50 7

Solution:

 x f Cumulative Frequency 1 5 20 9 5 25 8 5+9=14 30 1 14 + 8  =  22 40 10 22 + 1  =  23 50 7 23 + 10  =  33 33 + 7 =  40

Here, the total frequency, N = $\Sigma f$ = 40

N/2  =  40 / 2  =  20

The median is (N/2)th value = 20th value.

Now, the 20th value happens in the cumulative frequency 22, whose corresponding x value is 25.

Hence, the median = 25.

### Mode

is defined as the value that is found mostly in a data set. When the frequencies in the data keep repeating, the mode takes place. This is mainly used for taking out most of the averages. For example, if you want to calculate the average of how many students scored the most, you might want to use the mode.

### Solved Example of Mode

1. Find the mode for the following frequency table:

 x f 1 5 20 9 25 8 30 1 40 10 50 7

By observing the given data set, the number 40 occurs more often. That is 10 times.

Hence the mode is 40.

Mean  =  29.125

Mode  =  25 and

Mode  =  40.

## FAQs on Mean, Median and Mode

1. What is the best way to study for mean, median and mode?

Mean, median and mode are some of the calculation methods that students can use in examinations as well as in daily life to make calculations and results easy. The best way to study mean, mode and median are to make tabular forms with the three as the headings and write down the characteristics that differ from each other. Then, students can try to make formulas that fit the best in the table, according to their understanding. To get a mastery over the topic, the students can practice questions relating to mean, median and mode, which can be found on the official website of Vedantu.

2. How can I study mean better?

Mean is a type of average that can be better understood when the students try to practice it vigorously. Since mean is one of the most common forms of calculation in statistics, it is used in management by employers and for many daily life practices such as finding scores in matches of sports and also marks in a test. The best way to study the mean is to practice the sample questions regarding the topic which is mean. These questions are available on the official website of Vedantu.

3. How can I study the median better?

Practicing the questions makes you work your memory and the concept you know about the topic Mean already and what you need to know. It is better to take sample questions that are already solved but they should be revealed before the students attempt the questions. This helps in confirming whether or not the students' answers are correct. The sample question forms which have the answers solved and explained in detail are compiled in PDfs. These pdfs are available inaccessible and downloaded ready at the exposure for the students.

4. How can I study mode better?

Since mode happens to be the easiest form of calculation among all the three measures of central tendencies, students can practice the sample questions of mode provided along with answers that are explained step by step and in order. These answers are compiled neatly in PDFs that are created by high-quality experts of the subject matter. These answers can be easily accessed and downloaded from the official website of Vedantu.

5. How are mean, median, the mode used and applied?

Mean, median, and mode are the type of calculations that can be used in the daily life of a student. There are many professions that have the use of the statistics from measures of central tendency i.e mean, median and mode. These methods can be used in businesses, management, and other branches such as science, physics, and mathematics. They can also be used in several other daily life actions such as rankings and scores in sports such as cricket and marks in tests.