How to Find Arithmetic Mean in Statistic

The term measures of central tendency is represented as a single value which is used to define a collection of data by arranging the central position within that set of data. It is also categorized as a statistical summary. It helps to make the statistical summary of large organized  data. The most common method of measure of central tendency in statistics is Arithmetic Mean.


For Example: Have you ever observed the daily temperature records while reading the newspaper early morning. As we know temperature varies throughout the day, yet how can a single temperature represent the weather condition of the whole day.? Or when students get a scorecard in the examination, the best way to calculate the students performance is to calculate the aggregate percentage of all the subjects.


The representation of large amounts of data in a single value makes it easy to understand and analyze the collection of data or to get the required information out of it. Let us now understand arithmetic mean in statistics, how to find the arithmetic mean in statistics, arithmetic mean examples, arithmetic formula etc.

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What is Arithmetic Mean in Statistics

Arithmetic Mean is the most common measurement of central tendency. According to the layman, the mean of data represents an average of the given collection of the data. It is equivalent to the sum of all the observations of a given data divided by the total number of observations.


The mean of data for n values in a set of data namely p1, p2, p3 ------pn   is given by:


\[\overline{p}\] = p1, p2, p3------pn / n


For calculating the arithmetic when the number of observation along with the frequency of observation is given such that p1, p2, p3 ------pn  are the recorded observation and f1,f2, f3------ fn are the corresponding frequencies of the observation the,


\[\overline{x}\] = f1p1,f2p2, f3p3 ------fn pn / f1, f2, f3…….fn


The above Arithmetic mean formula is expressed as 

Σ fixi / Σ fi

The above- given arithmetic mean formula is used to calculate mean when the data given is ungrouped. For calculating the mean of group data , we calculate the class marks. Midpoints of the classmark is computed as:


Midpoint = Upper Limit + Lower limit / 2


The method discussed above is the calculation of arithmetic mean by the direct method.


Arithmetic Mean Formulas in Statistics

Arithmetic mean formula in statistics for grouped data 

\[\overline{x}\] = Σ fixi / Σ fi

Here,

 

\[\overline{x}\]= Arithmetic mean

 f = Frequency

 X = variable

  Σ f = Sum of frequencies

Arithmetic mean formula in statistics for ungrouped data 


Midpoints of class interval = Upper Limit + Lower limit / 2


Arithmetic mean of a collection of numbers (from x1to xn) is derived by the formula

x =  1/n \[\sum_{i=1}^{n}\] xi  =  x1 + x2 + x3…. + x/ n


How to Calculate Arithmetic Mean in Statistics?

There are two steps to find the arithmetic mean in statistics: sum up all the numbers given in a set.

and then divide it by the total number of items in your set. The arithmetic mean is found similarly as a sample mean.

For example: Calculate the arithmetic mean for the average driving speed for one bus over a 5hours journey. 50 mph , 23mph, 60mph, 65mph, 30mph


Step 1: Addition all the numbers in a data set : 52 + 23 + 60 + 65 + 30

Step 2: Divide by the total number of items in a given data set. There are 5 numbers in a  above set

Accordingly,

= 52+ 23 + 60 + 65 + 30 / 5

= 230/5

= 46

Hence, the average driving speed of a bus is 46 mph.


Arithmetic Mean Example in Statistics

Let us understand the concept of arithmetic mean clearly through an example:


1. In a class of 30 Students , Marks Obtained by the Students in Science out of 50 are Given Below in Tabular Form. Calculate the Arithmetic Mean of the Data.

Marks obtained 

10-20

20-30

30-40

40-50

No.of students

5

5

8

12


Solution:

Market Obtained

Number of Students (x)

Midpoints

Class Marks (f)

fixi

10 -20

5

15

75

20 -30

5

25

125

30 - 40

8

35

280

40 - 50

12

45

540


∑ fi = 30


∑ fixi = 1020



Midpoint formula  : Upper Value  + lower Value  / 2


The arithmetic mean of above data is

\[\overline{x}\] = ∑ fixi/ ∑ fi = 1020/ 30 = 34


What is meant in Statistics with Example?

Mean is simply the average of the given set of values in a data set. The mean is represented by \[\overline{X}\].

Mean = Sum of the given values in a data set/ Total number of values

Generally, mean is defined for the average of the sample, whereas the average denotes the addition of all the values to the total number of values.. Logically, average and mean both are similar terms.

For example: Calculate the mean of the given values: 5,6,3,2,1,8

Mean = ( 5 + 6+ 3+ 2+ 1+ 8) / 5

= 25/5

= 5

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Solved Example

1. Calculate the Arithmetic Mean From the Following Data

Price(Rs)

15-18

18-21

21-24

24-27

27-30

30-33

33-36

Quantity Demanded

28

23

17

18

8

4

2


Solution:

Price( Rs)

Midpoint(m)

Frequency (f)

fm

15-18

16.5

28

462

18-21

19.5

23

448.5

21-24

22.5

17

382.5

24-27

25.5

18

459

27-30

28.5

8

228

30-33

31.5

4

126

33-36

34.5

2

69



∑f = 100

∑ fm = 2175


Midpoint point for 1st  class interval = 15 + 18 /2 = 16.5

\[\overline{x}\] = ∑ fm/ ∑f

= 2175/100

= Rs. 21.75


2. If the arithmetic mean of the 14 different observations are 26, 12, 15, x, 17, 9, 11, 18, 16, 28, 20, 22, 8, is 17. Find the missing observation.


Solution: 

14 observations are = 26,12,15,x,17,9,11,18,16,28,20,22,8,

Arithmetic mean = 17

We know that,

Arithmetic Mean= Sum of total observations /Total number of observations

17 = (216 + X)/ 14

17 X 14 = 216 + x

X = 238-216

x= 22

Hence, missing observation is 22.


3. The marks obtained by 7 students in science class tests are 20, 22, 24, 26, 28, 30, 10. Find the arithmetic mean.


Solution: 

\[\overline{x}\]= 20+ 22+ 24+ 26+ 28+ 30+ 10 / 7

= 22.8 

Hence, arithmetic mean of 7 students is 22.8


Quiz Time

1. What should be the  value of 'a', if the arithmetic mean between a and 10 is 30 

  1. 45

  2. 60

  3. 50

  4. 53

    

  2. The series obtained by adding the term of an arithmetic sequence is known as,

  1. Arithmetic Series

  2. Harmonic Series

  3. Geometric Series

  4. Infinite Series


   3. What will be the arithmetic mean between 1 + x +x and 1-x+x ?

  1. 1-x

  2. 1+ x

  3. 2- x

  4. 2 + x


Fun Facts

  • Tycho Brahe was the first to use the concept of the  arithmetic mean.

  • An Indian Mathematician and astronomer Brahmagupta is the father of the arithmetic mean.

  • The word arithmetic is derived from the Greek noun arithmos meaning "number".

FAQ (Frequently Asked Questions)

1. What are the Merits and Demerits of Arithmetic Mean?

Merits of Arithmetic Means

  • The calculation of arithmetic mean is easy as it requires basic knowledge of Maths such as addition, subtraction, multiplication, and division of numbers.

  • The meaning of the arithmetic mean can be easily understood such as the value or unit or cost per unit etc.

  • The value of the arithmetic mean is always definite as it is defined rigidly.

  • It can be widely used in advanced statistical analysis as it has competency for further algebraic operation.

  • A comparison of the data of two or more groups can be easily done through arithmetic mean.

  • All the values of data are taken into consideration while calculating arithmetic mean .


Demerits of Arithmetic Mean

  • The calculation of arithmetic mean is not possible if all the items of the series are not available.

  • The calculation of the arithmetic mean cannot be done just by observing the series such as median or mode.

  • Arithmetic means cannot be represented on graph paper.

  • It is not possible to calculate the  arithmetic mean for qualitative data such as intelligence, honesty, victory etc.

  • It is not possible to compute arithmetic mean in case of open- end class distribution as it cannot be calculated without making assumptions about the class size.

  • Sometimes arithmetic means given illogical results. For example- if teachers say the average number of girls in a class is 28.97, it sounds illogical.

2. What are Some Basic Properties of Arithmetic Mean?

Here are some of the basic arithmetic mean properties:

  1. If X̅ is considered as the mean of n observations x1, x2 ……xn , then the mean of x1-a, x2-a …xn– a is X̅ – a, where a is any real number.

  2. If X̅ is considered as the mean of n observations x1, x2 ……xn , then the mean of x1/a, x2/a …xn/a is X̅ / a, where a is any non-zero.

  3. If X̅ is considered as the mean of x1, x2 ……xn , then the mean of ax1, ax2…axn is a  X̅ where a is any number which is different from zero. If every observation is multiplied by a  non-zero a. Then the mean will also be multiplied by a

If X̅ is considered as the mean of x1, x2 ……xn , then prove that Σni=1( xi - X̅) is equal to zero.