Arithmetic Mean Statistics

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Statistics, by its simplest understanding, is the analysis that involves collection, review, and the inference to be drawn from data. While, there is usually a large volume of data involved in this academic discipline, the concept of central tendency deviates from it. 


Central tendency focuses on a solitary value for the description of a given set of data. Such function is undertaken with the identification of central position located in the provided data set. There are three ways to measure central tendency – Mean, Median and Mode. It is an arithmetic mean statistics that are being elaborated further.  


What is Understood by Arithmetic Mean Statistics? 

Definition of arithmetic mean in Statistics simply covers the measurement of average. It involves the addition of a collective of numbers. The resulting sum is further divided with the count of numbers that are present in a given series. 


Simple arithmetic mean formula can be understood from the following example –

Say, within a series the numbers are – 36, 46, 58, and 80. The sum is 220. To arrive at arithmetic mean, the sum has to be divided by the count of numbers within the series. Hence, 220 is divided by 4, and the mean comes out to be 55. 


Arithmetic mean statistics includes the formula –

\[\bar{X}\] = \[\frac{(x_{1}+x_{2}+.....+x{n})}{n}\] = \[\frac{\sum_{i=1}^{n}xi}{n}\]

In the above equation,

X̄ = arithmetic mean symbol ___________________ (a)

X1,…,Xn =  mean of ‘n’ number of observations _____ (b)

∑ = summation ______________________________ ©


Concept of Arithmetic Mean Median Mode 

Even though arithmetic mean statistics has been elaborated, it can be better understood in the context of median and mode as well. 

Within a given data set –

  • average of data is mean;

  • most frequently occurring data is mode; and

  • the middle unit within the data set is median

Mean of a data set can comprise of several different series – (1) Individual, (2) Discrete, (3) Continuous, (4) Direct. On the other hand, for calculating the median, the data set has to be arranged in descending or ascending order. Mode covers such data which occurs the most number of times within a given series. The mode formula may be applicable in case of discrete, individual and continuous series. 


Finding Arithmetic Mean

Following example illustrates the application of arithmetic mean formula.

  • In a team comprising of 30 participants, scores achieved in an activity on the aggregate of 50 are indicated below. Find out the arithmetic mean of a given data set.  


Scores obtained

Number of Participants

10 - 20

5

20 - 30

5

30 - 40

8

40 - 50

12


The Arithmetic Mean Formula in Statistics is – 

\[\bar{X}\] = \[\frac{(x_{1}+x_{2}+.....+x{n})}{n}\] = \[\frac{\sum_{i=1}^{n}xi}{n}\]

In the first two steps, midpoints of values (f) and aggregate of such values (fi xi) have to be found out. 

Midpoint = (upper value) + (lower value) / 2


Scores Obtained

Number of Participants (x)

Midpoints of Scores (f)

(fi xi)

10 - 20

5

(20 + 10)/2 = 15

(15 X 5) = 75

20 - 30

5

(30 + 20)/2 = 25

(25 X 5) = 125

30 - 40

8

(40 + 30)/2 = 35

(35 X 8) = 280

40 - 50

12

(50 + 40)/2 = 45

(45 X 12) = 540


From the above table, it can be derived – 

∑ fi = 30  ………………………………… (i)

∑ fixi = 1020 …………………………… (ii)  

Therefore, the arithmetic means of given data amounts to –

X̄ = ∑ fixi / ∑ fi

= 1020/30

= 34

Arithmetic mean statistics can be a complex concept to grasp. You can have all your doubts clarified in Vedantu’s online classes. To find out more about this, download the app today!

FAQs (Frequently Asked Questions)

1. What is the Formula to Find Arithmetic Mean?

Ans. The arithmetic mean equation is –

X̄ = (x₁ + x₂ + …… + xₙ)/n = ∑ⁿᵢ₌₁ xi / n

In the above equation, x̄ denotes arithmetic mean, and the observations are indicated till n. The sigma symbol represents summation. The entire equations show that arithmetic mean statistics are arrived at by dividing the sum of observation with the number of observations.  

2. What is the Difference Between the Arithmetic Mean and Median?

Ans. The difference between the arithmetic mean and median is mostly in terms of applicability. Mean represents the average of a data set in measuring the central tendency. It is applicable for regular distributions. 


On the other hand, median functions to separate the higher and lower half of a sample, probability distribution or population. It is usually applied in case of skewed distributions. 

3. What is Understood By the Central Tendency in Statistics?

Ans. The central tendency in statistics is a representation of the middle value in a given data set. Such value identifies with the characteristics shown by units present in the rest of the distribution. The objective is to accurately describe the whole data set. It can be measured by mean, median and mode.

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