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The measures of central tendency enable us to make a statistical summary of the enormous organized data. One such method of measure of central tendency in statistics is the arithmetic mean. This condensation of a large amount of data into a single value is known as measures of central tendency.
For example, in the early morning while reading a newspaper, have you observed the daily temperature reports. Well, the temperature varies all day still how a single temperature can indicate the condition for the entire day? Or when you get your scorecard in exams, instead of analyzing your performance based on the percentage in all subjects, the performance is based upon the aggregate percentage.
The significance of indicating a single value for a large amount of data in real life makes it easy to study and analyze the collection of data and deduce important information out of it. Let us discuss the arithmetic mean in Statistics and examples in detail.
What is Arithmetic Mean in Statistics?
The most common measure of central tendency is the arithmetic mean. In layman’s terms, the mean of data indicates an average of the given collection of data. It is equal to the sum of all the values in the group of data divided by the total number of values.
For n values in a set of data namely as x1, x2, x3, … xn, the mean of data is given as:
It can also be denoted as:
For calculating the mean when the frequency of the observations is given, such that x1, x2, x3,… xn is the recorded observations, and f1, f2, f3 … fn is the respective frequencies of the observations then;
This can be expressed briefly as:
The above method of calculating the arithmetic mean is used when the data is ungrouped in nature. For calculating the mean of grouped data, we calculate the class mark. For this, the midpoints of the class intervals are calculated as:
After calculating the class mark, the mean is calculated as discussed earlier. This method of calculating the mean is known as the direct method.
What is Arithmetic Mean Formula?
To calculate the arithmetic mean of given observations, you just simply add all the given observations and divide the resultant sum by the total number of observations. The arithmetic mean formula to calculate the mean set of observations is given as:
Arithmetic Mean Formula
Arithmetic mean is the sum of all observations divided by a number of observations.
Arithmetic mean formula = {Sum of Observation}÷{Total numbers of Observations}
where i varies from 1 to n.
Mean Definition in Statistics
As we have understood about the arithmetic mean, now let us understand what does the mean stands for in statistics.
Mean is nothing but the average of the given values in a data set.
Mean = Sum of given values/Total number of values
Majorly the mean is defined for the average of the sample, whereas the average represents the sum of all the values divided by the number of values. But logically both mean and average is same.
For example, find the mean of given values: 2,3,4,5,6,6,
Mean = (2+3+4+5+6+6)/6 = 26/6 = 13/3
Examples of Arithmetic Mean in Statistics
Let us look into an example to understand this clearly.
Example 1:
In a class of 30 students, marks obtained by students in mathematics out of 50 is tabulated below. Calculate the mean of the data.
Solution:
The mean of the data given above is,
Thus, the mean of the given data is 34.
Example 2:
Calculate the arithmetic mean of the first 7 natural numbers.
Solution:
We know that the first 7 natural numbers are 1, 2, 3, 4, 5, 6, 7.
We know that,
Arithmetic Mean = Sum of all values / Total number of values.
Hence, the arithmetic mean of first 7 natural numbers = Sum of first 7 natural numbers/Total number of natural numbers.
Arithmetic Mean = (1+2+3+4+5+6+7)/7
AM = 28/7
AM = 4
Therefore, the arithmetic mean of the first 7 natural numbers is 4.
Example 3:
Determine the mean of the first 5 prime numbers.
Solution:
The first 5 prime numbers are 2, 3, 5, 7 and 11.
Hence, the mean of the first 5 prime numbers is calculated as follows:
Mean = Sum of first 5 prime numbers/Total number of prime numbers
Mean = (2+3+5+7+11)/5
Mean = 28/5 = 5.6.
Therefore, the mean of the first 5 prime numbers is 5.6.
Examples on Arithmetic Mean Formula
Example 1: The marks obtained by 8 students in a class test are 10, 19, 12, 21, 18, 20, 11, and 19. What is the arithmetic mean of the marks obtained by the students?
Solution:
To find: Arithmetic mean of the marks obtained by the students
Using the arithmetic mean formula,
Arithmetic mean = {Sum of Observation} ÷ {Total numbers of Observations}
Arithmetic mean = (10 + 19 + 12 + 21 + 18 + 20 + 11 + 19) ÷ 8
= 16.25
Answer: Arithmetic mean of the marks obtained by the students = 16.25.
Example 2: The heights of five students are 164 cm, 134 cm, 155 cm, 156 cm, and,172 cm respectively. Find the mean height of the students.
Solution:
To find: Mean height of the students
Using the arithmetic mean formula,
Arithmetic mean = {Sum of Observation}/{Total numbers of Observations}
Arithmetic mean = (164 + 134 + 155 + 156 + 172)/5
= 781/5 = 156.2 cm
Answer: Mean height of the students = 156.2 cm.
Example 3: The mean monthly salary of 5 workers of a group is $1400. One more worker whose monthly salary is $1550 has joined the group. Find the arithmetic mean of the monthly salary of 6 workers of the group.
Solution: Here, n = 5, x̄ =1400
Using the arithmetic mean formula,
x̄ = ∑xi/n
∴∑xi = x̄ × n
∑xi = 1400 × 5 = 7000
Total salary of 5 workers = $7000
Total salary of 6 workers = $7000 + 1550 = $8550
Average salary of 6 workers = 8550/6 = 1425
Answer: ∴ Average monthly salary of 6 workers = $1425
Frequently Asked Questions on Arithmetic Mean in Statistics
Q1
What is meant by Arithmetic Mean?
In Mathematics and Statistics, the Arithmetic Mean (AM) or Mean or Average is defined as the sum of all observations in the given data set divided by the total number of observations in the dataset.
Q2
What is the formula to calculate arithmetic mean?
The formula to calculate the arithmetic mean is:
Arithmetic Mean, AM = Sum of all Observations/Total Number of Observations.
Q3
How to calculate the arithmetic mean between two numbers?
The steps to calculate the arithmetic mean between 2 numbers are:
Step 1: Add the given two numbers.
Step 2: Divide the sum by 2.
Q4
What is the arithmetic mean between 2 and 6?
The arithmetic mean between 2 and 6 is 4.
(i.e) AM = (2+6)/2 = 8/2 = 4.
Q5
What is the arithmetic mean between 10 and 24?
The arithmetic mean between 10 and 24 is 17.
(i.e) AM = (10+24)/2 = 34/2 = 17.
FAQs on Arithmetic Mean Formula
How To Calculate the Arithmetic Mean Using Arithmetic Mean Formula?
If the set of ‘n’ number of observations is given then the arithmetic mean can be easily calculated by using a general arithmetic mean formula that is, Arithmetic Mean = {Sum of Observation} ÷ {Total numbers of Observations}.
How To Use the Arithmetic Mean Formula?
The general arithmetic mean formula is mathematically expressed as Arithmetic Mean = {Sum of Observation} ÷ {Total numbers of Observations}. Let us consider an example to understand how to use arithmetic mean formula.
Example: Find the arithmetic mean of (1, 2, 3, 4, 5).
Solution: Total number of observation = 5
Arithmetic mean formula = {Sum of Observation} ÷ {Total numbers of Observations}
Arithmetic Mean = (1 + 2 + 3 + 4 + 5) ÷ 5 = 15/5 = 3
Arithmetic Mean of (1, 2, 3, 4, 5) is 3
What Will Be the Arithmetic Mean Formula for n Observations?
Arithmetic mean formula for ‘n’ observations is expressed as, Arithmetic mean of n observations = {Sum of ‘n’ Observation} ÷ {Total numbers of ‘n’ Observations}
MATHS Related Links
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✅ Mean Deviation Formula ⭐️⭐️⭐️⭐️⭐️
✅ Mean Value Theorem Formula ⭐️⭐️⭐️⭐️⭐️
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