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Arithmetic mean is the simplest and most popular way to measure an average. Calculating it is simple for anyone with even a basic understanding of mathematics and finance.
- Arithmetic computations are made by adding all the numbers together, then dividing that amount by the total number of numbers in the series.
- The arithmetic mean, one of the most commonly used metrics of central tendency, is represented by the letter x.
- Arithmetic mean has the propensity to produce useful conclusions even with a massive grouping of numbers, which makes it a valuable measure of central tendency.
Read More: Degree of Polynomial
Key Terms: Arithmetic mean, Data, Observations, Frequency, Weighted arithmetic mean, Simple arithmetic mean.
What is Arithmetic Mean?
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The mean or arithmetic average are other names for the arithmetic mean.
- It is determined by adding up each number in a particular data set, then dividing the result by the overall number of items in the data set.
- For uniformly distributed integers, the middle number serves as the arithmetic mean (AM).
- Additionally, the AM is estimated using a variety of techniques dependent on the volume and distribution of the data.
Read More: Difference Between Variance and Standard Deviation
Arithmetic Mean Formula
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Let n be the operation's total number of observations, and n1, n2, n3,..., nn be the supplied numbers. According to the definition, arithmetic means the formula is the division of the total sum of the group's numbers by the total number of items.
A.M. = (n1 + n2 + n3 + n4 + … + nn)/n
So, the formula of arithmetic mean will be
A.M = \(\frac{1}{n} \sum ^n _{i=1} a_i\)
Read More: Dispersion
Properties of Arithmetic Mean
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Following are the important characteristics of Arithmetic Mean:
- The mean of n observations x1, x2, x3, x4,…..xn. is x̄. If each observation is increased by y, the mean of the new observations is (x̄+y).
- The mean of n observations x1, x2, x3, x4,…..xn. is x̄. If each observation is decreased by y, the mean of the new observations is (x̄–y).
- The mean of n observations x1, x2, x3, x4,…..xn. is x̄. If each observation is multiplied by a non-zero number y, the mean of the new observations is (y×x̄).
- The mean of n observations x1, x2, x3, x4,…..xn. is x̄. If each observation is divided by a non-zero number y, the mean of the new observations is (x̄y).
- If all of the observations in the provided data set have a value, let's say 'y', then 'y' is also the value of their arithmetic mean.
Read More: Variance
How to Calculate Arithmetic Mean for Ungrouped Data?
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By adding up all the values in a data set and dividing it by the number of values, it is easy to determine the arithmetic mean for ungrouped data.
Mean, x̄ = Sum of all values/Number of values
Example: Find the arithmetic mean of 7, 8, 10, 18, 20.
Solution: Given, 7, 8, 10, 18, 20 is the set of values.
The sum of values = 7+ 8+10+18+20 = 63
Number of values = 5
Mean = 63/5 = 12.6
If x1, x2, x3,……,xn be the observations with the frequencies f1, f2, f3,……,fn, then the arithmetic mean is given by:
x̄ = (x1f1 + x2f2 +……+ xnfn) / ∑fi
where ∑fi is the summation of all the frequencies.
Read More: Measures of Dispersion
How does Arithmetic Mean work in the field of Finance?
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The arithmetic mean is widely used in various aspects of finance.
- For instance, the average salary is just an arithmetic mean.
- Let's say one wants to know what the 12 analysts' average profit projections are for a particular stock.
- To calculate the arithmetic mean, all the approximations are added up and divided by 12.
- The average closing price of a stock for a specific month can also be found.
- Let's say that a month has 22 trading days.
- To find the arithmetic mean, all the prices are added up and then divided by 22 to get the arithmetic mean.
Read More: Statistics Formula
Advantages and Limitations of using Arithmetic Mean
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Following are the advantages of using Arithmetic Mean:
- Simplicity - The arithmetic mean is used most frequently since it is the easiest to compute and comprehend.
- Rigid Formula - The arithmetic mean's value is fixed because it is determined by a strict formula.
- Facilitates Further Study - Arithmetic mean can be utilized for further research, particularly for statistical analysis and algebraic calculations.
- Minimum Fluctuation - Variation in the sample has less of an impact on the arithmetic mean.
- No Need for Data Arrangement - Unlike other measures of central tendency, the arithmetic mean does not need the arranging and division of data (i.e. median and mode).
- Depends on Observation - The mathematical mean depicts the data rather than the order of terms because it is entirely dependent on observation.
Following are the limitations of using Arithmetic Mean:
- If any observational component is absent, the arithmetic mean cannot be calculated.
- The extreme values of the series have a significant impact on the arithmetic mean.
- Qualitative data cannot be analyzed using the arithmetic mean.
- The arithmetic mean cannot be determined when there are open-ended class intervals.
Arithmetic Range
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The term "range" refers to the distinction between the largest and smallest value of data. This aids in identifying the data's distribution range. For instance, a batch of ten students recently completed a test worth 100 points. Here, there are two possibilities.
First: 60, 53, 70, 61, 48, 93, 80, 72, 91, 50
Second: 71, 72, 70, 75, 73, 74, 75, 70, 74, 72
The difference between the biggest value, 93, and the smallest value, 48, in the first scenario serves as a representation of the range.
Range in the first set = 93 – 48 = 45
The difference between the two values—75 for the highest number and 70 for the lowest—represents the range in the second scenario.
Range in second set = 75 – 70 = 5
One may estimate the range over which the values are scattered by comparing the values of range in the two scenarios; the wider the range, the further apart the values are spread. This provides us with additional information that would otherwise not be transmitted.
Read More:
| Related Articles | ||
|---|---|---|
| Arithmetic Mean Formula | Difference between mean and median | |
| Mean, Median and Mode | Direct, Assumed Mean and Step-deviation Methods | |
Things to Remember
- The simplest and most popular way to measure a mean or average is the arithmetic mean. It simply entails adding up a collection of numbers, then dividing that total by the total number of numbers in the series.
- There are two types of Arithmetic Mean,
- Simple Arithmetic Mean.
- Weighted Arithmetic Mean.
- The standard deviation is primarily used to measure the variability of data and is usually used to determine the stock's volatility.
- The mean of a set of two or more numbers is essentially their average.
- Mean is essentially the data's simple average.
- A stock's volatility is gauged using the standard deviation.
- As it takes into account every observation in a set of data and has a very straightforward application compared to other measures of central tendency, the arithmetic mean is the most widely used measure of central tendency.
Previous Year Questions
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Sample Questions
Ques. Calculate the arithmetic mean of the first five prime numbers. (2 Marks)
Ans. The first five prime numbers are 2, 3, 5, 7, and 11. The mean or the average formula is = (2 + 3 + 5 + 7 + 11) / 5 = 5.6
Ques. Maria scored 40, 73, 68, 50, and 54 in 5 subjects respectively. Calculate the Arithmetic mean. (2 Marks)
Ans. Marks obtained by Maria in 5 Test Subjects are:40, 73, 68, 50 and 54
Thus, Mean = Total Marks / Number of subjects
Total Marks = 40 + 73 + 68 + 50 + 54 = 285
Number of Subjects = 5
Mean = 285/5 = 57
Ques. Determine the “k” value if the arithmetic mean of 9, 8, 10, k, 12 is 15. (3 Marks)
Ans. Given that, the mean of 5 numbers is 15.
i.e. (9 + 8 + 10 + k + 12)/5 = 15
Hence, we get
(39 + k)/5 = 15
Now, simplify the above equation, and we get
⇒ 39 + k = 15 × 5
⇒ 39 + k = 75
Now, subtract 39 on both sides, and we get
⇒ 39 – 39 + k = 75 – 39
⇒ k = 36
Hence, the value of k is 36.
Ques. The mean of 14 numbers is 6. What will be the new mean if 3 is added to every number? (4 Marks)
Ans. Assume that the given numbers be a1, a2, a3, ….. a14.
Therefore, the mean of these numbers = (a1 + a2 + a3+ ….. a14)/14
Therefore, (a1 + a2 + a3+ ….. a14)/14 = 6
The above equation can be written as:
⇒ a1 + a2 + a3 + ….. a14 = 84 …(1)
To find: New mean if 3 is added to every number.
Therefore, the new numbers are (a1 + 3), (a2 + 3), (a3 + 3), ….,(a14 + 3)
Mean of the new numbers = (a1 + 3) + (a2 + 3)+(a3 + 3)+ …. +(a14 + 3)/14
The above form can be written as
Mean of new numbers = [(a1 + a2 + a3+ ….. a14) + 42]/14
Now, use the equation (1), and we get
Mean of new numbers = (84 + 42)/14
Simplifying the values, we get
= 126/14
= 9
Therefore, the new mean obtained is 9.
Ques. The arithmetic mean of 40 numbers was found to be 38. It was then discovered that the number 56 had been misinterpreted as 36. Determine the correct mean of the numbers given. (2 Marks)
Ans. The calculated arithmetic mean of 40 numbers is 38.
Hence, the sum of these numbers = (38 × 40) = 1520.
Thus, the actual sum of these numbers = [1520 – (Incorrect value) + (correct value)]
Correct Mean = 1520 – 36 + 56
Correct Mean = 1540.
Hence, the correct mean of the given numbers = 1540/40 = 38.5.
Ques. The arithmetic mean of the first five numbers is 28. The mean is reduced by two when one of the numbers is removed. Find the number that isn’t included. (3 Marks)
Ans. Given that, the arithmetic mean of 5 numbers = 28.
Thus, the sum of these 5 numbers = (28 x 5) = 140.
Also, given that the mean is reduced by two when one of the numbers is removed.
i.e. The mean of the remaining 4 numbers = (28 – 2) =26.
Therefore, the sum of these remaining 4 numbers = (26 × 4) = 104.
Thus, the number that is not included = (sum of the given 5 numbers) – (sum of the remaining 4 numbers)
= 140 – 104
= 36
Hence, the excluded number is 36.
Ques. P and Q have an average monthly salary of Rs. 5050. Q and R have an average monthly income of Rs. 6250, while P and R have an average monthly income of Rs. 5200. Find the monthly salary of P. (3 Marks)
Ans. Assume that A, B and C represent the monthly income of P, Q, and R, respectively. So, we have
A + B = (5050×2) = 10100….(1)
B + C = (6250×2) = 12500…..(2)
A + C = (5200×2) = 10400….(3)
Now, add the equations 1, 2 and 3:
2(A+B+C) = 33000
The above equation can also be written as
A + B + C = 33000/2 = 16500…(4)
Now, subtract (2) from (4), we get A = 4000.
Therefore, P’s monthly income is Rs.4000.
Ques. A class of 16 boys has an average weight of 50.25 kg, while the remaining 8 boys have an average weight of 45.15 kg. Calculate the average weight of all boys in the class. (3 Marks)
Ans. 16 boys in a class have an average weight of 50.25kg
8 boys in a class have an average weight of 45.15kg
Thus, the required average = [(50.25×16) + (45.15×8)] / (16 + 8)
= ( 804+361.20) / 24
∴ Required average = 1165.20 / 24
= 48.55 kg
Therefore, the average weight of all boys in the class is 48.55 kg.
Ques. The marks of a student were entered incorrectly as 83 instead of 63. As a result, the class’s average marks increased by half (1/2). Find the number of students in the class. (3 Marks)
Ans. Let’s say the class has x students.
As a result, the overall increase in mark = (x × 1/2) = x/2.
x/2 = 83 – 63
x/2 = 20
x = 40.
Hence, the number of students in the class is 40.
Ques. 36 is the arithmetic mean of 25 observations. Find the 13th observation if the mean of the first observation is 32 and the final 13 observations are 39. (3 Marks)
Ans. The first 13 observations have a mean of 32.
The sum of the first 13 observations is 416, i.e. 32 × 13 = 416.
The average of the last 13 observations is 39.
The sum of the last 13 observations is 507, i.e. 39 × 13 = 507
The average of the 25 observations is 36.
The total of all 25 observations is (36 × 25) = 900.
As a result, the thirteenth observation equals (416 + 507 – 900) = 23.
Therefore, the thirteenth observation is 23.
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