Mean Mathematics
In mathematics, the mean is a calculation tool that allows you to summarize a list of numeric values into a single real number, regardless of the order in which the list is given. By default, this is the arithmetic mean, which is calculated as the sum of the terms in the list divided by the number of terms1. Other averages or mean mathematics may be more appropriate depending on the context.
The mean is one of the first statistical indicators for a series of numbers. When these numbers represent a quantity shared between individuals, the average expresses the value that each would have if the sharing were fair.
The notion of mean extends to functions with the mean value, in classical geometry with the barycenter and in probability theory with the expectation of a random variable.
Average or Mean of two values
Intermediate value
The notion of average is historically linked to that of intermediate value, also called mediety. Given two numbers a and b, how do we choose a value c so that a is c that c is b? The answer differs depending on the operation you choose to go from number to number.
For example, to go from 2 to 18, we can add twice 8, with a step in 10, or multiply twice by 3, with a step in 6. The first case describes an arithmetic mean, which is obtained by the fraction .
The second case is a geometric mean, which is obtained with the square root .
The usual remarkable identities make it possible to quickly show that the geometric mean of two positive numbers is always less than their arithmetic mean.
A proof of the arithmetical-geometric inequality on two values
If a and b are two reals such that a <b, of the identity of Legendre
we deduct
and we conclude by applying the square root function (which is strictly increasing).
Weighted average
The search for an average position can accommodate an imbalance between the initial data. For example, you might want the mean to be three times closer to the first value than to the second. Between 7 and 19, the number 10 is indeed three times closer to 7 (with a difference of 3) than to 19 (with a difference of 9). We then say that 10 is the weighted average of the numbers 7 and 19 with the coefficients 3 and 1. It is found by calculating the weighted sum which we divide by the sum of the coefficients { (3×7+1×19) / (3+1) } = 40/40 = 10
Fair sharing
Another way to define these averages is to accumulate the numbers chosen and then to find out how we can obtain the same result by accumulating the same value twice. It all depends on the cumulation procedure. With an addition, we find 2 + 18 = 20, which we could have obtained by setting 10 + 10 = 20. With a multiplication, we find 2 × 18 = 36, which we could have obtained with 6 × 6 = 36.
Other cumulation procedures on two numbers a and b allow to define the harmonic mean and the root mean square .
This approach allows you to define means for lists of more than two numbers.
Arithmetical-geometric mean mathematics
From two numbers a and b, the arithmetic mean and the geometric mean providing two new numbers, and we can iterate the process to obtain two adjacent sequences which converge towards an intermediate real (sometimes denoted by M (a, b)) called the arithmetic-geometric mean and which is related to the length of an ellipse.
Read also ? Math Symbol | What Does This Symbol Mean in Mathematics?
Questions and Answers of Mean Mathematics
1. What is the Mean of these numbers? 6, 11, 7
Answer:
Add the numbers: 6 + 11 + 7 = 24
Divide by how many numbers (there are 3 numbers): 24 / 3 = 8
The Mean is 8
2. Find the mean of the following data.
(a) 9, 7, 11, 13, 2, 4, 5, 5
(b) 16, 18, 19, 21, 23, 23, 27, 29, 29, 35
(c) 2.2, 10.2, 14.7, 5.9, 4.9, 11.1, 10.5 (d) 11/4, 21/2, 51/2, 31/4, 21/2
Answer:
(a) 7
(b) 24
(c) 8.5
(d) 3
3. The mean of 6, 8, x + 2, 10, 2x – 1, and 2 is 9. Find the value of x and also the value of the observation in the data.
Answer:
9, 11, 17.
4. The mean of 8, 11, 6, 14, x and 13 is 66. Find the value of the observation x.
Answer:
344.
5. Find the mean of the following distributions.
(a) The age of 20 boys in a locality is given below.
| Age in Years | 12 | 10 | 15 | 14 | 8 |
| Number of Boys | 5 | 3 | 2 | 6 | 4 |
(b) Marks obtained by 40 students in an exam are given below.
| Marks | 25 | 30 | 15 | 20 | 24 |
| Number of Students | 8 | 12 | 10 | 6 | 4 |
(c)
| xi | 1 | 2 | 3 | 4 | 5 |
| fi | 4 | 5 | 8 | 10 | 3 |
(d) The daily wages of 50 employees in an organization are given below:
| Daily wages (in $) | 100 – 150 | 150 – 200 | 200 – 250 | 250 – 300 |
| Number of Workers | 12 | 13 | 17 | 8 |
Find the mean daily wages.
Answer:
(a) 11.8
(b) 23.15
(c) 3.1
(d) $196
Source: Britannica
