In Maths, an average of a list of data is the expression of the central value of a set of data. Mathematically, it is defined as the ratio of summation of all the data to the number of units present in the list. In terms of statistics, the average of a given set of numerical data is also called mean. For example, the average of 2, 3 and 4 is (2+3+4)/3 = 9/3 =3. So here 3 is the central value of 2,3 and 4. Thus, the meaning of average is to find the mean value of a group of numbers.
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Average = Sum of Values/Number of Values
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Average Definition
The average is defined as the mean value which is equal to the ratio of the sum of the number of a given set of values to the total number of values present in the set.
The average formula has many applications in real life. Suppose if we have to find the average age of men or women in a group or average male height in India, then we calculate it by adding all the values and dividing it by the number of values.
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Symbol
The average is basically the mean of the values which are represented by xÌ„. It is also denoted by the symbol ‘μ’.
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Average Formula in Mathematics
The formula to find the average of given numbers or values is very easy. We just have to add all the numbers and then divide the result by the number of values given. Hence, the average formula in Mathematics is given as follows:
Average = ------------------------------------------------------------
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Suppose we have given with n number of values such as x , x , x …..., x . The average or the mean of the given data will be equal to:
Average x̄ = --------------------------------------------
Sum of Values
Number of values
1
2
3
1
n
x + x + x +......+ x
n
2
3
n
How to Calculate Average?
We can easily calculate the average for a given set of values. We just have to add all the values and divide the outcome by the number of given values.
Average can be calculated using three simple steps. They are:
Step 1: Sum of Numbers: The first step in finding the average of numbers is to find the sum of all the given numbers.
Step 2: Number of Observations: Next, we have to count how many numbers are in the given dataset.
Step 3: Average Calculation: The final step in calculating the average is to divide the sum by the number of observations.
An average or more accurately an arithmetic mean is, in crude terms, the sum of n different data divided by n.
For example, if a batsman scores 35, 45 and 37 runs in first, second and third innings respectively, then his average runs in 3 innings is equal to -------------------------- = 39 runs.
Therefore, the two mostly used formulas in this chapter are:
Average = ------------------------------------------ or -----------------------------------------------------
And Total = Average x No. of Data
Age Formula
Age of entrant = New average + No. of old members Increase
Weight Formula:
Weights of new person = weight of removed person + No. of person increases in average
Marks Formula:
Number of passed candidates = ------------------------------------------------------------------––-----------------------------
And number of failed candidates = –------------------------------------------------------------------------------------------------
Average Related to speed:
Theorem: If a person travels a distance at a speed of x km/hr and the same distance at a speed of y km/hr, then
the average speed during the whole journey is given by -----------
If half of the journey is travelled at a speed of x km/hr and the next half at a speed of
y km/hr, then average speed during the whole journey is ----------- km/hr.
If a man goes to a certain place at a speed of x km/hr and returns to the original place
at a speed of y km/hr. then the average speed during up-and down journey is --------- km/hr.
Theorem: If a person travels three equal distances at a speed of x km/hr, y km/hr and z km/hr
respectively, then the average speed the whole journey is ------------------------------ km/hr.
Proof: Let the three equal distance be A km.
Time taken at the speed of x km/hr = ------
Time taken at the speed of y km/hr = -------
Time taken at the speed of z km/hr = -------
Total distance travelled in time = ----- + ----- + ------ = 3A km
Average speed during the whole journey = ----------------------------- = --------------------------------------=-----------------------------
------ + -----+ ------
Unlike terms: - Terms that do not contain the same power of the same varies able are called unlike terms.
Eg : - 3x and 3y, 3x and 6x2
Factor theorem: P(x) is a polynomial and is a real number, if, p (a) = o then x – a is a factor of P (x)
35 + 45 + 37
3
Total of Data
Number of Data
Sum of Observations
Number of Observations
Total Candidate (Total Average - Failed Average)
Passed Average - Failed Average
Total Candidate (Passed Average - Total Average)
Passed Average - Failed Average
2xy
x + y
2xy
x + y
2xy
x + y
3xyz
xy + yz + zx
A
x
A
x
A
y
A
z
A
y
A
z
3A
A
x
A
y
A
z
3xyzA
xyz + Axz + Axy
3xyz
xy + yz + zx
Ex.1:Find the average of all prime numbers between 30 and 50?
Sol: There are five prime numbers between 30 and 50.
They are 31,37,41,43 and 47.
Therefore, the required average = (31 + 37 + 41 + 43 + 47)/5
= 199/5
= 39.8.
Ex.2. find the average of first 40 natural numbers?
Sol: Sum of first n natural numbers = n (n+1)/2
So, sum of 40 natural numbers = (40*41)/2
= 820.
Therefore, the required average = (820/40)
= 20.5.
Ex.3. find the average of first 20 multiples of 7?
Sol: Required average = 7 (1 + 2 + 3 + ……. + 20)/20
= (7*20*21)/ (20*2)
= (147/2)
=73.5.
Ex.4. the average of four consecutive even numbers is 27. find the largest of these numbers?
Sol: Let the numbers be x, x + 2, x + 4 and x + 6. then,
(x + (x + 2) + (x + 4) + (x + 6))/4) = 27
⇒ (4x+12)/4 = 27
⇒ x+3=27
⇒ x=24.
Therefore, the largest number (x + 6) = 24 + 6 = 30.
Ex.5. there are two sections A and B of a class consisting of 36 and 44 students respectively. If the average weight of section A is 40kg and that of section B is 35kg, find the average weight of the whole class?
Sol: Total weight of (36 + 44) students = (36*40+44*35) kg = 2980 kg.
Therefore, weight of the total class = (2980/80) kg
= 37.25kg.
Ex:6. Nine persons went to a hotel for taking their meals 8 of them spent Rs.12 each on their meals and the ninth spent Rs.8 more than the average expenditure of all the nine. What was the total money spent by them?
Sol: Let the average expenditure of all nine be Rs. x
Then 12*8 + (x + 8) = 9x
⇒ 8x = 104
⇒ x =13.
Total money spent = 9x
= Rs. (9*13)
= Rs.117.
Ex.7: Of the three numbers, second is twice the first and is also thrice the third. If the average of the three numbers is 44. Find the largest number.
Sol: Let the third number be x.
Then second number = 3x.
First number = 3x/2.
Therefore, x + 3x + (3x/2) = (44*3)
⇒ x=24
⇒ So largest number = 2nd number = 3x = 72.