¶ Average
Average(aka Arithmetic Mean) - The sum of all of the numbers in a list divided by the number of items in that list.
Average =Sum of observations/Number of observations
In mathematics and statistics, average refers to the sum of a group of values divided by n, where n is the number of values in the group. An average is also known as a mean. Like the median and the mode, the average is a measure of central tendency, meaning it reflects a typical value in a given set. Averages are used quite regularly to determine final grades over a term or semester. Averages are also used as measures of performance. For example, batting averages express how many runs a cricket player is likely to score when they are up to bat. Gas mileage expresses how far a vehicle will typically travel on a gallon of fuel.
Basics:
>The average is always calculated for a set of numbers.
>The average always lies above the lowest number of the set and below the highest number of the set.
>The net deficit due to the numbers below the average equals the net surplus due to the numbers above the average.
>If the average of some numbers is A, and if all the numbers are multiplied by N, then the new average equals to (A × N).
>If the average of some numbers is A, and if all these numbers are divided by N, then the new average equals to A/N.
>If the average of some numbers is A, and if N is added to each number, then the new average is (A + N).
>If the average of some numbers is A, and if N is subtracted from each number, then the new average is (A – N).
This also means
Am x M = Total of the set of numbers.
Calculating Average using General Formula:
A mathematical average is calculated by taking the sum of a group of values and dividing it by the number of values in the group. It is also known as an arithmetic mean. With a small set of values, calculating the average takes only a few simple steps.
For example, let us imagine we want to find the average age among a group of five people. Their respective ages are 12, 22, 24, 27, and 35.
First, we add up these values to find their sum: 12 + 22 + 24 + 27 + 35 = 120.
Then we take this sum and divide it by the number of values: 120 ÷ 5 = 24. The result 24 is the average age of the five individuals.
Calculating Average of Consecutive Numbers:
A sequence of consecutive numbers is called a series. Because a series consists of evenly-spaced numbers, the median and mean (average) of the series will be the same. For a short series of consecutive numbers, it is easy to find the average by finding the middle number in the sequence, or the median. There are three ways to calculate the average of consecutive numbers.
Method 1:
Averaging Any Short Series of Consecutive Numbers
Step 1: Count the number of terms in the series. This is the number of numbers in the sequence/series. Determine whether the series has an odd or even number of terms. For example, the sequence 3, 4, 5, 6, 7, 8, 9 has seven terms, an odd amount. The sequence 3, 4, 5, 6, 7, 8 has six terms, an even amount.
Step 2: Identify the middle number of a series with an odd number of terms. This is the number that has the same amount of terms on either side of it. This middle number will be the average of the series. For example, in the sequence 3, 4, 5, 6, 7, 8, 9, the middle number is 6. It has three numbers to the left of it, and three numbers to the right of it. So, in this series of numbers, 6 is the average.
Step 3: Average the middle numbers of a series with an even number of terms. To do this, find the pair of numbers that are in the middle. To find the average, add these two numbers together and divide them by two. Their average will be the average of the series. For example, in the sequences 3, 4, 5, 6, 7, 8, the middle pair is 5 and 6. It has two numbers to the left of it, and two numbers to the right of it. So, to calculate the average of the series, calculate the average of these two numbers:
(5+6)/2 = 11/2 = 5.5
So, in this series of numbers, 5.5 is the average.
Method 2:
Averaging Any Long Series of Consecutive Numbers:
For calculating the average of any long series of consecutive numbers, the formula is Average = (x1 + x2)/2; where x1 is the first number in the series and x2 is the last number in the series.
For example, if we were finding the average of sequential numbers beginning with 15 and ending with 45, the average will be
Average = (15+45)/2 = 30
Method 3:
Averaging Any Consecutive Series Beginning with 1 and with common difference 1
Step 1: Set up the formula for calculating the sum of a series of consecutive numbers. The formula is S = n (n + 1)/2; where s equals to the sum of all the numbers in the series, and n equals the number of terms (numbers) in the series.
Step 2: Count the number of terms in the series and calculate the value of S. Since the series begins with 1, the number of terms is equal to the last term in the series. So, this will be the value of n. For example, if you are finding the sum of consecutive numbers 1 through 25, you have 25 numbers in your sequence, so n = 25 and your formula will look like this:
S = 25 (25 + 1)/2 = 650/2 = 325
Step 3: Divide the sum by the number of terms in the series. This will give you the average of the series.
So, Average = 325/25 = 13
Calculating Average Using Deviation from Assumed Mean
In this method, we first assume any number in the middle to be the mean or average. Then, we calculate the deviation of each number in the series from this assumed mean.
For example, let us consider that we want to calculate the average of the numbers 4, 6, 7, 8, 10 and 13.
Let the assumed mean be 7. Therefore, the deviation from each number is
4 – 7 = -3
6 – 7 = -1
7 – 7 = 0
8 – 7 = 1
10 – 7 = 3
13 – 7 = 6
Next, we take the sum of all the deviations. (-3) + (-1) + 0 + 1 + 3 + 6 = 6. After that we divide the sum by the number of terms and add that value to that of the assumed mean. So, we divide 6 by 6. The result is 1. Then we add 1 to the assumed mean 7 and we will get the average.
So, the average is (7 + 1) = 8.
Change in average due to inclusion or exclusion of a term
If the old Average (average before inclusion or exclusion of the term) is A, the number of term before inclusion or exclusion of a new term is n, and the value of the new term is x, then the new average is:
For inclusion, New Average = A + (x – A)/ (n + 1)
For exclusion, New Average = A – (x – A)/ (n – 1)
For example, let the average marks of 9 students in the final exam be 78. The 10th student scored 100 in the final exam. What will be the average of all the students in the final exam?
Here, A = 78
x = 100
n = 9
Therefore, new average = A + (x – A)/ (n + 1)
= 78 + (100 – 78)/ (9 + 1)
= 78 + 22/10
= 78 + 2.2
= 80.2
Example:
The average weight of 29 students is 28 kg. By the admission of a new student, the average weight is reduced to 27.8 kg. The weight of the new student is:
A. 22 kg B. 21.6 kg C. 22.4 kg D. 21 kg E. None of these
Solution:
Here, A = 28
n = 29
New average = 27.8
We have to determine x.
According to question,
28 + (x – 28)/ (29 + 1) = 27.8
Or, (x – 28)/30 = (- 0.2)
Or, x – 28 = (- 6)
Or, x = 28 – 6
So, x = 22
Therefore, the correct answer is option (a) 22 kg.
Change in Average due to Replacement of a Term:
If x is the amount of change in average due to replacement of a term, then we can write,
x = (i – o)/ n (Easier to remember this formula as i = nx + o)
Where, x= the amount of change in average due to replacement of a term
i = the value of the new term
o = the value of the new term
n = number of terms
And if the old average is A, then the new average is: New Average = A + x
For example, the average marks of the students in a class is 40. While checking the copies of his students, Jawad mistakenly gave a student 45 marks instead of 35. What will be the average marks of the students in his class if the total number of students were 5?
Here, i = 45
o = 35
n = 5
Therefore, x = (45 – 35)/ 5 = 2
So, the new average = 40 + 2 = 42
Example
The average weight of 8 people increases by 2.5 kg when a new person comes in place of one of them weighing 65 kg. What might be the weight of the new person?
A. 76 kg B. 76.5 kg C. 85 kg D. 80 kg E. None of these
Solution:
Here, x = 2.5
o = 65
n = 8
We have to determine the value of i.
So, i = nx + o
Or, i = 8 X 2.5 + 65
Therefore, i = 85
So, the correct answer is option (c) 85 kg.
Average Speed Formulas
There is a single versatile formula for ALL average speed questions and that is
Average Speed = Total Distance/Total Time
Variations of this formula:
1. Average Speed = (a + b)/2; Applicable when one travels at speed a for half the time and speed b for the other half of the time. In this case, the average speed is the arithmetic mean of the two speeds.
2. Average Speed = 2ab/(a + b); Applicable when one travels at speed a for half the distance and speed b for the other half of the distance. In this case, the average speed is the harmonic mean of the two speeds. On similar lines, you can modify this formula for one-third distance.
3. Average Speed = 3abc/(ab + bc + ca); Applicable when one travels at speed a for one-third of the distance, at speed b for another one-third of the distance, and speed c for rest of the one-third of the distance.
Note that the generic Harmonic mean formula for n numbers is Harmonic Mean = n/(1/a + 1/b + 1/c + …).
Change in average due to inclusion or exclusion of an element
Inclusion of an element: Suppose, the average (Age/Weight) of N persons is A. If a new person having age/weight ‘P’ is included in this group, then its new average becomes M. The Age or Weight of included person= No. of persons previously in the group × (Increase in average) + New Average i.e.
P = N × (M – A) + M
Exclusion of an element: Suppose, the average (Age/Weight) of N persons is A. If an existing person having age/weight ‘P’ is excluded from this group, then its new average becomes M. The Age or Weight of excluded person = No. of persons previously in the group × (Decrease in average) + New Average I.e.
P = N × (A – M) + M
Mean median and mode
Mean: The mean is found by adding the numbers in a data set and dividing by how many numbers there are. The formula for mean is as follows: x̅ =(∑ x)/N
Median: The median is the middle number in a data set when the numbers are listed in either ascending or descending order. If the total number of numbers(n) is odd then the formula is given below:
Median =( (n + 1)/2)th term
If the total number of the numbers(n) is an even number, then the formula is given below:
Median ={(n/2)th term + ((n/2)+ 1)th term}/2
Mode: The mode is the value that occurs the most often in a data set.
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