Average

3. Mode

4. Median

5. Variance

1) Average or Arithmentic Mean

The average or mean or Arithmetic Mean is the sum of a list of numbers divided by the number of elements in the list.

For example , the average of 3 , 9 , 11 , 15 , 18 , 19 and 23 is :

Average or Mean or Arithmetic Mean = (3 + 9 + 11 + 15 + 8 + 19 + 23)/7 = 14

Example 1: Varun purchased 5 pen at the rate of RS 200 each, 6 pen at the rate of Rs 250 each and 9 pen at the rate of Rs 300 each. Calculate the average cost of 1 pen?

Price of 5 pen = 200 * 5 = 1000 Rs

Price of 6 pen = 250 * 6 = 1500 Rs

Price of 9 pen = 300 * 9 = 2700 Rs

Total Number of pen = 5 + 6 + 9 = 20

Note 1 :

Average of two or more groups taken together :

a) If the number of quantities in two groups be a and b and their average is x and y respectively , the combined average (average of all of them put together) is :

Example 2: The average weight of 24 students of section A of a class is 58 kg whereas the average weight of 26 students of section B of the same class is 60.5 . Find the average weight of all the 50 students of the class ?

Given : n1=24 , n2 = 26 , x=58 and y = 60.5

b) If the average of a quantities is x and the average of b quantities out of them is y , the average of remaining group (rest of the quantities) is :

Example 3: Average salary of all the 50 employees including 5 officers of a company is Rs 850 . If the average salary of the officers is Rs 2500 , find the average salary of the remaining staff of the company.

Given : n1= 50 , n2 = 5 , x = 850 and y = 2500

Note 2 :

If x is the average of x1, x2 ...... , xn then:

(a) The average of x1 + a , x2 + a , ...... , xn + a is x + a i.e. if each of the element is increased by a, then their average will also increase by a.

Example 4: The average value of six numbers 7 , 12 , 17 , 25 , 26 and 28 is 19 . If 8 is added to each number, what will be new average?

Given: x (current average) = 9

New value (a) added to each number = 8

The new average will be = x + a

= 19 + 8 = 27

(b) The average of x1 - a, x2 - a, ...., xn - a is x - a i.e. if each of the element is decreased by a, then their average will also, be decreased by a.

Example 5: The average of x numbers is 5x . If x - 2 is subtracted from each given number , what will be the new average ?

Given: x (current average) = 5x

New value (a) subtracted from each number = x - 2

The new average will be = x - a

= 5x - (x - 2)

= 4x + 2

(c) The average of ax1, ax2, ....,axn is ax, provided a not equal to 0 i.e. if each of the element is multiplied by a, then their average gets multiplied by a.

Example 6: The average of 8 numbers is 21 . If each of the number is multiplied by 8 , find the average of a new set of numbers ?

Given: x (current average) = 21

New value (a) multiplied from each number = 8

The new average will be = x a

= 21 x 8 = 168

(d) The average of x1/a, x2/a, ...., xn/a is x/a, provided a not equal to 0. i.e. if each of the element is divided by a, then their average gets divided by a

Note 3 :

The average of n items is equal to x . If one of the given item whose value is p , is replaced by a new item having value q , the average becomes y, Then value of new item q is

q = p + n(y - x)

Example 7: The average weight of 25 persons is increased by 2 kg when one of them whose weight is 60 kg is replaced by a new person . What is the weight of new person ?

Given: Difference between new and old average = y-x = 2

Total number of men (n) = 25

Weight of man who was replaced (p) = 60

So, weight of new man will be: q = p + n(y - x)

= 60 + 25(2) = 110 kg

Alternate Solution :

Assume that average weight of the 25 person is x

So the total weight of 25 person will be = 25 X x

Now given that one person whose weight is 60 kg is replaced by a new person .

Assume weight of new person is y.

So new weight will be : 25 X x - 60 + y -------------- (1)

Given that average weight is increased by 2 after new person came , so new average will be : x + 2 and new total weight will be :

25 (x + 2) ----------------- (2)

So , both the equation (1) and (2) gives the new total weight :

Hence : (1) = (2)

25 X x - 60 + y = 25(x+2)

25 X x - 60 + y = 25 x + 50

So y = 110

Note 4 :

a) The average of n items is equal to x . When a item is removed , the average becomes y . The value of the removed item is :

n( x - y ) + y

n = number of items

x = old average

y = new average

Example 8: The average age of 24 students and the class teacher is 16 years . If the class teacher's age is excluded , the average age reduces by 1 year . What is the age of the class teacher ?

Given:

Number of Students (n) = 24

Old Average (x) = 24

New Average (y) = 24 - 1 = 23 (reduced by 1 of the original average)

The Age of the teacher = n(x - y) + y

= 25 (16 - 15) + 15

= 40 years

b) The average of n item is equal to x . When a item is added , the average becomes y . The value of the new item is :

n(y - x) + y

n = number of items

x = old average

y = new average

Example 9: The average age of 30 children in a class is 9 years . If the teacher's age be included, the average age becomes 10 years. Find the teacher's age:

Given:

Number of Students (n) = 30

Old Average (x) = 9

New Average (y) = 10

The Age of teacher = n(y - x) + y

= 30 (10 - 9 ) + 10

= 40 years

Note 5 :

(a) The average of first n natural number is: (n + 1)/2

Example 10: Find the average of first 81 natural numbers .

The average of first n natural numbers is : (n+1 )/2

= (81+1 )/2 = 41

(b) The average of odd numbers from 1 to n is: (last odd number + 1)/2

Example 11: What is the average of odd numbers from 1 to 40 ?

The required average = (last odd number +1 )/2

= (39+1 )/2 = 20

(c) The average of even numbers from 1 to n is: (last odd number + 2) / 2

Example 12: What is the average of even numbers from 1 to 81 ?

The required average = (last even number + 2 )/2

= (80+2 )/2 = 41

Note 6 :

a) If n is odd :

The average of n consecutive numbers , consecutive even numbers or consecutive odd numbers is always the middle number.

Example 13: Find the average of 7 consecutive numbers 3 , 4 , 5 , 6 , 7 , 8 , 9 ?

Total number is 7 which is odd .

So the required average = middle number = 6

b) If n is even :

The average of n consecutive numbers , consecutive even numbers or consecutive odd numbers is always the average of middle two numbers.

Example 14: Find the average of consecutive odd numbers 21 , 23 , 25 , 27 , 29 , 31 , 33 , 35?

Total number is 8 which is even

So the required average = average of middle 2 numbers = average of 27 and 29

= (27+29)/2 = 28

c) The average of first n consecutive even numbers is (n + 1)

Example 15: Find the average of first 31 consecutive even numbers ?

The required average of first (n) consecutive even numbers is : (n + 1)

Given n = 31

Average = 31 + 1 = 32

d) The average of first n consecutive odd numbers is n.

Example 16: Find the average of first 50 consecutive odd numbers.

The required average of first (n) consecutive odd numbers is : n

Given n = 50

So required average = 50

e) If the average of n consecutive numbers is m , then the difference between the smallest and the largest number is : 2(n-1)

Example 17: If the average of 6 consecutive numbers is 48 , what is the difference between the smallest and the largest number ?

We know that If the average of n consecutive numbers is m , then the difference between the smallest and the largest number is : 2(n-1)

Given : n = 6

So the required difference between smallest and largest number is : 2 (n - 1 ) = 2(6 - 1) = 10

2) Geometric Mean or Geometric Average

Geometric mean of x1, x2,...., xn is denoted by:

Geometric mean is useful in calculating averages of ratios such as averages of ratios such as average population growth rate, average percentage increase and So, on.

Example 18: The production of a company for three successive years has increased by 10% , 20% and 40% respectively . What is the average annual increase of production ?

We know that Geometric Mean is used for calculating average percentage increage or decrease .

So Average Increase by Geometric Mean of x , y and z = (x * y * z) 1/3

Average = (10 * 20 * 40)1/3

Average = 20 %

Example 19: The population of a city in two successive years increases at the rates of 16% and 4% , respectively . Find the average increase of two years ?

In case of population increase, the geometric mean is required.

So Geometric Mean of 16% and 4% is = (16*4)1/2 = 8 %

Important: Geometric Mean is always less than or equal to Arithmetic Mean.

3) Mode :

The mode in a dataset is the value that is most frequent in a dataset.

For example, the mode of the dataset S = 1,2,3,3,3,3,3,4,4,4,5,5,6,7, is 3 since it occurs the maximum number of times in the set S.

4) Median:

The Median is the 'middle value' in the list.

a) When the totals of the list are odd, the median is the middle entry in the list after sorting the list into increasing order.

b) When the totals of the list are even, the median is equal to the sum of the two middle (after sorting the list into increasing order) numbers divided by two.

Example 20: Find the Median of: 9, 3, 44, 17, 15 (Odd amount of numbers)

Order the number in ascending order : 3, 9, 15, 17, 44 (smallest to largest)

The Median is: 15 (The number in the middle as per point a above)

Example 20: Find the Median of: 8, 3, 44, 17, 12, 6 (Even amount of numbers)

Order the number in ascending order: 3, 6, 8, 12, 17, 44

Add the 2 middles numbers and divide by 2 : (8 + 12) / 2 = 20 / 2 = 10

The Median is 10.

5) Variance

The Variance is defined as the average of the squared differences from the Mean.

6) Standard Deviation

The Standard Deviation is a measure of how spread out numbers are.

- A standard deviation close to 0 indicates that the data points tend to be very close to the mean of the set

- A high standard deviation indicates that the data points are spread out over a wider range of values

Example 21: Varun measured the heights of dogs (in millimeters) .The heights (at the shoulders) are: 600mm, 470mm, 170mm, 430mm and 300mm. Find out the Mean, the Variance, and the Standard Deviation.

Average or Mean or Arithmetic Mean = (Sum of Elements )/(Total Number of Elements)

= (600 + 470 + 170 + 430 + 300 )/5 = 394

so the mean (average) height is 394 mm.

Let's plot this on the chart:

To calculate the Variance, take each difference, square it, and then average the result: