AVERAGE TOPIC

AVERAGE:-

Definition:-

The average is defined as the mean value which is equal to the ratio of sum of number of a given set of values to the total number of values present.

Mean formula:-

 According to the definition of mean, the formula to calculate the mean is given by,

Mean = Sum of Given Data / Total Number of Data

The three different types of mean are:

• Arithmetic Mean (AM)
• Harmonic Mean (HM)
• Geometric Mean (GM)

QUESTIONS:-

1) Find the average of 6, 13, 17, 21, 23.

ANS :- 16

2)Find the mean for the given data set: 2, 6, 4, 5, 8

ANS:- 5

3) If the age of 9 students in a team is 12, 13, 11, 12, 13, 12, 11, 12, 12. Then find the average age of students in the team.

ANS :- 12

4) If the heights of males in a group are 5.5, 5.3, 5.7, 5.9, 6, 5.10, 5.8, 5.6, 5.4, 6. Then find the average height.

 ANS :- 5.63

5)One day a supermarket received a delivery of 25 frozen turkeys. if the average weight of turkey was 14.2 pounds, what was the total weight, in pounds, of all the turkeys?

 ANS :- 355

6) If the average of 25, 31, and x is 37 , what is the value of x?

ANS :- 55 

WEIGHTED AVERAGE:-

Weighted average is a calculation that takes into account the varying degrees of importance of the numbers in a data set. In calculating a weighted average, each number in the data set is multiplied by a predetermined weight before the final calculation is made.

EXAMPLE:-

Given two school classes, one with 20 students, and one with 30 students, the grades in each class on a test were:

Morning class = 62, 67, 71, 74, 76, 77, 78, 79, 79, 80, 80, 81, 81, 82, 83, 84, 86, 89, 93, 98

Afternoon class = 81, 82, 83, 84, 85, 86, 87, 87, 88, 88, 89, 89, 89, 90, 90, 90, 90, 91, 91, 91, 92, 92, 93, 93, 94, 95, 96, 97, 98, 99

The mean for the morning class is 80 and the mean of the afternoon class is 90. The unweighted mean of the two means is 85. However, this does not account for the difference in number of students in each class (20 versus 30); hence the value of 85 does not reflect the average student grade (independent of class). The average student grade can be obtained by averaging all the grades, without regard to classes (add all the grades up and divide by the total number of students):

${\displaystyle {\bar {x}}={\frac {4300}{50}}=86.}$

Or, this can be accomplished by weighting the class means by the number of students in each class. The larger class is given more "weight":

${\displaystyle {\bar {x}}={\frac {(20\times 80)+(30\times 90)}{20+30}}=86.}$

STANDARD DEVIATIONS:-

FIND the standard deviations of the following five pieces of Data :

 

10 , 20 , 30, 50 , 65 

MEAN = 10+20+30+50+65 / 5 = 175 /5 = 35

FOR Standard deviations :-

(10- 35 )²= 625

(20- 35 )²= 225

(30- 35 )²= 25

(50- 35 )²= 225

(65- 35 )²= 900

 

ADD all the terms= 625+225+25+225+900/5 = 2000/5 = 400

 Square root of 400 is 20.

STANDERD DEVIATION = 20

MEDIAN :-

The median of a finite list of numbers is the "middle" number, when those numbers are listed in order from smallest to greatest.

If the data set has an odd number of observations, the middle one is selected.

1, 3, 3, 6, 7, 8, 9

has the median of 6, which is the fourth value.

MODE:-

A mode is defined as the value that has a higher frequency in a given set of values. It is the value that appears the most number of times. 

Example: In the given set of data: 2, 4, 5, 5, 6, 7, the mode of the data set is 5 since it has appeared in the set twice.

RANGE :-

Range = Maximum Value – Minimum Value

EXAMPLE:-

1) Find the mean, median, mode, and range for the following list of values:

13, 18, 13, 14, 13, 16, 14, 21, 13

2) The maximum mark in an examination is 100 and the minimum is 0. The average mark of seven students such that no two of them have scored the same marks is 88. If the median score is 92 and all the marks are integers, what is the maximum possible difference between the highest and the least mark obtained by these seven students?

1. 11
2. 46
3. 99
4. 54
5. 100

3) The average (arithmetic mean) of two numbers x and y is 15. If x, y, and z are non-negative integers such that x < z < y, what is the minimum possible average of x, y, and z?

1. 1013$\frac{1}{3}$
2. 1023$\frac{2}{3}$
3. 15
4. 11
5. 10

4 If the average (arithmetic mean) of 5 positive integers is 11, what is the maximum possible difference between the largest and the smallest of these 5 numbers?

1. 50
2. 44
3. 39
4. 4
5. 20

5)The average weight of the women in a group is 60 kg and that of the men is 72 kg. If the average weight of the group is 68 kg, what is the ratio of women to men in the group?

1. 1 : 3
2. 2 : 3
3. 3 : 2
4. 1 : 2
5. 2 : 1

6) If the median of -10, 29, 6, 11, 31 and x is 20. What is the least possible average of the 6 numbers?

1. 1613$\frac{1}{3}$
2. 1623$\frac{2}{3}$
3. 1616$\frac{1}{6}$
4. 16
5. 1116