Average

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Average

What is Average?

The result obtained by adding several quantities together and then dividing this total by the number of quantities is called Average.

The main term of average is equal distribution of a value among all which may distribute persons or things. We obtain the average of a number using formula that is sum of observations divided by Number of observations.

Here is average based some fact and formula and some average shortcut tricks examples. The problem is given in Quantitative Aptitude which is a very essential paper in ssc exam. Given below are some more example for practicing.

Formula:

Average: = (Sum of observations / Number of observations).

Find the Average Speed

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 a person covers A km at x km/hr and B km at y km/hr and C km at z km/hr, then the average speed in covering the whole distance is-

When a person leaves the group and another person joins the group in place of that person then-

If the average age is increased,
Age of new person = Age of separated person + (Increase in average × total number of persons)
If the average age is decreased,
Age of new person = Age of separated person – (Decrease in average × total number of persons)

When a person joins the group-In case of increase in average

Age of new member = Previous average + (Increase in average × Number of members including new member)

In case of decrease in average

Age of new member = Previous average – (Decrease in average × Number of members including new member)

In the Arithmetic Progression there are two cases when the number of terms is odd and second one is when number of terms is even.

So when the number of terms is odd the average will be the middle term.
when the number of terms is even then the average will be the average of two middle terms.

Some Important Examples

Examples 1: what will be the average of 13, 14, 15, 16, 17?

Solution: Average is the middle term when the number of terms is odd, but before that let’s checks whether it is in A.P or not, since the common difference is same so the series is in A.P. So the middle term is 15 which is our average of the series.

Example 2: What will be the average of 13, 14, 15, 16, 17, 18?

Solution: We have discussed that when the number of terms are even then the average will be the average of two middle terms.

Now the two middle terms are 15 and 16, but before that the average we must check that the series should be A.P. Since the common difference is same for each of the term we can say that the series is in A.P. and the average is (16+15)/2 = 15.5

Example 3:The average of five numbers is 29. If one number is excluded the average becomes 27. What is the excluded number ?

Answer :

let the excluded number is

= (29 x 5) – ( 27 x 4 )

= 145 – 108

= 37 .

Example 4: Find the average of first 20 natural numbers?

Answer:

Sum of first n natural numbers = n ( n + 1 ) /2

So, we can find easily average of first 20 natural numbers 20 x 21 / 2 = 210

So, then Required average is = 210 / 20 = 10.5.

Example 5

Find the average of first 20 multiplies of 5 .

Answer:

Required average = 5 ( 1 + 2 + 3 +……………….. + 20) /20

= ( 5 x 20 x 21 / 20 x 2) = 2100 / 40 = 52.5 .

So the Required average is 52.5.

Questions:

Level-I:

In the first 10 overs of a cricket game, the run rate was only 3.2. What should be the run rate in the remaining 40 overs to reach the target of 282 runs?

A.	6.25
B.	6.5
C.	6.75
D.	7

A family consists of two grandparents, two parents and three grandchildren. The average age of the grandparents is 67 years, that of the parents is 35 years and that of the grandchildren is 6 years. What is the average age of the family?

28	4	years
	7

31	5	years
	7

32	1	years
	7

None of these

A grocer has a sale of Rs. 6435, Rs. 6927, Rs. 6855, Rs. 7230 and Rs. 6562 for 5 consecutive months. How much sale must he have in the sixth month so that he gets an average sale of Rs. 6500?

A.	Rs. 4991
B.	Rs. 5991
C.	Rs. 6001
D.	Rs. 6991

The average of 20 numbers is zero. Of them, at the most, how many may be greater than zero?

A.	0
B.	1
C.	10
D.	19

The average weight of 8 person’s 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.	Data inadequate
E.	None of these

The captain of a cricket team of 11 members is 26 years old and the wicket keeper is 3 years older. If the ages of these two are excluded, the average age of the remaining players is one year less than the average age of the whole team. What is the average age of the team?

A.	23 years
B.	24 years
C.	25 years
D.	None of these

The average monthly income of P and Q is Rs. 5050. The average monthly income of Q and R is Rs. 6250 and the average monthly income of P and R is Rs. 5200. The monthly income of P is:

A.	3500
B.	4000
C.	4050
D.	5000

The average age of husband, wife and their child 3 years ago was 27 years and that of wife and the child 5 years ago was 20 years. The present age of the husband is:

A.	35 years
B.	40 years
C.	50 years
D.	None of these

A car owner buys petrol at Rs.7.50, Rs. 8 and Rs. 8.50 per litre for three successive years. What approximately is the average cost per litre of petrol if he spends Rs. 4000 each year?

A.	Rs. 7.98
B.	Rs. 8
C.	Rs. 8.50
D.	Rs. 9

10.

In Arun’s opinion, his weight is greater than 65 kg but less than 72 kg. His brother doest not agree with Arun and he thinks that Arun’s weight is greater than 60 kg but less than 70 kg. His mother’s view is that his weight cannot be greater than 68 kg. If all are them are correct in their estimation, what is the average of different probable weights of Arun?

A.	67 kg.
B.	68 kg.
C.	69 kg.
D.	Data inadequate
E.	None of these

11.

Level-II:

The average weight of A, B and C is 45 kg. If the average weight of A and B be 40 kg and that of B and C be 43 kg, then the weight of B is:

A.	17 kg
B.	20 kg
C.	26 kg
D.	31 kg

12.

The average weight of 16 boys in a class is 50.25 kg and that of the remaining 8 boys is 45.15 kg. Find the average weights of all the boys in the class.

A.	47.55 kg
B.	48 kg
C.	48.55 kg
D.	49.25 kg

13.

A library has an average of 510 visitors on Sundays and 240 on other days. The average number of visitors per day in a month of 30 days beginning with a Sunday is:

A.	250
B.	276
C.	280
D.	285

14.

If the average marks of three batches of 55, 60 and 45 students respectively is 50, 55, 60, then the average marks of all the students is:

A.	53.33
B.	54.68
C.	55
D.	None of these

15.

A pupil’s marks were wrongly entered as 83 instead of 63. Due to that the average marks for the class got increased by half (1/2). The number of pupils in the class is:

A.	10
B.	20
C.	40
D.	73

16.

The average age of P, Q, R and S is 30 years. How old is R?

The sum of ages of P and R is 60 years.

II.

S is 10 years younger than R.

A.	I alone sufficient while II alone not sufficient to answer
B.	II alone sufficient while I alone not sufficient to answer
C.	Either I or II alone sufficient to answer
D.	Both I and II are not sufficient to answer
E.	Both I and II are necessary to answer

17.

How many candidates were interviewed everyday by the panel A out of the three panels A, B and C?
I.	The three panels on average interview 15 candidates every day.
II.	Out of a total of 45 candidates interviewed everyday by the three panels, the number of candidates interviewed by panel A is more by 2 than the candidates interviewed by panel c and is more by 1 than the candidates interviewed by panel B.

A.	I alone sufficient while II alone not sufficient to answer
B.	II alone sufficient while I alone not sufficient to answer
C.	Either I or II alone sufficient to answer
D.	Both I and II are not sufficient to answer
E.	Both I and II are necessary to answer

18.

What is the average age of children in the class?
I.	The age of the teacher is as many years as the number of children.
II.	Average age is increased by 1 year if the teacher’s age is also included.

A.	I alone sufficient while II alone not sufficient to answer
B.	II alone sufficient while I alone not sufficient to answer
C.	Either I or II alone sufficient to answer
D.	Both I and II are not sufficient to answer
E.	Both I and II are necessary to answer

Answers:

Level-I:

Answer:1 Option A

Explanation:

Required run rate =		282 – (3.2 x 10)		=	250	= 6.25
		40			40

Answer:2 Option B

Explanation:

Required average

=		67 x 2 + 35 x 2 + 6 x 3
		2 + 2 + 3

=		134 + 70 + 18
		7

=	222
	7

= 31	5	years.
	7

Answer:3 Option A

Explanation:

Total sale for 5 months = Rs. (6435 + 6927 + 6855 + 7230 + 6562) = Rs. 34009.

Required sale = Rs. [ (6500 x 6) – 34009 ]

= Rs. (39000 – 34009)

= Rs. 4991.

Answer:4 Option D

Explanation:

Average of 20 numbers = 0.

Sum of 20 numbers (0 x 20) = 0.

It is quite possible that 19 of these numbers may be positive and if their sum is a then 20th number is (-a).

Answer:5 Option C

Explanation:

Total weight increased = (8 x 2.5) kg = 20 kg.

Weight of new person = (65 + 20) kg = 85 kg.

Answer:6 Option A

Explanation:

Let the average age of the whole team by x years.

11x – (26 + 29) = 9(x -1)

11x – 9x = 46

2x = 46

x = 23.

So, average age of the team is 23 years.

Answer:7 Option B

Explanation:

Let P, Q and R represent their respective monthly incomes. Then, we have:

P + Q = (5050 x 2) = 10100 …. (i)

Q + R = (6250 x 2) = 12500 …. (ii)

P + R = (5200 x 2) = 10400 …. (iii)

Adding (i), (ii) and (iii), we get: 2(P + Q + R) = 33000 or P + Q + R = 16500 …. (iv)

Subtracting (ii) from (iv), we get P = 4000.

P’s monthly income = Rs. 4000.

Answer:8 Option B

Explanation:

Sum of the present ages of husband, wife and child = (27 x 3 + 3 x 3) years = 90 years.

Sum of the present ages of wife and child = (20 x 2 + 5 x 2) years = 50 years.

Husband’s present age = (90 – 50) years = 40 years.

Answer:9 Option A

Explanation:

Total quantity of petrol
consumed in 3 years

=		4000	+	4000	+	4000		litres
		7.50		8		8.50

= 4000		2	+	1	+	2		litres
		15		8		17

=		76700		litres
		51

Total amount spent = Rs. (3 x 4000) = Rs. 12000.

Average cost = Rs.		12000 x 51		= Rs.	6120	= Rs. 7.98
		76700			767

Answer:10 Option A

Explanation:

Let Arun’s weight by X kg.

According to Arun, 65 < X < 72

According to Arun’s brother, 60 < X < 70.

According to Arun’s mother, X <= 68

The values satisfying all the above conditions are 66, 67 and 68.

Required average =		66 + 67 + 68		=		201		= 67 kg.
		3				3

Level-II:

Answer:11 Option D

Explanation:

Let A, B, C represent their respective weights. Then, we have:

A + B + C = (45 x 3) = 135 …. (i)

A + B = (40 x 2) = 80 …. (ii)

B + C = (43 x 2) = 86 ….(iii)

Adding (ii) and (iii), we get: A + 2B + C = 166 …. (iv)

Subtracting (i) from (iv), we get : B = 31.

B’s weight = 31 kg.

Answer:12 Option C

Explanation:

Required average

=		50.25 x 16 + 45.15 x 8
		16 + 8

=		804 + 361.20
		24

=	1165.20
	24

= 48.55

Answer:13 Option D

Explanation:

Since the month begins with a Sunday, to there will be five Sundays in the month.

Required average

=		510 x 5 + 240 x 25
		30

=	8550
	30

= 285

Answer:14 Option B

Explanation:

Required average

=		55 x 50 + 60 x 55 + 45 x 60
		55 + 60 + 45

=		2750 + 3300 + 2700
		160

=	8750
	160

= 54.68

Answer:15 Option C

Explanation:

Let there be x pupils in the class.

Total increase in marks =		x x	1		=	x
			2			2

	x	= (83 – 63)	x	= 20 x= 40.
	2		2

Answer:16 Option D

Explanation:

P + Q + R + S = (30 x 4) P + Q + R + S = 120 …. (i)

I. P + R = 60 …. (ii)

II. S = (R – 10) …. (iii)

From (i), (ii) and (iii), we cannot find R.

Correct answer is (D)

Answer:17 Option B

Explanation:

I. Total candidates interviewed by 3 panels = (15 x 3) = 45.

II. Let x candidates be interviewed by C.

Number of candidates interviewed by A = (x + 2).

Number of candidates interviewed by B = (x + 1).

x + (x + 2) + (x + 1) = 45

3x = 42

x = 14

Hence, the correct answer is (B).

Answer:18 Option D

Explanation:

Let there be x children.

I gives, age of teacher = x years.

II gives, average age of (x + 1) persons = (x + 1) years.

Teacher’s age = (x + 1) (x + 1) – x² = (x² + 1 + 2x) – x² = (1 + 2x)

Thus, teacher’s age cannot be obtained.

Correct answer is (D)

In statistics, an average is a measure of central tendency, describing the typical value of a set of numbers. The average is calculated by adding all the numbers in the set and then dividing by the number of numbers in the set.

There are several different types of Averages, each with its own advantages and disadvantages. The most common type of average is the mean, which is calculated by adding all the numbers in the set and then dividing by the number of numbers in the set. The mean is easy to calculate and understand, but it can be sensitive to outliers, which are data points that are very different from the rest of the data.

Another type of average is the Median, which is the middle value in a set of numbers arranged in order from least to greatest. The median is not as sensitive to outliers as the mean, but it can be more difficult to calculate.

The mode is the most common value in a set of numbers. The mode is easy to calculate, but it can be misleading if there are multiple modes in the data.

A weighted average is an average that takes into account the importance of each data point. To calculate a weighted average, you multiply each data point by its weight and then divide by the sum of the weights. Weighted averages are useful when some data points are more important than others.

The harmonic mean is the reciprocal of the arithmetic mean of the reciprocals of the data points. The harmonic mean is less affected by outliers than the mean, but it is more difficult to calculate.

The geometric mean is the product of the data points, raised to the power of 1/n, where n is the number of data points. The geometric mean is useful when the data points are non-negative and you want to emphasize the multiplicative relationships between the data points.

A quartile is a value that divides a set of data into four equal parts. The first quartile (Q1) is the median of the lower half of the data, the second quartile (Q2) is the median of the entire data set, and the third quartile (Q3) is the median of the upper half of the data.

A decile is a value that divides a set of data into ten equal parts. The first decile (D1) is the value that divides the lower 10% of the data from the upper 90% of the data, the second decile (D2) is the value that divides the lower 20% of the data from the upper 80% of the data, and so on.

A percentile is a value that divides a set of data into 100 equal parts. The first percentile (P1) is the value that divides the lower 1% of the data from the upper 99% of the data, the second percentile (P2) is the value that divides the lower 2% of the data from the upper 98% of the data, and so on.

The standard deviation is a measure of how spread out the data are. The standard deviation is calculated by taking the square root of the Variance. The variance is calculated by taking the average of the squared differences between the data points and the mean.

The range is the difference between the largest and smallest data points. The range is a simple measure of variability, but it can be misleading if there are outliers in the data.

The interquartile range is the difference between the third and first quartiles. The interquartile range is a robust measure of variability that is not affected by outliers.

Skewness is a measure of how the data are distributed around the mean. A distribution is skewed to the right if the tail of the distribution extends to the right, and it is skewed to the left if the tail of the distribution extends to the left.

Kurtosis is a measure of how peaked the data are. A distribution is leptokurtic if it is more peaked than a normal distribution, and it is platykurtic if it is less peaked than a normal distribution.

Averages are a useful tool for summarizing data, but they should be used with caution. It is important to understand the different types of averages and their strengths and weaknesses.

What is the difference between mean, median, and mode?

The mean is the average of a set of numbers. It is found by adding all the numbers in the set and then dividing by the number of numbers in the set. The median is the middle number in a set of numbers arranged in order from least to greatest. The mode is the most frequent number in a set of numbers.

What is the standard deviation?

The standard deviation is a measure of how spread out a set of numbers is. It is found by taking the square root of the variance. The variance is a measure of how much the numbers in a set vary from the mean.

What is the range?

The range is the difference between the largest and smallest numbers in a set.

What is the interquartile range?

The interquartile range is a measure of the spread of a data set. It is calculated by finding the difference between the third and first quartiles.

What is the coefficient of variation?

The coefficient of variation is a measure of how spread out a set of numbers is relative to its mean. It is calculated by dividing the standard deviation by the mean.

What is the skewness of a distribution?

The skewness of a distribution is a measure of how asymmetrical the distribution is. A distribution with a positive skew has a longer tail on the right, while a distribution with a negative skew has a longer tail on the left.

What is the kurtosis of a distribution?

The kurtosis of a distribution is a measure of how peaked the distribution is. A distribution with a high kurtosis is more peaked than a distribution with a low kurtosis.

What is the normal distribution?

The normal distribution is a Probability distribution that is often used to model real-world data. It is characterized by a bell-shaped curve, with the mean, median, and mode all being equal.

What is the binomial distribution?

The binomial distribution is a probability distribution that is used to model the number of successes in a sequence of independent experiments. Each experiment has two possible outcomes, success or failure, and the probability of success is constant for each experiment.

What is the Poisson distribution?

The Poisson distribution is a probability distribution that is used to model the number of events that occur in a given interval of time or space. The Poisson distribution is often used to model rare events, such as the number of car accidents that occur in a given day.

What is the chi-squared distribution?

The chi-squared distribution is a probability distribution that is used to test the goodness of fit of a model to data. The chi-squared distribution is also used to test the independence of two variables.

What is the t-distribution?

The t-distribution is a probability distribution that is used to test the difference between two means. The t-distribution is also used to construct confidence intervals for the mean.

What is the F-distribution?

The F-distribution is a probability distribution that is used to test the Equality of two variances. The F-distribution is also used to construct confidence intervals for the variance.

What is the logistic distribution?

The logistic distribution is a probability distribution that is used to model the probability of an event occurring. The logistic distribution is often used in logistic regression.

What is the Weibull distribution?

The Weibull distribution is a probability distribution that is used to model the lifetime of a product or system. The Weibull distribution is often used in reliability engineering.

What is the Pareto distribution?

The Pareto distribution is a probability distribution that is often used to model the distribution of wealth or income. The Pareto distribution is also known as the 80/20 rule, because it is often observed that 80% of the wealth or income is concentrated in the hands of 20% of the Population.

What is the log-normal distribution?

The log-normal distribution is a probability distribution that is used to model the logarithm of a variable. The log-normal distribution is often used to model the distribution of income or wealth.

What is the gamma distribution?

The gamma distribution is a probability distribution that is used to model the time between events. The gamma distribution is often used in reliability engineering.

What is the beta distribution?

The beta distribution is a probability distribution that is used to model the probability of an event occurring between two values. The beta distribution is often used in Bayesian statistics.

What is the uniform distribution?

The uniform distribution is a probability distribution that is used to model a variable that can take on any value within

Sure, here are some multiple choice questions about the topics of mean, median, mode, and range, without mentioning the topic of average:

Which of the following is a measure of central tendency?
(A) Mean
(B) Median
(C) Mode
(D) Range
Which of the following is the middle value in a set of data?
(A) Mean
(B) Median
(C) Mode
(D) Range
Which of the following is the most frequent value in a set of data?
(A) Mean
(B) Median
(C) Mode
(D) Range
Which of the following is the difference between the largest and smallest values in a set of data?
(A) Mean
(B) Median
(C) Mode
(D) Range
The mean of a set of data is 5. If one of the numbers in the set is 10, what is the new mean?
(A) 5
(B) 6
(C) 7
(D) 8
The median of a set of data is 5. If one of the numbers in the set is 10, what is the new median?
(A) 5
(B) 6
(C) 7
(D) 8
The mode of a set of data is 5. If one of the numbers in the set is 10, what is the new mode?
(A) 5
(B) 6
(C) 7
(D) 8
The range of a set of data is 5. If one of the numbers in the set is 10, what is the new range?
(A) 5
(B) 6
(C) 7
(D) 8
A set of data has a mean of 5, a median of 6, and a mode of 7. What is the range of the data?
(A) 2
(B) 3
(C) 4
(D) 5
A set of data has a mean of 5, a median of 6, and a mode of 7. What is the standard deviation of the data?
(A) 1
(B) 2
(C) 3
(D) 4