Average

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.

Introduction

The average of a number is a measure of the central tendency of a set of numbers. In other words, it is an estimate of where the centre point of a set of numbers lies.
The basic formula for the average of n numbers x1 , x2 , x3,…xn is

Avg = (x1+x2+ …. + xn)/n

The average [also known as arithmetic mean (AM)] of a set of numbers can also be defined as the number by which we can replace each and every number of the set without changing the total of the set of numbers.

Illustration

The average of 4 numbers 12, 13, 17 and 18 is:
Avg=(12+13+17+18)/4 = 60/4= 15
This means that if each of the 4 numbers of the set were replaced by 15 each, there would be no change in the total. This is an important way to look at averages. This view is a very important way to visualise averages.
This can be visualised as
12 → +3 →15
13 → +2 →15
17 → -2 →15
18 → -3 →15
60 → 0 → 60
Average of numbers is always greater than the smallest number in a set and is always less than the largest number.
In above example, 12 < Avg < 18 (i.e. Average is greater than 12 and less than 18)
The net deficit due to the numbers below the average always equals the net surplus due to the numbers above the average.

Ages and averages:

If the average age of a group of persons is x years today then after n years their average age will be (x + n).
Also, n years ago their average age would have been (x – n). This happens due to the fact that for a group of people, 1 year is added to each person’s age every year.

Understanding the Mean

The mean, often referred to as the average, is calculated by summing up all values in a dataset and dividing by the number of values. It’s highly sensitive to outliers and can be influenced by extreme values.

Exploring the Median

The median is the middle value in a dataset when it’s arranged in ascending or descending order. It’s less affected by outliers and is a robust measure of central tendency.

Diving into the Mode

The mode represents the most frequently occurring value in a dataset. It’s particularly useful for categorical data and can sometimes provide multiple modes.

Comparing Mean, Median, and Mode

Each of these measures has its strengths and weaknesses. Understanding when to use mean, median, or mode depends on the dataset’s characteristics and the insights you seek.

Weighted Averages: Adding Complexity

Weighted averages assign different levels of importance to various values. This is especially useful when dealing with datasets where some values hold more significance than others.

Harmonic Mean: When Rates Matter

The harmonic mean is essential for situations involving rates or ratios. It’s calculated by dividing the number of values by the reciprocal of each value’s weight.

Geometric Mean: Unveiling Growth Rates

The geometric mean is ideal for calculating average growth rates or returns. It considers the compounding effect of percentages.

Central Limit Theorem: Averaging in Samples

The central limit theorem states that the distribution of sample means approaches a normal distribution, regardless of the original data’s distribution. It’s crucial for inferential statistics.

Moving Average: Trends and Fluctuations

A moving average smooths out fluctuations in data, revealing trends over time. It’s widely used in analyzing financial data, stock prices, and market trends.

Exponential Moving Average: Reacting to Recent Data

The exponential moving average places more weight on recent data points, making it highly responsive to changes in trends or patterns.

Weighted Moving Average: Assigning Significance

In a weighted moving average, different weights are assigned to different data points. This technique is useful for emphasizing recent or crucial data.

Trimmed Mean: Trimming Extremes for Stability

A trimmed mean involves removing a certain percentage of extreme values from a dataset before calculating the average. It’s valuable when outliers significantly affect the mean.

Interquartile Range: Dispersion Around the Median

The interquartile range (IQR) represents the spread of data around the median. It’s calculated as the difference between the third and first quartiles.

Practice Questions

1. The average of the first 50 natural numbers is
a. 12.25
b. 21.15
c. 25
d. 25.5

2. In a class, there are three groups A, B and C. If one student from group A and two students from group B are shifted to group C, then what happens to the average weight of the students of the class?
a. Increases
b. Decreases
c. Remain same
d. Data insufficient

3. The average age of 10 boys and the principal is 15 years. When the principal’s age is excluded, the average age decreases by 1 years. What is the age of the principal?
a. 29
b. 31
c. 25
d. 33

4. The average of a batsman after 25 innings was 56 runs per innings. If after the 26th inning his average increased by 2 runs, then what was his score in the 26th inning?
a. 108
b. 109
c. 110
d. 107

5. If we take four numbers, the average of the first three is 20 and that of the last three is 10. If the first number is 20, the last number is
a. -20
b. -10
c. -30
d. -15

6. The average height of 30 girls out of a class of 40 is 160 cm and that of the remaining girls is 156 cm. The average height of the whole class is?
a. 155 cm
b. 157 cm
c. 159 cm
d. None of these

7. Ram bought 2 toys for Rs. 5.50 each, 3 toys for Rs.3.66 each and 6 toys for Rs.1.833 each. The average price per toy is (in Rs)
a. 3
b. 10
c. 5
d. 9

8. 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

9. 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

10. Find the average of numbers 87, 84, 86, 90, 82, 88, 78.
a. 85
b. 84
c. 83
d. 82

11. The average monthly income of a person in a certain family of 5 Rs. 10,000. What will be the average monthly income of a person in the same family if the income of one person increased by Rs. 1,20,000 per year? (UPSC 2016)
a. Rs.12,000
b. Rs.16,000
c. Rs.20.000
d. Rs.34.000

12. The average marks of 100 student are given to be 40. It was found later that marks of one student were 53 which were misread as 83. The corrected mean marks are (UPSC 2019)
a. 39
b. 39.7
c. 40
d. 40.3

13. A family has two children along with their parents. The average of the weights of the children and their mother is 50 kg. The average of the weights of the children and their father is 52 kg. If the weight of the father is 60 kg, then what is the weight of the mother?
(UPSC 2019)
a. 48 kg
b. 50 kg
c. 52 kg
d. 54 kg

14. The average score of a batman after his 50th innings was 46.4. after 60th innings, his average score increases by 2.6. What was his average score in the last ten innings?(UPSC 2020)
a. 122
b. 91
c. 62
d. 49

15. In a class, there are three groups A, B and C. If one student form group B are shifted to group C, then what happens to the average weight of the students of the class? (UPSC 2020)
a. It increases
b. It decreases
c. It remains the same
d. No conclusion can be drawn due to the insufficient data

16. The average age of a teacher and three students is 20 years. If all the three students are of same age and the difference between the age of the teacher and each student is 20 years, then what is the age of the teacher? (UPSC 2020)
a. 25 years
b. 30 years
c. 35 years
d. 45 years

17. 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

18. The average of first 10 natural numbers is?
a. 5
b. 5.5
c. 6
d. 6.5

**19. There are two classes A and B having 25 and 30 students respectively. In class-A the highest score is 21 and lowest score is 17. In class-B the highest score is 30 and lowest score is 22. Four students are shifted form class-A to class-B. (UPSC 2021)

Consider the following statements:

The average score of class-B will definitely decrease.**
2. The average score of class-A will definitely increase.
Which of the above statements is/are correct?
a. 1 only
b. 2 only
c. Both 1 and 2
d. Neither 1 nor 2

20. Without any stoppage, a person travels a certain distance at an average speed of 42 km/h, and with stoppages he covers the same distance at an average speed of 28 km/h. How many minutes per hour does he stop?
a. 14 minutes
b. 15 minutes
c. 23 minutes
d. 28 minutes

21. The average age of 10 men is increased by 3 years when one of them, whose age is 54 years is replaced by a woman. What is age of the woman?
a. 68 years
b. 82 years
c. 72 years
d. 84 years

22. The average weight of A, B, C is 40 kg, the average weight of B, D, E is 42 kg and the weight of F is equal to that of B. what is the average weight of A, B, C, D, E and F?
(UPSC 2022)
a. 40.5 kg
b. 40.8 kg
c. 41 kg
d. Cannot be determined as data is inadequate

23. Three maths classes: X, Y and Z take an algebra test. The average score of class X is 83. The average score of class Y is 76. The average score of class Z is 85. The average score of class X and Y is 79 and average score of class Y and Z is 81. What is the average score of classes X, Y, Z?
a. 81.5
b. 80.5
c. 83
d. 78

24. The average marks of a group of 20 students on a test is reduced by 4 when the topper who scored 90 marks is replaced by a new student. How many marks did the new student have?
a. 30
b. 10
c. 20
d. 40

25. The average of 9 numbers is 14. If each number is increased by 4, the new average will be
a. 16
b. 17
c. 15
d. 18

26. The average score of a cricketer in three matches is 44 runs and two other matches, it is 33 runs. Find the average in all the five matches.
a. 39.6
b. 29
c. 49.6
d. 39

27. The average of the first six multiples of 5 is
a. 18.50
b. 21
c. 28
d. 17.50

28. The average of 17 numbers is 10.9. If the average of first nine numbers is 10.5 and that of the last nine numbers is 11.4, the middle number is
a. 11.8
b. 11.4
c. 10.9
d. 11.7

29. The average of the first ten even numbers is
a. 18
b. 22
c. 9
d. 11

30. Find average of natural numbers from 1 to 67?
a. 33.5
b. 34
c. 50
d. 67