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What is the weighted mean of first 10 natural numbers?

  • Last Updated : 13 Oct, 2021

In daily lives, people encounter various situations where dealing with numerical facts or information is required. These facts can pertain to anything, be it a teacher grading students’ papers, data about the ages of people in one’s locality, monthly consumption of groceries by a family for one whole year, etc. It is only natural that a teacher would like to know the class average or the total percentage of students who cleared the test. A person collecting data on age groups might be interested in knowing how many people around him are of the same age group as him. A family which prepares such a chart showing monthly consumption might be interested in comparing it with the previous year or to cut down certain expenses.

All those activities discussed are termed as ‘analysis’. It is imperative to note that they collected all such facts first, organized them in a meaningful pattern, then analyzed them to form an interpretation, and then take necessary action. Hence, one cannot do such an interpretation without collecting numerical facts first. Also, it is naturally not possible to collect the data and expect it to yield some meaningful conclusion. One needs to use some tools or procedures in order to arrive at a conclusion. This is when the concept of statistics comes into the picture.

What is Statistics?

In simple words, statistics implies the process of gathering, sorting, examine, interpret and then present the data in an understandable manner so as to enable one to form an opinion of it and take necessary action, if necessary. Examples:

  • A teacher collecting students’ marks, organizing them in ascending or descending manner, and calculating the average class marks, or finding the number of students who failed, informing them so that they start working hard.
  • Government officials collecting data for the census, and comparing it with previous records to see whether population growth is in control or not.
  • Analyzing the number of followers of a particular religion of a country.

Statistical Tools

The most popular tools of statistics are as follows:

  • Arithmetic Mean: Also known as average, the arithmetic mean for a given set of data is calculated by adding up the numbers in the data and dividing the sum so obtained with the number of observations.
  • Median: Such a value as separates the higher and lower values of a given set of statistical data is called the median.
  • Mode: Such a value as occurs most frequently in a given series of statistical data is called the mode.
  • Standard Deviation: Such a value as indicates the extent to which certain values of a statistical series tend to vary or disperse from its mean or median is called standard deviation.
  • Range: Such a value depicts the difference between the highest and lowest values in a series.
  • Correlation: Such a statistical tool as helps study the relationship between two variables is called correlation.

Arithmetic Mean

Arithmetic mean also known as average, arithmetic mean for a given set of data is calculated by adding up the numbers in the data and dividing the sum so obtained with the number of observations. It is the most popular method of central tendency.

Properties of Arithmetic Mean

  • Deviations from the arithmetic mean of all items in a statistical series would always add up to zero, i.e. ∑(x – X) = 0.
  • The squared deviations from the arithmetic mean is always minimum, i.e., less than the sum of such square deviations from other values like the median, mode, or another tool.
  • Replacing all the items in a statistical series with its arithmetic mean has no effect on the sum of the said items.

Types of Arithmetic Mean

There are two types of arithmetic mean, namely:

  • Simple Arithmetic Mean
  • Weighted Arithmetic Mean

Simple Arithmetic Mean

Simple arithmetic mean for a given set of data is calculated by adding up the numbers in the data and dividing the sum so obtained with the number of observations. It is the most popular method of central tendency.

Formula

The arithmetic mean is calculated using the following formula,

Sum of observation/ Number of observations

Mean of the series = x̄ = Σx/ N.



Weighted Arithmetic Mean

Sometimes, in a statistical series, certain items tend to contribute more to the arithmetic mean than other items. On the basis of the extent of influence, these items have over the arithmetic mean, such items are then assigned certain values, called ‘weights’ which represent the magnitude of such influence. Such arithmetic mean which is calculated by taking into consideration these weights assigned to statistical observations is called weighted arithmetic mean.

Formula

W = \frac{Σw_1x_1}{Σw_1}

Here, W represents the weighted arithmetic mean, w1 denotes the weights assigned to different items in the given statistical series and x1 denotes the various items in the given data set.

What is the weighted mean of the first 10 natural numbers?

Solution: 

Considering that the weights of the given numbers are corresponding to their magnitudes, we have:

X

W

XW



1

1

1

2

2

4

3

3

9

4

4

16

5

5

25

6

6

36

7

7



49

8

8

64

9

9

81

10

10

100

 

ΣW = 55

ΣXW = 385

Weighted Average(W) = \frac{ΣXW}{ΣW}

\frac{385}{55}

= 7

Hence the weighted average of given numbers is 7.

Conceptual Questions

Question 1. Find the weighted arithmetic mean of the following observations: 1, 2, 3, 4 if their weights are 73, 378, 459, and 90.

Solution:

X



W

XW

1

73

73

2

378

756

3

459

1377

4

90

360

 

ΣW  = 1000

ΣXW = 2566

Weighted Average(W) = \frac{ΣXW}{ΣW}

\frac{2566}{1000}

= 2.566

Hence the weighted average of given numbers is 2.566

Question 2. Find the weighted arithmetic mean of the following observations: 20, 30, 40, 50 if their weights are 2, 3, 4, and 1.

Solution:

X

W

XW

20

2

40

30



3

90

40

4

160

50

1

50

 

ΣW = 10

ΣXW = 340

Weighted Average(W) = \frac{ΣXW}{ΣW}

\frac{340}{10}

= 34

Hence the weighted average of the given numbers is 34.

Question 3. Find the weighted arithmetic mean of the following observations: 7, 5, 8, 4 if their weights are 9, 3, 2, and 1.

Solution:

X

W

XW

7

9

63

5

3

15

8

2

16

4

1

4

 

ΣW = 15

ΣW = 98

Weighted Average(W) = \frac{ΣXW}{ΣW}

\frac{98}{15}

= 6.53

Hence the weighted average of the given numbers is 6.53

Question 4. Find the weighted arithmetic mean of the following observations: 50, 55, 60, 65, 70, 75 if their weights are 4, 10, 8, 20, 5, and 3.



Solution:

X

W

XW

50

4

200

55

10

550

60

8

480

65

20

1300

70

5

350

75

3

225

 

ΣW = 50

ΣXW = 3105

Weighted Average(W) = \frac{ΣXW}{ΣW}

= \frac{3105}{50}

= 62.1

Hence the weighted average of the given numbers is 62.1


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