Vedanta Economics Book 10 Final (2077)

2. Discrete Series:

Step I: Arrange the data in ascending or descending order. Change the
corresponding frequency along with the items.

Step II: Find cumulative frequency.

Step III: Find the size of N+1/2 th item where N=∑f.

Step IV: The cumulative frequency greater or equal to N+1/2 th item is taken and

the variable corresponding to this cumulative value is median.

Example 1. Calculate median from the following observation.

X 5 12 15 20 25

F 73 2 34

Solution:

x f C. f.
5 7 7
12 3
15 2 10
20 3 12
25 4 15
Total N=19 19

Median size N +1 In discrete frequency distribu-
= 192+ 1 tion, cumulative frequency table
= 2 KEY has to be prepared to compute
IDEA Median

= 1011 item.

Here cf equals to 1011 item is 10.

So, median =l2(corresponding value of 10)

Example 2: Find median weight from the following data.

Weight(Kg) 40 60 50 70 30 80

Frequency 2 24 5 6 2

Solution

Weight(kg) Frequency (F) C.f.
30 6 6
40 2 8
50 4 12
60 2 14
70 5 19
80 2 21
21
Total



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Solution: Median size Points to note

N +1 th item • The median is the middle score
= 2 th item for a set of data that has been
arranged in order of magnitude.
21 + 1
= 2 • The median is less affected by
outliers and skewed data
= 11th item.
• Outliers are extreme
Here cf greater than 1111 item is 12. observations in a distribution

So, median=50 (corresponding value of 12) • Median can also be computed
in open end distribution
3. Continuous Series:

Step I: Find cumulative frequency.

Step II: For median class, find N/2 th item where N=∑f

Step III: Take cumulative frequency greater or equal to N/2 th item and the

corresponding class of this cumulative frequency is median class.

Step IV: For exact value of median, following formula is used.

Median = L + N – c.f ×h
Where, 2

f

L = lower limit of the median class. N = ∑f

cf = preceding cumulative frequency of median class.

f = frequency of median class. h = class interval.

Example: Find median of the data given below.

Marks 0-20 20-40 40-60 60-80 80-100
No. of students 8 12 16 10 4
Solution:
c.f.
Marks No. of students(f) 8
0-20 8
20
20-40 12 36
40-60 16 46
60-80 10 50
80-100
Total 4
N=50

N 50 Median is based on position
Size of 2 = 2 th item and not on the values. Positional
KEY averages are based on the posi-
= 25 th item IDEA tion of the given observation in a
series, arranged in an ascending
Cf just greater than 25 is 36. or descending order.

So, median class=40-60 202
Vedanta High School Economics - 10

Median = L + N – c.f ×h Key Takeaways
2
• The median is that value which
f divides the group into two equal
parts, one part comprising all
Here, L=40, cf=36, f=16, h=20 values greater, and the other,
all values less than median.
Median =40+ ( 25 – 20 ) × 20
16 • In other words, the median of
a sequence of numbers is the
= 40 + 6.25 value that lies in the middle of
the sequence when sorted from
= 46.25 smallest to largest or vice versa.

Merits of Median

1. It is easy to understand.

2. It is quite rigidly defined.

3. It does not require all the observations for its determination.

4. Median is useful in case of open-ended series such as income distribution since median
is based on the position and not on the values of items.

5. Median is easier to compute than mean in case of unequal class-intervals.

6. It is not affected by extreme values.

7. It is most suitable average for dealing with qualitative data i.e., where ranks are given.

8. It can be determined graphically.

9. It is capable of being expressed in qualitative form as it is not computed but located.

10. It gives a value, which very much exists in the series, and is a round figure in most of
the cases.

Limitations of Median

1. It requires the arrangement of data in ascending/descending order.

2. It is not rigidly defined and as such its value cannot be computed but located.

3. It is not based on all items of the series.

4. It is not capable of algebraic treatment. Its formula cannot be extended to calculate
combined Median of two or more related groups.

5. It is affected more by sampling fluctuations than the value of mean.

6. The computation formula of a median is in a way an interpolation under the assumption
that the items in the median class are uniformly distributed, which is not very true, i.e. in
case of continuous series, the median is estimated, but not calculated.

7. Typical representative of the observations cannot be computed if the distribution of item
is irregular. For example 1, 2, 3, 100 and 500, the median is 3.

8. It ignores the extreme items.

9. Where the number of items is large, the pre-requisite process of arranging the items is a
difficult process.

203 Vedanta High School Economics - 10

10.3 MODE

Mode is the item which has the highest frequency. Mode is often said to be that value
in a series which occurs most frequently or which has the greatest frequency. Mode is
that value of the set of numbers which is repeated maximum numbers of time. It is also
called the most typical or fashionable value of a distribution because it is the value
which has the greatest frequency density in its immediate neighbourhood.

According to Boddington, “Mode may be defined as the predominant kind, type or size
or item or the position of greatest density”.

Calculation of Mode

1. Individual Series:

The terms are arranged in any order, ascending or descending. If each term of the series
occurs once, then there is no mode, otherwise the value that occurs maximum number
of times is known as Mode.

Method to calculate Mode: Inspection Method

1. Arrange the terms in ascending or descending order

2. Note the term occurring maximum number of times if any.

3. This term is mode. Mode is the value which re-
Example 1. Find mode from the given set of data.
2,4,5,6,5,3,2,4,5,6,8,9,6,3,2,5,4,2,6.2,7,2,8 peats the maximum times in
Solution: Here the most repeated value is 2.
So, mode=2 KEY a diatribution. It is an actual
IDEA value, which has the highest

concentration of items in and

around it.

Note: If two number are repeated maximum time it is the case of bim­ odal and in case of
more than two numbers which have highest frequency is the case of multi modal. In this
case the following formula is used. Mode = 3 median - 2 mean

Example 2. Income of 12 worker is 40,30,20,40,60,50,30,20,30,40

Solution: Here 30 and 40 are repeated equally maximum time, in this case we use the
formula: Mode=3 Median-2 ean

Income No. of workers(f) fx cf

20 2 40 2
30 3 90 5
40 3 120 8
50 1 50 9
60 1 60 10
Total N=10 ∑fx=360

∑fx In discrete series, the value
Mean =N corresponding to the maximum
KEY frequency is the mode. The mode
360 IDEA of the data is the value that is
= 10 most common in the dataset.

= 36

Vedanta High School Economics - 10 204

Points to note
N +1
• Mode is the most fashionable
Size of 2 10 + 1 value in a distribution
= 2
• The items tend to concentrate
= 5.5 th item. most heavily around mode

Cf just greater than 5.5 is 8. • We get the value of Mode by
Interpolation as is the case
So, median = 40 (corresponding value of 8) with Median

Mode = 3 median- 2 mode • Mode can also be computed
in open end distribution
= 3 × 40 - 2 × 36

= 120-72

= 48

2. Discrete Series:

Under Inspection method Mode is that value of the variable which has the highest
frequency. But this method is applicable if the following conditions are fulfilled:

1. There must be one item which has maximum frequency.
2. The maximum frequency should be located at most in the middle or centre of the series.
3. The frequencies should follow a particular pattern, i.e., they should gradually rise and

reach maximum level and thereafter decline.

Note: In case of bi-­modal class mode is computed by relation method. The following
formula is used Mode = 3 median - 2 mean

Example: Find the mode of the distribution.

x 4 56 7 8

f 2 35 3 1

Solution: In the table maximum frequency is 5. That is 6 is repeated maximum times.
So, Mode= 6

3. Continuous Series:

Modal class is the class which has the highest frequency. We use the following formula

to calculate mode for continuous series. Points to note

Mode=L + f1 – f0 ×h • We get the value of Mode by
Where, 2f1 – f0 – f2 Interpolation as is the case
with Median
L= lower limit of the lower class.
• In computing mode, the classes
fo= frequency of preceding class of modal class. have to be converted into
f,= frequency of modal class exclusive classes

fz= frequency of next class of modal class. • Mode can also be computed
h=class interval in open end distribution

• Mode can also be computed
as Mode = 3Median-2Mean

205 Vedanta High School Economics - 10

Example : Find the Mode from the following distribution.

Marks 0-20 20-40 40-60 60-80 80-100
8 12 16 10 4
No. of students

Solution: By inspection, the maximum frequency is 16 and its corresponding class is

40-60. Key Takeaways
So, Modal class=40-60
Here, L=40, f1=16, fo=12, fz=10 and h=20 • The mean is the average value
of all the data in the set.

Using formula, f1 – f0 ×h • The median is the middle value
Mode = L + 2f1 – f0 – f2 in a data set that has been
arranged in numerical order
= 40 + 16-12 × 20 so that exactly half the data is
= 40 + 2×16-12-10 above the median and half is
4 below it.
10 × 20
• The mode is the value that
occurs most frequently in the
set.

= 40+8 • In a normal distribution, mean,
= 48 median and mode are identical
in value.

Merits of Mode

1. It is easy to understand as well as easy to calculate. In certain cases, it can be found out
by inspection

2. It is usually an actual value as it occurs at the highest frequency in the series.

3. In certain situations mode is the only suitable average, e.g., modal size of garments,
modal size of shoes, modal wages, etc.

4. It is not affected by extreme values.

5. Its value can be determined in open-end distribution without ascertaining the class-
limits.

6. It can be used to describe qualitative phenomenon. For e.g. in market research to
determine the consumer preferences for different types of products, modal preference
can be used.

7. Its value can be determined graphically.

8. It indicates the point of maximum concentration in case of highly skewed or non-
normal distributions.

Limitations of Mode

1. In case of bi-modal/multimodal series, mode cannot be determined.

2. It is not capable of further algebraic treatment. For example, combined mode of two or
more series cannot be calculated.

3. It is not based on all the items of series.

4. It is not rigidly defined measure because different formulae give somewhat different
answers.

5. Its value is affected significantly by the size of the class-intervals.

6. It is stable only when the sample is large.
Vedanta High School Economics - 10
206

10.4 PRICE INDEX NUMBER

The value of all goods and services are measured in terms of money. But the value of
money cannot be expressed in money itself. Hence, the change in the value of money is
measured on the basis of the average of prices of some selected commodities. The average
of such prices is called price level and the series of these prices is called index number.

The value of money changes now and then. It depends upon the price level. This change in
price level is measured by the price index number. In brief, index number is the statistical
method of measuring the change in value of money between any two periods.

According to Secrist, “The index number is that series from which the change in time or
place of any fact is measured”.

In the words of L. C. Chandler, “An index number of prices is a figure showing the height
of average prices at one time relative to their height at some other time that is taken as the
base period.”

STEPS TO FOLLOW WHILE CONSTRUCTING A PRICE INDEX NUMBER

We need to observe the following steps in sequence while constructing a price index
number.

1. Statement of the Purpose: The purpose of index number must be clearly stated before
doing anything else. This is because we cannot construct an index number that fulfills all
purposes. Every index number has got its own uses. Moreover, nature of the information
to be collected and the method to be followed for the construction of the index number
depend mainly on the purpose of the index number.

2. Selection of the Base Period: A base year is one with reference to which price changes
in the current year are expressed. The base year must be selected with care. A base year
should be a normal year free from heavy and unnatural price level fluctuations due to the
occurrence of wars, earthquake, floods, violent political movements, etc.

3. Selection of Commodities: The number or types of commodities to be selected for the
construction of an index number depends largely on the purpose and scope of the index
number. The commodities selected should be representative of the tastes, habits, customs
and necessities of the people to whom the index number relates.

4. Collection of Market Prices: Generally we have two types of market price: wholesale and
retail. Wholesale prices are generally more stable than retail prices. Which prices are
taken depends upon the purpose or type of index number. Again, the prices should be
collected from those market places which are most frequently visited by the people to
whom the index number relates.

5. Selection of Appropriate Average: Index number can be constructed with the help of
mean. The geometric mean is the most appropriate average for the construction of the
index number in order to measure the percentage change in the price level.

6. Selection of Method: There are two methods of constructing a price index unweighted
or simple and weighted. If all commodities selected are equally important, then a simple
index number can be constructed and if some commodities are more important than the
others, then a weighted index should be computed. Weights are determined with reference
to the relative amounts of income spent on the commodities by the consumers. More
weights are given to more important commodities.

207 Vedanta High School Economics - 10

CONSTRUCTION OF A SIMPLE PRICE INDEX NUMBER

A simple price index number can be computed by using the following formula:

01 = ∑ 1 × 100
∑ 0

where,

P01 = price index for the current year with reference to the base year,
P1 = Price of current year and
P0 = Price of base year.
∑=Summation.
An example:

Calculate a simple price index from the following data. Use 2075 as the base year.

Items Rice LPG Electricity Newspaper Entertainment
8 5 100
Prices in 50 1000
2075 (Rs) 10 10 120

Prices in 70 1300
2076 (Rs)

Solution:

Computation of the Price Index

Items Base year Prices (P0) Current Year Prices (P1)
50 70
Rice 1000 1300
8 10
Gas 5 10
100 120
Electricity ∑P0=1163 ∑P1=1510

Newspaper The price index number meas-
ures the change in the price level
Entertainment KEY It can be weighted or unweight-
IDEA ed
N=5

Now using the formula,

01 = ∑ 1 × 100
∑ 0
1510
= 1163 × 100

= 129.8%

This implies that the price level in 2076 has risen by 29.8 % (i,e 129.8%-100%)

PROBLEMS IN THE CONSTRUCTION OF PRICE INDEX NUMBER

Economists and statisticians face a lot of problems while constructing a price index
number. The following are the major problems:

1. Selection of the Base Year: The index number is based on the base year. The base year
should be a normal year when prices are stable. Hence, it is difficult to select a base year.

2. Representative Commodities: The accuracy of index number depends on the selection

Vedanta High School Economics - 10 208

of representative commodities. People consume different commodities at different time.
Due to change in taste and fashion some old commodities may not be consumed and some
new commodities may be consumed. Hence, it is difficult to select the representative
commodities.
3. Problem of Weight: The commodities should be given proper weight for the accuracy and
reliability of index number. There is no definite basis of giving weight. Hence, it becomes
arbitrary. It is thus difficult to determine the importance and weight of a commodity.
There is great difficulty to decide how much of each commodity to select. Hence, there are
chances of weights being fixed in an arbitrary manner.
4. Difficulty of Calculating Average: The average should be calculated to construct an index
number. There are different methods of calculation. Which give different results, Hence,
it is difficult to compare averages.
5. Selection of Price: There are wholesale price as well as retail priceto construct an index
number. But the ordinary consumers purchase goods and services in retail prices. It is
really difficult to collect information about prices. The problem is particularly visible in
case of retail prices due to their variation even in the same market.
6. Collection of Data: There is difficulty in collection of data. Many people are reluctant to
give information regarding their expenditure pattern. Even when they are willing to give
information they are often unwilling to admit the full amount they spend on things such
as wine, cigarette, and entertainments.

EXTRA READINGS

209 Vedanta High School Economics - 10

CONCEPTS FOR REVIEW

Variable Discrete variable Continuous variable

Frequency Individual series Discrete series

Continuous series Tabulation Stub

Exclusive series Inclusive series Caption

Histogram Central tendency Average

Mean Median Mode

UNIT OVERVIEW

Parts of Statistical table shows the height.
Measure of central tendency
1. Title
2. Captions and stubs The combined term “Measures of central
3. Body of the table tendency “means the methods of finding
4. Headnote out the central value or average value of
5. Footnote a statistical series or any other series of
6. Source quantitative information. It is also called
Advantages of Diagrams and Graphs average.
Types of average
1. Attractive and effective presentation
2. Simple presentation of complex data 1. Mean
3. Remembrance for long period
4. Universal utility 2. Median
5. Information as well as entertainment
6. Saves time 3. Mode
7. Helpful in predictions
8. Useful in Comparison Characteristics of an Average
Types of diagrams 1. It is a single figure expressed in some

1. One dimensional diagrams such as bar quantitative form.
diagrams. 2. It lies between the extreme values of a

2. Two dimensional diagrams such as series.
rectangles, squares, circles. 3. It is a typical value that represents all the

3. Three dimensional diagrams such as values in a series.
cylinders, cubes. 4. It is capable of giving a central idea about

Types of Bar Diagrams the series it represents.
5. It is determined by some method or

procedure.

A. Simple bar diagram Requisites of an Ideal Average
B. Sub divided diagram 1. It should be rigidly defined.
C. Multiple bar diagram 2. It should be easy to understand.
D. Percentage bar diagram 3. It should be simple to compute.

Pie chart 4. Its definition should be in the form of

Pie diagram is a circle divided into different mathematical formula.
sectors with each sector representing a 5. It should be based on all the items in the
particular component. The total magnitude is
taken as 360o and the individual components distribution.
are expressed in angular terms. 6. Any single item or a group of items should
Histogram
not unduly influence it.
Histogram is most popular tools for graphical 7. It should be capable of further algebraic
representation. In case of continuous series
we draw histogram. The class interval treatment.
represents the width of the bar and frequency 8. It should facilitate further statistical

computation.
9. It should have sampling stability.

Vedanta High School Economics - 10 210

QUESTIONS FOR REVIEW
Answer the following questions:
1. What are the requisites of an ideal average?
2. What are the objectives of an average?
3. What are the advantages of an average?
4. What are the merits and demerits of mean
5. What are the merits and demerits of median?
6. What are the merits and demerits of mode?
7. What are the steps in the construction of Price Index Number?
8. What are the difficulties in the construction of Price Index Number?
9. Calculate average marks from the following data.

Marks 0-10 10-20 20-30 30-40 40-50
No. of Students
4 6 15 25 10

10. If 10,20, 40, 30, 50 and 60 are the marks secured by the students, find the median
marks.

11. Find the median marks of the class 10 students of a School in final exam of Economics.

Marks 40-50 50-60 60-70 70-80 80-90 90-100
4 7 3
No of Students 6 5 8

12. Find the mode from the following data:

a. 34, 35,36,33,38, 38, 36, 34 & 38.

b. 20, 30, 25, 30, 35, 35, 35, 25, 30, 36, 20 & 36

13. Find the mode from the following distribution of number of shoes sold in a day.

Size of shoes 4 5 67 89

No of shoes sold 3 5 7 10 3 2

14. The marks obtained by 50 students in economics are given below. Construct frequency
distribution table taking class interval 10 and find the mean.

42, 34, 33, 29, 27, 37, 59, 53, 41, 53, 51, 39, 21, 31, 42, 21, 51, 37, 42, 37, 38, 42, 49, 52,
38, 53, 14, 39, 42, 44, 59, 39, 71, 17, 33, 61, 47, 57, 57, 7, 27, 19, 54, 61, 43, 42, 16, 37,
67, 62

211 Vedanta High School Economics - 10

15. Calculate the median and mode from the following data,

Value Frequency

less than 10 4

less than 20 20

less than 30 35

less than 40 55

less than 50 62

less than 60 67

16. Calculate the mode for the marks distribution of 58 students from the data given below;

Marks 30-40 40-50 50-60 60-70 70-80 80-90
8
Frequency 10 7 12 12 9

17. Find the missing frequency of the following data, if the mean is 33.

Marks 0-10 10-20 20-30 30-40 40-50 50-60

Frequency 5 10 ? 30 20 10

18. Find the arithmetic mean of the following data using step-deviation method.

Daily wages 100-200 200-300 300-400 400-500 500-600
No of workers 20 30 45 40 10

19. Construct a Simple Price Index Number from the information given below:

Items Price in 2074 (Rs) Price in 2075 (Rs)
Rice Per Kg 50 60
Oil Per Litre 180 210
Sugar Per Kg 60 75
Lentil Per Kg 140 150

Vedanta High School Economics - 10 212

ADDITIONAL QUESTIONS FOR PRACTICE

1. Construct a simple bar diagram from the following data.

Year 2072 2073 2074 2075 2076
8500
Production(in quintal) 4000 5000 7000 6500

2. Represent the given data by percentage bar diagram.

Items Food Education Rent Entertainment Miscellaneous

Expenditure (Rs.) 6000 4000 3000 2000 3000

3. The following data shows expenditure of two families: family A and family B. Represent
this data by suitable diagram.

Items Food Education Rent Entertainment Miscellaneous
5000 8000
Expenditure Family A 6000 4000 2000 1000 12000

(Rs.) Family B 4000 10000 2000

4. Construct a multiple bar diagram to represent the following data.

Investment(Rs.`0000)

Year Primary sector Secondary sector Tertiary sector
2074 250 340 200
2075 200 460 400

2076 300 500 250

5. Represent the given data by pie diagram.

Items Cement Timber Bricks Labor Steel Miscellaneous

Expenditure(Rs. 180 90 135 225 135 135
‘000)

6. Expenditure of a school in extracurricular activities in different academic year is below.
Represent this by time series graph.

Year 2062 2063 2064 2065 2066

Expenditure(Rs.) 4000 8000 12000 9000 15000

7. Represent the following data by histogram. 100-150 150-200 200-250
7 8 4
Income(Rs.) 0-50 50-100
frequency 6
5

213 Vedanta High School Economics - 10

8. The following table shows the marks obtained by 60 students.

Marks 20 30 40 50 60 70
No. of students 8 12 20 10 6 4

Calculate arithmetic mean by (i) direct method. (ii) shortcut method. (iii) step deviation

method. (Ans: 41)

9. Calculate arithmetic mean from the following data by (i) direct method.

(ii) shortcut method. (iii) step deviation method. (Ans: 41.81)

Class 20-30 30-40 40-50 50-60 60-70 70-80
8 4 2
Frequency 12 12 6

10 Calculate median height. (Ans: 180)

Height 120 140 160 180 200 220

No. of students 2 4 8 12 31
11. Calculate exact value of median from the following data. (Ans: 300)

Class interval 0-100 100- 200- 300- 400- 500­
200 300 400 500 600

frequency 22 2 22 2
(Ans: 31.66)
12. Calculate median from the following data.

Marks Less Less Less Less Less Less
than 10 than 20 than 30 than 40 than 50 than 60

No. of students 25 40 60 75 95 125

13. Find mode of: (i) 55, 45, 60.55, 40, 44, 55, 60, 30, 50

(ii) 2, 3, 4, 5,4, 2. 4, 5,6, 8, 4, 3, 2,9,2 (Ans: i. 55 ii. 70)

14. Calculate mode from the given data. , (Ans: 50)

Marks 20 30 40 50 60 70
6
No. of students 4 5 8 12 10
80-90
15. Calculate mode from the following data. (Ans: 70) 3

Weight(kg) 40-50 50-60 60-70 70-80
No. of students 10 12 25 25

16. Find arithmetic mean, median and mode from the following data.

Vedanta High School Economics - 10 214

(Ans: Mean=42.77, Median = 44, Mode = 55)

X 15 25 35 45 55 65

F 24648 3

17. Find arithmetic mean, median and mode from the following data.
(Ans: Mean=29.61. Median = 26.25, Mode = 22.5)

Class interval 0-10 10-20 20-30 30-40 40-50 50-60
Frequency 2 6 I8 2 4 4

215 Vedanta High School Economics - 10

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Mankiw, N. Gregory. (2009). Principles of Microeconomics. South Western Cengage Learning (5th edition)

Dwivedi, D.N. (2001). Macroeconomic Theory and Policy. Tata McGraw-Hill Publishing Company Limited,
New Delhi

Todaro, M.P, Smith C.E, Economic Development, (11th Edition), Addison Wesley (Pearson)

Samuelson, P.A, Nordhaus (2012), Macroeconomics, Tata McGraw Hill, Mew Delhi

Paul, R.R, Money Banking and International Trade Kalyani Publishers, New Delhi

Kuznets, Simon, Prize Lecture

Mankiw, N. Gregory. (2009). Principles of Macroeconomics. South Western Cengage Learning (5th edition)

Koutsoyiannis, A. (1991). Modern Microeconomics. Hongkong: ELBS

Mansfield, E (1996). Managerial Economics. New York, W.W Norton &Co.

M. Parkin, Microeconomics- Tenth Edition(2012)- Pearson New Delhi

Dwivedi, D.N. (2009). Microeconomic Theory and Applications. New Delhi, Pearson Education Ltd,

Parkin-Powell-Matthews (2005). Economics: (Sixth Edition) Pearson Education Limited New Delhi

Frank, Robert H & Bernanke, Ben S (2009). Principles of Microeconomics (Fourth Edition) Mc Graw Hill New
Delhi

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Approach Paper, 15th Plan, National Planning Commission, GON

Current Macroeconomic and Financial Situation, Nepal Rastra Bank

Nepal Living Standard Survey (Various), Central Bureau of Statistics

Human Development Report 2019, UNDP

Annual Report,(Various) Nepal Electric Authority, GON

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