UNIT 3 STATISTICAL TOOLS ANDINTERPRETATION of Cen… · Measures of Central Tendency (Important...

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UNIT – 3 STATISTICAL TOOLS AND INTERPRETATION :-

1. Measures of central Tendency.

2. Measures of Dispersion

3. Co-relation and Measures of correlation

4. Introduction to Index Numbers.

Measures of Central Tendency

(Important Terms & Concepts) :-

1. Average or measures of central Tendency :- It is a value which is a typical orrepresentative of a set of data. Averages are also called measures of centraltendency, since they tend to lie centrally, with in a set of data arranged accordingto magnitude.

2. Functions of Average :-

Average helps to get a representative value to the entire set of data.

It Facilitates Comparison.

It is a useful tool in decision- making.

3. Essentials of a Good Average / good measures of Central Tendency : -

Simplicity in calculation

Easy to understand

Rigidly Defined

Precise Value

Based upon all observations

Unaffected by extreme values.

Capable for further statistical calculations.

4. Types of Averages / measures of central Tendency :- There are three averages,which are in common use- Arithmetic Mean, Median and Mode.

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5. Arithmetic Means ( X ):-

It is the most common type of measures of central tendency. It is obtained bydividing the sum of all observations in a series by the total numbers ofobservations.

6. Calculation of Arithmetic Mean :-

For Individual series / ungrouped Data :-

(i)N

XX Direct Method

(ii)N

dAX Assumed Mean Method / Short Cut Method.

(iii) iN

dAX '

(step Deviation method.)

For Discrete and Continuous Series / Grouped Data:

(i)

f

fmor

f

fxX ( Direct method )

(ii)N

fdAX (Assumed Mean Method / Short-cut method.)

(iii) iN

fdAX '

(step deviation method)

7. Mathematical Properties Of Arithmetic Mean :-

The algebraic sum of deviations of items from arithmetic mean X is alwayss

Zero, i.e. 0– XX .

The Sum of the squared deviations of the item from A.M. is minimum i.e.

2––2

AXXX .

8. Merits of Arithmetic Mean:-

(i) It is easy to calculate and simple to understand.(ii) It is rigidly defined.(iii) It is a calculated value not a positional value.(iv) It is based on all observations.



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9. Demerits of Arithmetic Mean:-

It is affected by presence of extreme values

It cannot be calculated in open-ended series

It cannot be ascertained graphically.

It sometimes gives misleading and surprising results.

10. MEDIAN:-

It is defined as the middle value of the series when the data is arranged inascending or descending order. In other words, median is that value of thedistribution which divides the group into two equal parts, one part comprisingall greater values and the other comprising values less than the median.

11. Calculation of median:-

For Individual and Discrete series :-

thN

ofSizeM

2

1 item.

For Continuous series:-

Median itemth

NofSize

2 item

iF

fcN

lM .–

21

12. Merits of median :-

It is easy to understand and easy to compute.

It is not unduly affected by extreme observations

Median can be located graphically with the help of ogives.

It is the most appropriate average in case of open-ended classes.

It is the most suitable average for qualitative measurement such asintelligence, beauty etc.

It is a positional value and not a calculated value.



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13. Demerits of Median(I) It is not based on all observations of the series since it is a positional average.(II) It requires arrangement of data, but other averages do not need it.(III) It can not be computed exactly where the number of items in a series is

even.14. Related Measures Of Median- Quartiles & Percentiles:-

Quartiles are the measures which divide the data into four equal parts, eachpart contains equal number of observations. There are three quartiles - Q1, Q2, andQ3. The 1st quartile denoted by Q1 is called lower quartile and 25% of the item ofthe distribution are below it and 75% of the items are greater than it.

The second quartile is known as median and is denoted by Q2. It has 50% of theitem above it. The third Quartile is known as Q3 and it is also called upper Quartileand 75% of the items are below it and 25% of the items are above it. Thus , Q1 andQ3 denote the two items with which central 50% of item lie.

Percentiles divide the series into hundred equal parts. For any series, thereare 99 percentiles denoted by P1, P2, P3,……, P99. P50 is the median value.15. Calculation of Quartiles and percentiles

For individual and discrete series:-

Q1=Size ofth

N

4

1 item.

Q3= Size of 3th

N

4

1 item.

For continuous series :-

Q1= Size ofth

N

4 item

Q1 = L1 if

fcN

.–

4

Q3=Size of 3th

N

4 item

Q3 = L1 + if

fcN

.–

43



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16 MODE (Z) :-

It is defined as the value which occurred most frequently in a series. In other words,it is the value which has highest Frequency in a distribution. For example : Mode inthe series: 20,21,23,23,23,23,25,26,26 would be 23 as this value occurs mostFrequently than any other value. There is greatest concentration of items aroundthis value.

17. CALCULATION OF MODE :-

For Individual Series :-

(i) To identify the value that occurs most frequently in a series.

(ii) By conversion into discrete series and then identify the value correspondingto which there is highest frequency.

For Discrete Series:-

(i) By Inspection method.

(ii) Grouping method.; By preparing grouping table and then preparinganalysis table.

For Continuous Series :-

(I) Determination of modal class interval by inspection method or groupingtable and analysis table.

(II) Applying the Formula:

ifff

ffLZ

201

011 ––2

–

Where, l1= Lower limit of modal class.

F1= frequency of the modal class.

F2= frequency of the class succeeding modal class.

F0= frequency of the class preceding modal class.

i-= size of the class interval.



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18. Merits of mode :-

(i) It is easy to understand and simple to calculate

(ii) It is not affected by the presence of extreme values .

(iii) It can be located graphically with the help of histogram.

(iv) It can be easily calculated in case of open-ended classes.

19. Demerits of mode:-

(i) It is not rigidly defined.

(ii) When frequencies of all items are identical, It is difficult to identify the ModalValue.

(iii) It is not based on all observations.

(iv) Mode is not capable of further algebraic treatment.

20. Relative position of mean, median and mode:-

The relative position of X , M and Z depends upon the shape of the frequencydistribution which is discussed below.

(i) In case of symmetrical distribution, mean median and mode are identicali.e.

ZMX .

(ii) In a moderately asymmetrical (skewed) distribution, mean median andmode are not equal, i.e. ZMX .

(a) When the distribution is positively skewed, i.e. skewed to the right, then

ZMX .

(b) When the distribution is negatively skewed, i.e. skewed to the left, then

ZMX .

Note :- The median (m) always lie between arithmetic mean X and mode (Z).

21. Empirical Relationship between x,m and z:

In a moderately asymmetrical distribution, the values of mean, median and modeare observed to have the following relationship :

Mode = 3 median – 2 mean.



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22. Questions with Answer

1 Mark Questions :-

1. Define an average?

Ans. An average is a single value that represents the whole group.

2. Name the measures of central tendency?

Ans. Three important types of statistical averages are :-

Arithmetic Mean, median and mode.

3. What is median?

Ans. It is defined as the middle value of the series when arranged either inascending order or in descending order.

4. What is mode ?

Ans. It is defined as the value which occurs most frequently in a series.

5. Can mode be graphically located ?

Ans : Mode can be located graphically with the help of histogram

6. Average daily wage of 50 workers of a factory was Rs. 200. Each worker isgiven a raise of Rs 20. What is the new average daily wage ?

Ans. Increase is wages of each worker=Rs20.

Total increase in wages = 50x20= Rs 1000.

Total wages before increase in wages = 50x200=Rs 10,000.

Total wages after increase in wages= 10,000+1000= Rs 11000.

New average wages 220.50

000,11Rs

N

X

Thus mean wage will increase by Rs 20.

7. What relationship exists between mean, median and mode in case of asymmetrical distribution ?

Ans. In a symmetrical distribution, ZMX



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8. What relationship exists between MX and Z in moderately negativeeswewed distribution ?

Ans. In a moderately negative skewed distribution. ZMX .

3 / 4 marks Questions : -

9. Explain the characteristics of a good average?

Ans. Characteristics of a good average/measures of central tendency

(i) It should be easy to understand.

(ii) It should be simple to compute.

(iii) It should be rigidly (well) defined.

(iv) It should be based on all the observations.

(v) It should not be unduly affected by the extreme values.

(vi) It should be capable of further algebraic treatment.

10. “Arithmetic mean is affected by very large and very small values but medianand mode are not affected by them.” Explain.

Ans. Median is the value of the middle item of a series arranged in ascending orin descending order of magnitude. Mode only takes values at the pointsaround which the items tend to be most heavily concentrated. Arithmeticmean takes into account the value of all items (i.e. very large and very small)in a series. Thus it is only the arithmetic mean which is affected by extremevalues in the series.

11. Which average would be suitable in the following cases?

(i) Average size of readymade garments.

(ii) Average intelligence of students in a class.

(iii) Average Production in a factory per shift.

(iv) Average wages in an industrial concern.

(v) When the sum of absolute deviations from average is least.

(vi) In case of open-ended frequency distribution

Ans.

(i) Mode, (ii) Median, (iii) Mean(iv) Mean, (V) Median, (vi) Median or mode.



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23. Some Numerical Questions : -

1. Calculate Arithmetic mean from the following data using direct and short cutmethod / Assumed mean method:

Size: 10 20 30 40 50 60

Frequency 7 8 12 15 5 3

2. Calculate Arithmetic mean from the following data using step- deviationmethod:-

Size 20-29 30-39 40-49 50-59 60-69

Frequency 10 8 6 4 2

3. Find median of the following observations:

20,15,25,28,18, 16,30

4. Calculate median of the following data.

Marks : 11-15 16-20 21-25 26-30 31-35 36-40 41-45 46-50

No of Students: 7 10 13 26 35 22 11 5

5. Calculate Q1 and Q3 From the following data

Marks: 10 20 30 40 50 60

No of students: 4 10 20 8 6 3

6. Calculate the value of median and Q1 from the following data :

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

No of Students: 5 8 10 4 3

7. Calculate the mode of the following data:

4,6,5,7,9,8,10,4,7,6,5,8,7,7,9

8. Calculate mode from the following data

Marks: 0-10 10-20 20-30 30-40 40-50 50-60 60-70

No of Students : 2 5 8 10 8 5 2



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UNIT 3 STATISTICAL TOOLS ANDINTERPRETATION of Cen… · Measures of Central Tendency (Important...

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