chapter 14 amit bajaj · pdf file · 2011-01-142011-01-14 · if n is an...

MATHEMATICS–IX STATISTICS 173

CHAPTER 14STATISTICS

Points to Remember :1. Facts or figures, collected with a definite pupose, are called Data.2. Statistics is the area of study dealing with the collection, presentantion, analysis and interpretation of

data.3. The data collected by the investigator himself with a definite objective in mind are known as Primary

data.4. The data collected by somone else, other than the investigator, are known as Secondary data.5. Any character which is capable of taking reversal different values is called a variable.6. Each group into which the raw data are condensed is known as class-interval. Each class is bounded by

two figures known as its limits. The figure on the left is lower limit and figure on the right is upper limit.7. The difference between true upper limit and true lower limit of a class is known as its class-size.

8. Mid-value of a class (or class mark) = 2

limitlowerlimitupper

9. Class size is the difference between any two successive class marks (mid-values).10. The difference between the maximum value and the minimum value of the variable is known as Range.11. The count of number of observations in a particular class is known as its Frequency.12. The data can be presented graphically in the form of bar graphs, histograms and frequency polygons.13. The three measures of central tendency for an ungrouped data are :

(i) Mean : It is found by adding all the values of the observations and dividing it by the total number ofobservations. It is denoted by x .

So,n

x

nx........xxx

n

ii

n

121 .

For an ungrouped frequency distribution,

n

ii

n

iii

n

nn

f

xf

f........ffxf......xfxfx

1

1

21

2211

(ii) Median : It is the value of the middle-most observation(s).

If n is an odd number, then median = value of the thn

21

observation.

and, if n is an even number, then median = mean of the values of thth

12

and2

nn

observations.

(iii) Mode : The mode is the most frequently occurring observation.Empirical formula for calculating mode is given by, Mode = 3 (Median) – 2 (Mean)

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174 STATISTICS MATHEMATICS–IX

ILLUSTRATIVE EXAMPLES

Example 1. The relative humidity (in %) of a certain city for a month of 30 days was as follows :98.1 98.6 99.2 90.3 86.5 95.3 92.9 96.3 94.2 95.189.2 92.3 97.1 93.5 82.7 95.1 97.2 93.3 95.2 97.396.2 92.1 84.9 90.2 95.7 98.3 97.3 96.1 92.1 89.0 (i) Construct a grouped frequency distribution table with classes 84-86, 86-88 etc. (ii) Which month or season do you think this data is about ?(iii) What is the range of this data? —NCERT

Solution. (i) Frequency distribution table

30Total41009879896696947949229290290881888618684

FrequencyMarksTally%)(inhumidityRelative

|||||||||||||||||||||

||||||

(ii) Month-June or season-Monsoon(iii) Range = Maximum observation – minimum observation

= 99.2 – 84.9 = 14.3Example 2. The value of upto 50 decimal places is given below :

3.14159265358979323846264338327950288419716939937510(i) List the digits from 0 to 9 and make a frequency distribution of the digits after the decimal point.(ii) What are the most and the least frequency occurring digits? —NCERT

Solution. (i) Frequency distribution table

5089584746554483525120

Total|||||||

||||||||||||||||||||

|||||||||||||||||

FrequencyMarksTallyDigit

(ii) Most frequency occuring digits are 3 and 9, and least occurring digit is 0.

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Example 3. The following table gives the life times of 400 neon lamps:

481000-90062900-80074800-70086700-60060600-50056500-40014400-300

)( lampsofNo.hoursintimeLife

(i) Represent the given information with the help of a histogram.(ii) How many lamps have a life time of more than 700 hours?

—NCERTSolution. (i)

(ii) No. of lamps having life time more than 700 hours= 74 + 62 + 48 = 184

Example 4. The following two tables give the distribution of students of two sections according to the marksobtained by them :

Section A Section BMarks Frequency Marks Frequency

0-10 3 0-10 510-20 9 10-20 1920-30 17 20-30 1530-40 12 30-40 1040-50 9 40-50 1

Represent the marks of the students of both the sections on the same graph by frequencypolygon.

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Solution. Required frquency polygon is as follows :

Example 5. Draw histogram of the weekly pocket expenses of 125 students of a school given below :

05100-903090-602560-504050-301530-201020-10

)( studentsofNo.Rs.inensesexppocketWeekly

Solution. Here, we observe that class intervals are unequal, so we will first adjust the frequencies of eachclass interval. Here, the minimum class size is 10.We know, Adjusted frequency of a class interval

classtheoffrequencysizeclass

sizeclassMinimum

The adjusted frequency of each class interval is given below :

055101005100-90

103030103090-60

252510102560-50

204020104050-30

151510101530-20

101010101020-10

frequencyAdjustedFrequencyexpensespocketWeekly

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So, required histogram is given below.

Example 6. In a mathematics test given to 15 students, the following marks (out of 100) are recorded :41, 48, 39, 46, 52, 54, 62, 40, 96, 52, 98, 40, 42, 52, 60.Find the mean, median and mode of the above marks.

Solution. (i) Mean 15

)(

15

1 i

ixx

1560524240985296406256545246394841

85415822 .

(ii) Arranging the data in the ascending order :

39, 40, 40, 41, 42, 46, 48, 52, 52, 52, 54, 60, 62, 96, 98

here, n = 15, which is odd.

median = value of thn

21 observation

th

2

115 observation = 8th observation = 52

(iii) Since, 52 occurs most frequently i.e. 3 times, so mode is 52.AMIT B

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Example 7. Find the mean salary of 60 workers of a factory from the following table: —NCERT

11000039000480006700086000

105000124000163000

)( workersofNo.RsinSalary

Solution.

30500060Total100001100002700039000320004800042000670004800086000500001050004800012400048000163000

)(

iii

iiii

xff

xffworkersofNo.x.RsinSalary

Mean Ans.5083.33Rs.

60

305000)(i

ii

fxf

x

Example 8. The mean of 5 numbers is 18. If one number is excluded, their mean is 16. Find the excludednumber.

Solution. Here, n = 5, .x 18

Now, 90185

ii x

nxx

So, total of 5 numbers is 90.Let the excluded number be a. Then, total of 4 numbers is 90 – a.

Mean of 4 numbers 4

90 a

164

90

a( Given, new mean = 16)

90 – a = 64 a = 26 the excluded number is 26. Ans.

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Example 9. The median of the observations 11, 12, 14, 18, x + 2, x + 4, 30, 32, 35, 41 arranged in ascending orderis 24. Find x.

Solution. Here, n = 10. Since n is even :

Median 2

nobservatio12

nobservatio2

thth nn

2nobservatio6thnobservatio5th24

324242

622

)4()2(24

xxxx

x = 21 Ans.Example 10. Find the mode for the following data :

14, 25, 28, 14, 18, 17, 18, 14, 23, 22, 14, 18.Solution. Arranging the given data in ascending order:

14, 14, 14, 14, 17, 18, 18, 18, 22, 23, 25, 28Since, 14 occurs maximum number of times (4 times), 14 is the required mode.

PRACTICE EXERCISE1. Construct a frequency table for the following ages (in years) of 30 students using equal class intervals,

one of them being 9-12, where 12 is not included.18, 12, 7, 6, 11, 15, 21, 9, 13, 8, 15, 17, 19, 22, 14, 21, 8, 23, 12, 17, 6, 18, 15, 23, 16,22, 9, 21, 16, 11

2. The electricity bills (in Rs.) of 40 houses in a locality are given below :116, 127, 100, 107, 80, 82, 65, 91, 101, 95, 87, 105, 81, 129, 92, 75, 78, 89, 61, 121, 128, 63, 76, 84, 62, 98, 65, 95,108, 115, 101, 65, 52, 59, 81, 87, 130, 118, 108, 116Construct a grouped frequency table.

3. For the following data of weekly wages (in Rs.) received by 30 workers in a factory, construct a groupedfrequency distribution table.258, 215, 320, 300, 290, 311, 242, 272, 268, 210, 242, 258, 268, 220, 210, 240, 280, 316, 306,215, 236, 319, 304, 278, 254, 292, 306, 332, 318, 300

4. Construct a frequency table, with equal class-intervals from the following data on the weekly wages(in Rs.) of 25 labourers working in a factory, taking one of the class-intervals as 460-500 (500 notincluded).580, 625, 485, 537, 540, 425, 637, 605, 607, 430, 611, 632, 600, 640, 638, 612, 584, 440, 536,515, 449, 480, 556, 561, 508

5. Given below are two cumulative frequency distribution tables. Form a frequency distribution table foreach of these.

(i) (ii)

8060Below7350Below6040Below3930Below2820Below1510Belowpersonsof.No)yearsin(Ages

6060thanMore5250thanMore3940thanMore2730thanMore1720thanMore010thanMoreStudentsof.NoobtainedMarksAMIT B

AJAJ


6. On a certain day, the temperature in a city was recorded as under :

1822262420)(p.m.6p.m.3a.m.11a.m.8a.m.5

CineTemperaturTime

Draw a bar graph to represent the above data.7. Read the bar graph given below and answer the questions given below :

(i) What information is given by the bar graph? (ii) In which year was the production maximum?(iii) After which year was there a sudden fall in the production?(iv) Find the ratio between the maximum and minimum production during the given period.

8. The table given below shows the number of blinds in a village :

30Total21008048060

106040940205200

blindsof.NogroupAge

Draw a histogram to represent the above data.9. The following table shows the average daily earnings of 40 general stores in a market, during a certain

week.

40Total4900-850

12850-8007800-7502750-700

10700-6505650-600

)( storesof.No.RsinearningDaily

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Draw a histogram to represent the above data.10. The following table gives the heights of 50 students of a class. Draw a frequency polygon to represent

this.

50Total1591-1807801-16523651-15018501-1351351-120

)( Studentsof.NocminHeight

11. In a study of diabetic patients in a village, the following observations were noted. Represent the givendata by a frequency polygon.

53Total470-60960-502050-401240-30630-20220-10

)( Patientsof.NoYearsinAge

12. Draw a histogram and a frequency polygon on the same graph to represent the following data :

135Total1009-803008-704007-602506-503050-40

)( Personsof.NocminWeight

13. Draw a histogram to represent the following frequency distribution.

1808-50805-30

1003-20902-15615-10

FrequencyIntervalClassAMIT BAJA

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14. Draw a histogram for the marks of students given below :

74Total606-50

1005-451854-303203-10801-0Studentsof.NoMarks

15. The runs scored by two teams A and B on the first 120 balls in a cricket match are given below :

420120-1081216108-9620896-8412684-72101272-6081060-48

182048-3616436-2421224-1241012-0

BTeamATeamballsofNumber

Represent the data of both the teams on the same graph with the help of frequency polygons.16. Find the mean for each of the following sets of numbers :

(i) 25, 12, 37, 19, 43, 40, 11 (ii) 6.2, 4.9, 7.1, 2.9, 5.7, 8.3(iii) 12, 22, 32, 42, 52, 62 (iv) 13, 23, 33, 43, 53

17. Calculate the mean ( x ) for each of the following distribution :

(i) 62531108642

fx

(ii) 31523630252015105

fx

18. The following table shows the number of accidents met by 120 workers in a factory during a month :

1325213436543210

workersofNo.accidentsofNo.

Find the average number of accidents per workers.19. The marks obtained out of 50 by 80 students in a test are given below :

437181216785353833302524222015

StudentsofNo.Marks

Calculate the average marks.20. If the mean of the following data is 18.75, find the value of p.

28710530251510

i

i

fpxAMIT B

AJAJ


21. The average of height of 30 boys out of a class of 50, is 160 cm. If the average height of the remainingboys is 165 cm, find the average height of the whole class.

22. The average of six numbers is 30. If the average of first four is 25 and that of last three is 35, find thefourth number.

23. The mean of 100 observations was calculated as 40. It was found later on that one of the observationswas misread as 83 instead of 53. Find the corrected mean.

24. The mean of 10 numbers is 18. If 3 is subtracted from every number, what will be the new mean?

25. If x is the mean of n observations x1, x2, ...., xn, then prove that

n

ii xx

10)( i.e. the algebraic sum of

deviations from mean is zero.26. Find the median of following data :

(i) 14, 6, 18, 9, 23, 22, 10, 19, 24(ii) 17, 13, 28, 19, 23, 22, 12, 32

27. The numbers 5, 7, 10, 12, 2x – 8, 2x + 10, 35, 41, 42, 50 are arranged in ascending order. If their median is25, find the value of x.

28. Find the median of the following observations :46, 64, 58, 87, 41, 77, 35, 55, 90, 92, 33. If 92 is replaced by 99 and 41 by 43 in the above data, find the newmedian.

29. Find out the mode of the following data :(i) 14, 28, 19, 25, 14, 31, 17, 14, 12, 27(ii) 8.3, 8.9, 8.1, 8.7, 8.9, 7.9, 8.7, 8.9, 8.1

30. Given below is the number of pairs of shoes of different sizes sold in a day by the owner of the shop.

196554322987654321

soldpairsof.NoshoeofSize

What is the modal shoe size?

PRACTICE TEST

MM : 15 Time : 21 hour

General Instructions :Each question carry 3 marks.

1. Three coins were tossed 30 times simultaneously. Each time the number of heads occuring was noteddown as follows:0 1 2 2 1 2 3 1 3 01 3 1 1 2 2 0 1 2 13 0 0 1 1 2 3 2 2 0Prepare a frequency distribution table for this data.

2. A random survey of the number of children of various age groups playing in a park was found as follows:

41091263517-1515-1010-77-55-33-22-1

childrenof.No)yearsin(Age

Draw a histogram to represent the data above.AMIT B

AJAJ


3. Find mean ( x ) for the following distribution:

137423353025201510

fx

4. The following observations have been arranged in ascending order. If the median of the data is 63, findthe value of x.29, 32, 48, 50, x, x + 2, 72, 78, 84, 95

5. The marks obtained by 80 students in a test are given. Find the modal marks.

815241510844362820124

studentsof.NoMarks

ANSWERS OF PRACTICE EXERCISE

1.737445

24-2121-1818-1515-1212-99-6Frequency

Class

2.144758362140-130130-120120-110110-100100-9090-8080-7070-6060-50

FrequencyClass

3.2934624

340-320320-300300-280280-260260-240240-220220-200

workersofNo.

.)Rsin(WagesWeekly

4.573424

660-620620-580580-540540-500500-460460-420

workersofNo.

.)Rsin(WagesWeekly

5. (i) 7132111131560-5050-4040-3030-2020-1010-0)(

personsofNo.yearinAge

(ii) 171012138100-8080-6060-4040-2020-0

StudentsofNo.obtainedMarksAMIT B

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6.

7. (i) The given bar graph shows the production (in million tonnes) of food grains during the period from2000 to 2004.

(ii) 2002 (iii) 2000 (iv) 5 : 2

8.

9.

10. 11.

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12. 13.

14.

15.

0

6

9

5

2

3

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16. (i) 26.71 (ii) 5.85 (iii) 15.16 (iv) 4517. (i) 7.05 (ii) 15.25 18. 1.4 19. 26.7 (approx) 20. 2021. 162 cm 22. 25 23. 39.7 24. 1526. (i) 18 (ii) 20.5 27. 12 28. 58, 5829. (i) 14 (ii) 8.9 30. 8

ANSWERS OF PRACTICE TEST3. 22 4. 62 5. 28

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chapter 14 amit bajaj · pdf file · 2011-01-142011-01-14 · if n is an...

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