NCERT Class 9 Solutions: Statistics (Chapter 14) Exercise 14.3-Part 2

Frequency polygon formed by joining midpoints of tops of histogram bars

Histogram and Frequency Polygon

Frequency polygon formed by joining midpoints of tops of histogram bars

Q-4 The length of 40 leaves of a plant are measured correct to one millimeter, and the obtained data is represented in the following table:

Length of leaves vs. corresponding number
Length of leaves of a plant are measured correct to one millimeter and obtained data is represented

Length (in mm)

Number of leaves

118-126

3

127-135

5

136-144

9

145-153

12

154-162

5

163-171

4

172-180

2

  1. Draw a histogram to represent the given data.

  2. Is there any other suitable graphical representation for the same data?

  3. Is it correct to conclude that the maximum number of leaves is 153 mm long?

    Why?

Solution (I):

  • The length of leaves is represented in a discontinuous class interval of 1 in between them.

  • So add, 12=0.5 to each upper class limit

  • And subtract 0.5 from the lower class limits

Converting the data to continuous intervals
Converting the discontinuous invervals in above table to continuous intervals

Length (in mm)

Number of leaves

117.5-126.5

3

126.5-135.5

5

135.5-144.5

9

144.5-153.5

12

153.5-162.5

5

162.5-171.5

4

171.5-180.5

2

Solution (II):

  • Other suitable graphical representation of this data is frequency polygon.

Solution (III):

  • No, all we know is the frequency distribution between specific ranges. We cannot draw conclusion about a particular values. We can say that maximum number of leaves were found to have lengths between 144.5 mm and 153.5 mm. We cannot say that for a particular value in this range (say 153 mm)

Q-5 The following table gives the life times of neon lamps:

Life times fo neon lamps
Lifetime in hours vs number of lamps (frequency)

Length (in hours)

Number of lamps

300-400

14

400-500

56

500-600

60

600-700

86

700-800

74

800-900

62

900-1000

48

Title: Neon lamps

Description: The life times of neon lamps, also give length (in hours) and number of lamps.

  1. Represent the given information with the help of a histogram.

  2. How many lamps have a lifetime of more than 700 hours?

Solution (I):

  • With neon lamps on x-axis and the number of lamps on y-axis

  • The intervals are continuous so we don’t need to change them

  • The histogram of the given information can be draw below:

{"chart":{"plotBackgroundColor":null,"plotShadow":true,"backgroundColor":"transparent","width":600},"title":{"text":"Life time (in hours) Vs Number of lamp"},"tooltip":{"pointFormat":"{series.name}: {point.y}"},"credits":{"enabled":false},"xAxis":{"categories":["300","400","500","600","700","800","900","1000"],"title":{"text":"Life time (in hours)"}},"yAxis":{"type":"linear","title":{"text":"Number of lamp"},"stackLabels":{"enabled":true,"style":{"fontWeight":"bold","color":"gray"}}},"plotOptions":{"series":{"allowPointSelect":true,"cursor":"pointer"}},"legend":{"floating":false,"backgroundColor":"white","borderColor":"silver","borderWidth":1,"shadow":true,"align":"right"},"series":[{"type":"column","name":"Life time (in hours) Vs Number of lamp","data":[10,20,30,40,50,60,70,80,90,100],"colors":[{"linearGradient":{"x1":0,"y1":0,"x2":0,"y2":1},"stops":[[0,"#f15c80"],[1,"rgb(165,16,52)"]]},{"linearGradient":{"x1":0,"y1":0,"x2":0,"y2":1},"stops":[[0,"#e4d354"],[1,"rgb(152,135,8)"]]},{"linearGradient":{"x1":0,"y1":0,"x2":0,"y2":1},"stops":[[0,"#2b908f"],[1,"rgb(0,68,67)"]]},{"linearGradient":{"x1":0,"y1":0,"x2":0,"y2":1},"stops":[[0,"#f45b5b"],[1,"rgb(168,15,15)"]]},{"linearGradient":{"x1":0,"y1":0,"x2":0,"y2":1},"stops":[[0,"#91e8e1"],[1,"rgb(69,156,149)"]]},{"linearGradient":{"x1":0,"y1":0,"x2":0,"y2":1},"stops":[[0,"#7cb5ec"],[1,"rgb(48,105,160)"]]},{"linearGradient":{"x1":0,"y1":0,"x2":0,"y2":1},"stops":[[0,"#434348"],[1,"rgb(0,0,0)"]]},{"linearGradient":{"x1":0,"y1":0,"x2":0,"y2":1},"stops":[[0,"#90ed7d"],[1,"rgb(68,161,49)"]]},{"linearGradient":{"x1":0,"y1":0,"x2":0,"y2":1},"stops":[[0,"#f7a35c"],[1,"rgb(171,87,16)"]]},{"linearGradient":{"x1":0,"y1":0,"x2":0,"y2":1},"stops":[[0,"#8085e9"],[1,"rgb(52,57,157)"]]}],"dataLabels":{"enabled":true,"style":{"color":"black"},"connectorColor":"silver"},"stacking":"normal","stack":"0"}]}
Created with Highcharts 4.2.5Life time (in hours)Number of lampLife time (in hours) Vs Number of lamp102030405060708090100102030405060708090100Life time (in hours) Vs Number of lamp3004005006007008009001000890255075100125800Life time (in hours) Vs Number of lamp: 60

Solution (II):

  • It can be concluded that the number of neon lamps having their lifetime more than 700 is the sum of the number of neon lamps having their lifetime as 700-800, 800-900, and 900-1000.

  • So, neon lamp having lifetime more than 700 hours is (74 + 62 + 48 = 184).

Q-6 The following table gives the distribution of students of two sections according to the mark obtained by them:

Student's marks in section A and section B and their frequency
The table dipicts student's marks in section A and section B and their frequency

Section A

Section B

Marks

Frequency

Marks

Frequency

0-10

3

0-10

5

10-20

9

10-20

19

20-30

17

20-30

15

30-40

12

30-40

10

40-50

9

40-50

1

Represent the marks of the students of both the sections on the same graph by two frequency polygons. From the two polygons compare the performance of students in the two sections.

Solution:

The class marks of the given class intervals are calculated by using the formula.

Class mark = Upperclasslimit+lowerclasslimit2 . For example, for the range 0-10 class mark would be 0+102=5 . Therefore we can put the class marks in the above table and redraw it as:

Student's marks in section A and section B with class marks and their frequency
We have added class marks to above table and kept the rest of the data the same

Section A

Section B

Marks

Class marks

Frequency

Mark

Class marks

Frequency

0-10

5

3

0-10

5

5

10-20

15

9

10-20

15

19

20-30

25

17

20-30

25

15

30-40

35

12

30-40

35

10

40-50

45

9

40-50

45

1

We draw the graph with following properties:

  • On x-axis we put the class marks and on y-axis put the frequency.

  • Scale is kept so that 1 mark on frequency is given 1 unit on y-axis.

  • We get following frequency polygon.

Chart3.txt

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