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Page No 127:
Question 1:
The frequency distribution of marks obtained by students in a class test is given below:
Marks | : | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 |
No. of Students | : | 3 | 10 | 14 | 10 | 3 |
Answer:
Histogram is a joining rectangular diagram with equal class interval of size 10.
Page No 128:
Question 2:
Present the data given in the table below in the form of a Histogram :
Mid-points | : | 115 | 125 | 135 | 145 | 155 | 165 | 175 | 185 | 195 |
Frequency | : | 6 | 25 | 48 | 72 | 116 | 60 | 38 | 22 | 3 |
Answer:
The two-dimensional diagrams that depict the frequency distribution of a continuous series by the means of rectangles are called histograms.
In order to construct a histogram, we first require the class intervals corresponding to the various mid-points, which is calculated using the following formula.
The value obtained is then added to the mid point to obtain the upper limit and subtracted from the mid-point to obtain the lower limit.
For the given data, the class interval is calculated by the following value of adjustment.
Thus, we add and subtract 5 to each mid-point to obtain the class interval.
For instance:
The lower limit of first class = 115 – 5 = 110
Upper limit of first class = 115 + 5 = 120.
Thus, the first class interval is (110-120). Similarly, we can calculate the remaining class intervals.
Mid-points | Class Interval | Frequency |
115 125 135 145 155 165 175 185 195 |
110 − 120 120 − 130 130 − 140 140 − 150 150 − 160 160 − 170 170 − 180 180 − 190 190 − 200 |
6 25 48 72 116 60 38 22 3 |
Page No 128:
Question 3:
Make a frequency Polygon and Histogram using the given data:
Marks obtained | : | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 | 60-70 |
No. of Students | : | 5 | 12 | 15 | 22 | 14 | 4 |
Answer:
Page No 128:
Question 4:
Draw Histogram from the following data:
Marks obtained | : | 10-20 | 20-30 | 30-40 | 40-50 | 50-70 | 70-100 |
No. of Students | : | 6 | 10 | 15 | 10 | 6 | 3 |
Answer:
The data is given in the form of unequal class interval. So, we will first make appropriate adjustment in the frequencies to make the class intervals equal. The general formula for the adjustment of the frequency is as follows.
Marks | No. of Students | Adjusted Frequency |
10−20 | 6 | − |
20−30 | 10 | − |
30−40 | 15 | − |
40−50 | 10 | − |
50−70 | 6 | |
70−100 | 3 |
Page No 128:
Question 5:
In a certain colony a sample of 40 households was selected . The data on daily income for this sample are given as follows:
200 | 120 | 350 | 550 | 400 | 140 | 350 | 85 |
180 | 110 | 110 | 600 | 350 | 500 | 450 | 200 |
170 | 90 | 170 | 800 | 190 | 700 | 630 | 170 |
210 | 185 | 250 | 120 | 180 | 350 | 110 | 250 |
430 | 140 | 200 | 400 | 200 | 400 | 210 | 300 |
(b) Show that the area under the polygon is equal to the area under the histogram.
Answer:
Income | Tally Marks | Frequency | Area for each class |
0 − 100 | 2 | 100 × 2 = 200 | |
100 − 200 | 14 | 100 × 14 = 1400 | |
200 − 300 | 8 | 100 × 8 = 800 | |
300 − 400 | 5 | 100 × 5 = 500 | |
400 − 500 | 5 | 100 × 5 = 500 | |
500 − 600 | 2 | 100 × 2 = 200 | |
600 − 700 | 2 | 100 × 2 = 200 | |
700 − 800 | 1 | 100 × 1 = 100 | |
800 − 900 | 1 | 100 × 1 = 100 | |
40 | Total area =4000 |
(b) The area under a histogram and under a frequency polygon is the same (i.e equal to 4000) because of the fact that we extend the first class interval to the left by half the size of class interval as the starting point of the frequency polygon. Similarly, the last class interval is extended to the right by the same amount as the end point of the frequency polygon. This ensures that the area that was excluded while joining the mid-points is included in the frequency polygon such that the area under the frequency polygon and the area of histogram is the same.
Page No 128:
Question 6:
Present the data given in the table below in Histogram:
Marks | : | 25-29 | 30-34 | 35-39 | 40-44 | 45-49 | 50-54 | 55-59 |
Frequency | : | 4 | 5 | 23 | 31 | 10 | 8 | 5 |
Answer:
Before proceeding to construct histogram, we first need to convert the given inclusive series into an exclusive series using the following formula.
The value of adjustment as calculated is then added to the upper limit of each class and subtracted from the lower limit of each class.
Therefore, we add 0.5 to the upper limit and subtract 0.5 from the lower limit of each class.
Marks | Frequency |
24.5 − 29.5 29.5 − 34.5 34.5 − 39.5 39.5 − 44.5 44.5 − 49.5 49.5 − 54.5 54.5 − 59.5 |
4 5 23 31 10 8 5 |
Page No 128:
Question 7:
A survey showed that the average daily expenditures (in rupees) of 30 households in a city were:
11, 12, 14, 16, 16, 17, 18, 18, 20, 20, 20, 21, 21, 22, 22, 23, 23, 24, 25, 25, 26, 27, 28, 28, 31, 32, 32, 33, 36, 36.
(a) Prepare a frequency distribution using class intervals:
10-14, | 15-19, | 20-24, | 25-29, | 30-34 | and | 35-39. |
(b) What percent of the households spend more than â¹ 29 each day?
(c) Draw a frequency histogram for the above data.
Answer:
(a)
Class Interval | Tally Marks | Frequency |
10 − 14 15 − 19 20 − 24 25 − 29 30 − 34 35 − 39 |
3 5 10 6 4 2 |
|
30 |
(b) Households that spend more than 29 each day
(c) To construct histogram, we first need to convert the given inclusive series into an exclusive series using the following formula.
The value of adjustment as calculated is then added to the upper limit of each class and subtracted from the lower limit of each class.
Therefore, we add 0.5 to the upper limit and subtract 0.5 from the lower limit of each class.
Class Interval | Frequency |
9.5 − 14.5 14.5 − 19.5 19.5 − 24.5 24.5 − 29.5 29.5 − 34.5 34.5 − 39.5 |
3 5 10 6 4 2 |
30 |
Page No 128:
Question 8:
Draw Histogram from a given data relating to monthly pocket money allowance of the students of class XII in a school:
Size of classes( in â¹ ) | 0-5 | 5-10 | 10-15 | 15-20 | 20-25 | 25-30 | 30-35 | 35-40 |
No of students | 5 | 10 | 15 | 20 | 25 | 15 | 10 | 5 |
Answer:
Page No 128:
Question 9:
Draw a Histogram and Frequency Polygon of the following information.
Wages in Rupees | : | 75-80 | 80-85 | 85-90 | 90-95 | 95-100 | 100-105 | 105-110 | 110-115 |
No. of workers | : | 9 | 12 | 15 | 11 | 20 | 20 | 11 | 2 |
Answer:
Page No 128:
Question 10:
Draw ogive(a) less than, and (b) more than of the folowing data:
Weekly wages of No of workers | : | 100-105 |
105-110 |
110-115 | 115-120 | 120-125 | 125-130 | 130-135 |
Workers (â¹) | : | 200 | 210 | 230 | 320 | 350 | 520 | 410 |
Answer:
In order to construct the ogives, we first need to calculate the less than and the more than cumulative frequencies as as follows.
Weekly Wages | Cumulative Frequency |
Less than 105 Less than 110 Less than 115 Less than 120 Less than 125 Less than 130 Less than 135 |
200 200 + 210 = 410 410 + 230 = 640 640 + 320 = 960 960 + 350 = 1410 1410 + 520 = 1930 1930 + 410 = 2240 |
Weekly Wages | Cumulative Frequency |
More than 100 More than 105 More than 110 More than 115 More than 120 More than 125 More than 130 |
2240 2240200 = 2040 2040210 = 1830 1830230 = 1600 1600320 = 1280 1280350 = 930 930520 = 410 |
Page No 129:
Question 11:
Prepare a less than ogive from the following data:
Class | : | 0-6 | 6-12 | 12-18 | 18-24 | 24-30 | 30-36 |
Frequency | : | 4 | 8 | 15 | 20 | 12 | 6 |
Answer:
In order to prepare a less than ogive, we first need to calculate the less than cumulative frequency distribution as follows:
Class | Cumulative Frequency |
Less than 6 Less than 12 Less than 18 Less than 24 Less than 30 Less than 36 |
4 4 + 8 = 12 12 + 15 = 27 27 + 20 = 47 47 + 12 = 59 59 + 6 = 65 |
We now plot the cumulative frequencies against the upper limit of the class intervals. The curve obtained on joining the points so plotted is known as the less than ogive.
Page No 129:
Question 12:
From the following frequency distribution prepare a 'less than ogive'.
Capital (â¹ in lakhs) | : | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 | 60-70 | 70-80 |
No. of companies | : | 2 | 3 | 7 | 11 | 15 | 7 | 2 | 3 |
Answer:
For constructing a less than ogive, we convert the frequency distribution into a less than cumulative frequency distribution as follows:
Capital | Cumulative Frequency |
Less than 10 Less than 20 Less than 30 Less than 40 Less than 50 Less than 60 Less than 70 Less than 80 |
2 2 + 3 = 5 5 + 7 = 12 12 + 11 = 23 23 + 15 = 38 38 + 7 = 45 45 + 2 = 47 47 + 3 = 50 |
We now plot the cumulative frequencies against the upper limit of the class intervals. The curve obtained on joining the points so plotted is known as the less than ogive.
Page No 129:
Question 13:
Arrange the following information on a time-series graph:
Year | : | 2009-10 | 2010-11 | 2011-12 | 2012-13 | 2013-14 | 2014-15 | 2015-16 |
NDP(â¹ in '000 crores) | : | 35 | 36 | 37 | 40 | 41 | 44 | 44 |
Answer:
Time Series Graph
Page No 129:
Question 14:
Plot the following data of annual profits of a firm on a time series graph.
Year | : | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 |
Profits(â¹ in thousand) | : | 60 | 72 | 75 | 65 | 80 | 95 |
Answer:
Time Series Graph
Page No 129:
Question 15:
Prepare a suitable graph from the following data relating to export and import.
Year | 2010-11 | 2011-12 | 2012-13 | 2013-14 | 2014-15 | 2015-16 | 2016-17 |
Exports (â¹ in crores) | 1505 | 2265 | 2070 | 1805 | 1632 | 1527 | 1845 |
Imports( â¹ in crores) | 1005 | 1145 | 1980 | 1335 | 1547 | 1435 | 1740 |
Answer:
The given data can be presented in the form of a time series graph as follows:
Here, the smallest value is 1,005 which is far higher than zero. Therefore, in this case we use a false base line starting from 1,000.
Page No 129:
Question 16:
Present graphically the following sales of Delhi branch of USHA FANS.
Year | : | 2008 | 2009 | 2010 | 2011 | 2012 | 2013 | 2014 | 2015 | 2016 |
Sales (â¹ in '000) |
: | 13 | 15 | 12 | 19 | 25 | 31 | 29 | 27 | 35 |
Answer:
The given data can be presented in the form of a time series graph as follows:
Page No 129:
Question 17:
Prepare a graph showing total cost and total production of a scooter manufacturing company.
Year | : | 2012 | 2013 | 2014 | 2015 | 2016 |
Production (in units) | : | 8500 | 9990 | 11700 | 13300 | 15600 |
Total cost (â¹ in lakh) | : | 24 | 29 | 34 | 45 | 49 |
Answer:
The given data can be presented in the form of a time series graph (i.e multiple y-axis graph) as follows:
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