$A$ teacher wanted to analyze the performance of two sections of students in a mathematics test of $100$ marks. Looking at their performances,she found that a few students got under $20$ marks and a few got $70$ marks or above. So she decided to group them into intervals of varying sizes as follows: $0-20, 20-30, ..., 60-70, 70-100$. Then she formed the following table:
Marks Number of students $0-20$ $7$ $20-30$ $10$ $30-40$ $10$ $40-50$ $20$ $50-60$ $20$ $60-70$ $15$ $70-100$ $8$ Total $90$
$A$ histogram for this table was prepared by a student as shown in Fig. Carefully examine this graphical representation. Do you think that it correctly represents the data?
| Marks | Number of students |
| $0-20$ | $7$ |
| $20-30$ | $10$ |
| $30-40$ | $10$ |
| $40-50$ | $20$ |
| $50-60$ | $20$ |
| $60-70$ | $15$ |
| $70-100$ | $8$ |
| Total | $90$ |
$A$ histogram for this table was prepared by a student as shown in Fig. Carefully examine this graphical representation. Do you think that it correctly represents the data?
(N/A) No,the graph is giving us a misleading picture. In a histogram,the areas of the rectangles are proportional to the frequencies. This condition is satisfied when the widths of all class intervals are equal. However,in this case,the widths of the rectangles are varying,so the histogram does not give a correct representation. For example,it shows a greater frequency in the interval $70-100$ than in $60-70$,which is incorrect.
To correct this,we must modify the lengths of the rectangles so that their areas become proportional to the frequencies. The steps are:
$1$. Select the minimum class size,which is $10$ in this case.
$2$. Modify the lengths of the rectangles to be proportionate to the class size of $10$. For a class size of $20$,the length is $7$. For a class size of $10$,the length becomes $\frac{7}{20} \times 10 = 3.5$.
Following this method,we get the modified table:
These lengths represent the "proportion of students per $10$ marks interval". Plotting these values gives the correct histogram.
To correct this,we must modify the lengths of the rectangles so that their areas become proportional to the frequencies. The steps are:
$1$. Select the minimum class size,which is $10$ in this case.
$2$. Modify the lengths of the rectangles to be proportionate to the class size of $10$. For a class size of $20$,the length is $7$. For a class size of $10$,the length becomes $\frac{7}{20} \times 10 = 3.5$.
Following this method,we get the modified table:
| Marks | Frequency | Width | Length of rectangle |
| $0-20$ | $7$ | $20$ | $\frac{7}{20} \times 10 = 3.5$ |
| $20-30$ | $10$ | $10$ | $\frac{10}{10} \times 10 = 10$ |
| $30-40$ | $10$ | $10$ | $\frac{10}{10} \times 10 = 10$ |
| $40-50$ | $20$ | $10$ | $\frac{20}{10} \times 10 = 20$ |
| $50-60$ | $20$ | $10$ | $\frac{20}{10} \times 10 = 20$ |
| $60-70$ | $15$ | $10$ | $\frac{15}{10} \times 10 = 15$ |
| $70-100$ | $8$ | $30$ | $\frac{8}{30} \times 10 = 2.67$ |
These lengths represent the "proportion of students per $10$ marks interval". Plotting these values gives the correct histogram.
Explore More
Similar Questions
The value of $\pi$ up to $50$ decimal places is given below:
$3.14159265358979323846264338327950288419716939937510$
$(i)$ Make a frequency distribution of the digits from $0$ to $9$ after the decimal point.
$(ii)$ What are the most and the least frequently occurring digits?
$3.14159265358979323846264338327950288419716939937510$
$(i)$ Make a frequency distribution of the digits from $0$ to $9$ after the decimal point.
$(ii)$ What are the most and the least frequently occurring digits?
Difficult
View SolutionThe blood groups of $30$ students of Class $VIII$ are recorded as follows:
$A, B, O, O, AB, O, A, O, B, A, O, B, A, O, O,$
$A, AB, O, A, A, O, O, AB, B, A, O, B, A, B, O.$
Represent this data in the form of a frequency distribution table. Which is the most common,and which is the rarest,blood group among these students?
$A, B, O, O, AB, O, A, O, B, A, O, B, A, O, O,$
$A, AB, O, A, A, O, O, AB, B, A, O, B, A, B, O.$
Represent this data in the form of a frequency distribution table. Which is the most common,and which is the rarest,blood group among these students?
Medium
View SolutionLet us consider the following frequency distribution table which gives the weights of $38$ students of a class:
Weights (in $kg$) Number of students $31-35$ $9$ $36-40$ $5$ $41-45$ $14$ $46-50$ $3$ $51-55$ $1$ $56-60$ $2$ $61-65$ $2$ $66-70$ $1$ $71-75$ $1$ Total $38$
If two new students of weights $35.5\, kg$ and $40.5\, kg$ are admitted to this class,how should the frequency distribution table be adjusted to include them?
| Weights (in $kg$) | Number of students |
| $31-35$ | $9$ |
| $36-40$ | $5$ |
| $41-45$ | $14$ |
| $46-50$ | $3$ |
| $51-55$ | $1$ |
| $56-60$ | $2$ |
| $61-65$ | $2$ |
| $66-70$ | $1$ |
| $71-75$ | $1$ |
| Total | $38$ |
If two new students of weights $35.5\, kg$ and $40.5\, kg$ are admitted to this class,how should the frequency distribution table be adjusted to include them?
Difficult
View SolutionThree coins were tossed $30$ times simultaneously. Each time the number of heads occurring was noted down as follows:
$\begin{array}{llllllllll}0 & 1 & 2 & 2 & 1 & 2 & 3 & 1 & 3 & 0 \\ 1 & 3 & 1 & 1 & 2 & 2 & 0 & 1 & 2 & 1 \\ 3 & 0 & 0 & 1 & 1 & 2 & 3 & 2 & 2 & 0\end{array}$
Prepare a frequency distribution table for the data given above.
$\begin{array}{llllllllll}0 & 1 & 2 & 2 & 1 & 2 & 3 & 1 & 3 & 0 \\ 1 & 3 & 1 & 1 & 2 & 2 & 0 & 1 & 2 & 1 \\ 3 & 0 & 0 & 1 & 1 & 2 & 3 & 2 & 2 & 0\end{array}$
Prepare a frequency distribution table for the data given above.
Easy
View SolutionThe relative humidity (in $\%$) of a certain city for a month of $30$ days was as follows:
$98.1$ $98.6$ $99.2$ $90.3$ $86.5$ $95.3$ $92.9$ $96.3$ $94.2$ $95.1$ $89.2$ $92.3$ $97.1$ $93.5$ $92.7$ $95.1$ $97.2$ $93.3$ $95.2$ $97.3$ $96.2$ $92.1$ $84.9$ $90.2$ $95.7$ $98.3$ $97.3$ $96.1$ $92.1$ $89$
$(i)$ 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?
| $98.1$ | $98.6$ | $99.2$ | $90.3$ | $86.5$ | $95.3$ | $92.9$ | $96.3$ | $94.2$ | $95.1$ |
| $89.2$ | $92.3$ | $97.1$ | $93.5$ | $92.7$ | $95.1$ | $97.2$ | $93.3$ | $95.2$ | $97.3$ |
| $96.2$ | $92.1$ | $84.9$ | $90.2$ | $95.7$ | $98.3$ | $97.3$ | $96.1$ | $92.1$ | $89$ |
$(i)$ 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?
Medium
View SolutionVedclass Products
For Students
Vedclass Test Series
Mock tests in real JEE/NEET style with performance analysis. 5-day free trial.
Start Free TrialFor Teachers
Exam Paper Generator
Generate Set A/B/C/D exam papers from 7.5L+ questions in 2 minutes. 3 chapters free.
Try FreeFor Institutes
Online Exam Module
Live online exams with unlimited students, 360° analytics & white-label branding.
See Demo