Ex.14.1 Q5 Statistics Solution - NCERT Maths Class 10

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Question

In a retail market, fruit vendors were selling mangoes kept in packing boxes. These boxes contained varying number of mangoes. The following was the distribution of mangoes according to the number of boxes.

Number of mangoes \(50 – 52\) \(53 – 55\) \(56 – 58\) \(59 – 61\) \(62 – 64\)
Number of boxes \(15\) \(110\) \(135\) \(115\) \(25\)

Find the mean number of mangoes kept in a packing box. Which method of finding the mean did you choose?

Text Solution

 

What is known?

The distribution of mangoes according to the number of boxes.

What is unknown?

The mean number of mangoes kept in a packing box.

Reasoning:

We solve this question by step deviation method. 

Hence, the given class interval is not continuous. First we have to make it continuous. There is a gap of \(1\) between two class interval. Therefore, \(0.5\) has to be added to the upper class limit and \(0.5\) has to be subtracted from the lower class limit of each interval.

\[\text{Mean,}\; \overline x = a + \left( {\frac{{\sum {{f_i}{u_i}} }}{{\sum {{f_i}} }}} \right) \times h\]

Solution:

We know that,

\(\text{Class mark,} \;{x_i} = \frac{{{\text{Upper class limit }} + {\text{ Lower class limit}}}}{2}\)

\(\text{Class size,} h = 3\)

\(\text{Taking assumed mean,} a = 57\)

Class interval No of house hold \((f_i)\) \( x_i\) \( d_i = x_i -a \) \(\begin{align}{u_i}=\frac{ d_i}{h}\end{align}\) \( F_iu_i \)
\(49.5-52.5\) \(15\) \(51\) \(-6\) \(-2\) \(-30\)
\(52.5-55.5\) \(110\) \(54\) \(-3\) \(1\) \(-110\)
\(55.5-58.5\) \(135\) \(57(a)\) \(0\) \(0\) \(0\)
\(58.5-61.5\) \(115\) \(60\) \(3\) \(1\) \(115\)
\(61.5-64.5\) \(25\) \(63\) \(6\) \(2\) \(50\)
  \(\Sigma f_i=400 \)       \(\Sigma f_iu_i=25\)

From the table, we obtain

\[\begin{align}
\sum {{f_i} = 400} \\
\sum {{f_i}{u_i}}  = 25
\end{align}\]

Mean, \(\overline x  = a + \left( {\frac{{\sum {{f_i}{u_i}} }}{{\sum {{f_i}} }}} \right) \times h\)

\[\begin{align}
& = 57 + \left( {\frac{{25}}{{400}}} \right) \times 3\\
 &= 57 + \frac{1}{{16}} \times 3\\
 &= 57 + \frac{3}{{16}}\\
 &= 57 + 0.19\\
 &= 57.19
\end{align}\]

The mean number of mangoes kept in a packing box are \(57\).\(19\)

  
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