# Ex.14.3 Q5 Statistics Solution - NCERT Maths Class 10

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## Question

The following table gives the distribution of the life time of $$400$$ neon lamps:

 Life time (in hours) Number of lamps $$1500 - 2000$$ $$14$$ $$2000 - 2500$$ $$56$$ $$2500 - 3000$$ $$60$$ $$3000 - 3500$$ $$86$$ $$3500 - 4000$$ $$74$$ $$4000 - 4500$$ $$62$$ $$4500 - 5000$$ $$48$$

Find the median life time of a lamp.

## Text Solution

What is known?

The life time of $$400$$ neon lamps.

What is unknown?

The median life time of a lamp.

Reasoning:

Median Class is the class having Cumulativefrequency $$(cf)$$ just greater than $$\frac n{2}$$

Median $$= l + \left( {\frac{{\frac{n}{2} - cf}}{f}} \right) \times h$$

Class size,$$h$$

Number of observations,$$n$$

Lower limit of median class,$$l$$

Frequency of median class,$$f$$

Cumulative frequency of class preceding median class,$$cf$$

Steps:

 Life time (in hours) Number of lamps \begin{align}\mathbf{f}_{\mathbf{i}}\end{align} Cumulative frequency $$1500 - 2000$$ $$14$$ $$14$$ $$2000 - 2500$$ $$56$$ $$14 +56 =70$$ $$2500 - 3000$$ $$60$$ $$70 +60 =130$$ $$3000 - 3500$$ $$86$$ $$130 +86 =216$$ $$3500 - 4000$$ $$74$$ $$216 + 74 = 290$$ $$4000 - 4500$$ $$62$$ $$290 + 62 = 352$$ $$4500 - 5000$$ $$48$$ $$352 + 48 = 400$$ Total (n) = 400

From the table, it can be observed that

$$n = 400{\rm{ }} \Rightarrow \frac{n}{2} = 200$$

Cumulative frequency $$(cf)$$ just greater than $$200$$ is $$216,$$belonging to class $$3000 – 3500.$$

Therefore, median class $$=3000 – 3500$$

Class size ($$h$$) $$=500$$.

Lower limit ($$l$$) of median class $$=3000$$.

Frequency ($$f$$) of median class $$=86$$.

Cumulative frequency ($$cf$$) of class preceding median class $$=130$$.

\begin{align}\begin{aligned} \text {Median} &=l+\left(\frac{\frac{n}{2}-c f}{f}\right) \times h \\ &=3000+\left(\frac{200-130}{86}\right) \times 500 \\ &=3000+\frac{70 \times 500}{86} \\ &=3406.976 \end{aligned}\end{align}

Therefore, median life time of lamps is $$3406.98$$ hours.

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