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Q5:

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

Find the median life time of a lamp.

Solution :

Given: The frequency distribution of the lifetime of 400 neon lamps as follows:

Lifetime (hours) Number of Lamps ($f_i$)
1500 - 2000 14
2000 - 2500 56
2500 - 3000 60
3000 - 3500 86
3500 - 4000 74
4000 - 4500 62
4500 - 5000 48

To Find: The median lifetime of the neon lamps.

Step 1: Construct the Cumulative Frequency Table

To find the median, we first calculate the cumulative frequency ($cf$) for each class interval.

Lifetime (hours) Frequency ($f_i$) Cumulative Frequency ($cf$)
1500 - 2000 14 14
2000 - 2500 56 70
2500 - 3000 60 130
3000 - 3500 86 216
3500 - 4000 74 290
4000 - 4500 62 352
4500 - 5000 48 400
Total $N = 400$ -

Step 2: Identify the Median Class

The total number of observations is $N = 400$.
We calculate $\frac{N}{2} = \frac{400}{2} = 200$.
Looking at the cumulative frequency column, the first cumulative frequency greater than 200 is 216, which corresponds to the class interval $3000 - 3500$.
Therefore, the median class is $3000 - 3500$.

Step 3: Apply the Median Formula

The formula for the median of a grouped frequency distribution is:
$\text{Median} = l + \left( \frac{\frac{N}{2} - cf}{f} \right) \times h$
Where:
$l$ (lower limit of median class) = $3000$
$N$ (total frequency) = $400$
$cf$ (cumulative frequency of the class preceding the median class) = $130$
$f$ (frequency of the median class) = $86$
$h$ (class size) = $500$

Step 4: Perform the Calculation

Substitute the values into the formula:
$\text{Median} = 3000 + \left( \frac{200 - 130}{86} \right) \times 500$
$\text{Median} = 3000 + \left( \frac{70}{86} \right) \times 500$
$\text{Median} = 3000 + \left( \frac{35000}{86} \right)$ [Simplifying the fraction: $\frac{70}{86} = \frac{35}{43}$]
$\text{Median} = 3000 + 406.9767...$
$\text{Median} \approx 3406.98$

Final Answer: The median lifetime of a lamp is 3406.98 hours.


More Questions from Class 10 Mathematics Statistics EXERCISE 13.3


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