Q) The following table gives the distribution of the life time of 400 neon lamps.

table gives the distribution of the life time of 400 neon lamps

Find the median lifetime of a lamp.

Ans: Let’s re-organize the data in the frequency table to find out each part:

table gives the distribution of the life time of 400 neon lamps

To find the median, we need to identify middle value of the data. Let’s rearrange the data:

  • First, we need to find the cumulative frequency in the frequency table to find the median. Its shown in last column.
  • Next, Total number of sub-divisions or Sum of the frequencies = 400. It shown in the last row of middle column.
  • Next, we need to identify Median Class. Since the Median class is the class where the cumulative frequency crosses 50% of the half the total number of sub-divisions, here in the table, Cumulative frequency of 115 is crossing 50% of frequency i.e. 50, at class “3000 – 3500”. Hence, our Median class = 3000 – 3500
  • Next, To find the median, we use the formula:

Median = L+\left[\frac{\frac{n}{2}-c_f}{f}\right]x h

Here:

L = Lower boundary of the median class = 3000

n = Total number of sub-divisions = 400

{c_f} = Cumulative frequency of the class before the median class = 130

f = Frequency of the median class = 86

h = Class width = 3500 – 3000 = 500

hence, the Median = 3000 + \left[\frac{\frac{400}{2} - 130}{86}\right]x 500

⇒ 3000 + [(200 – 130)] x \frac{500}{86}

⇒ 3000 + \frac{70 \time 700}{86}

⇒ 3000 + 406.98

⇒ 3406.98 hours

Therefore, Median life of the lamp is 3,406.98 hours.

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