Question

The following table gives the life times of $400$ neon lamps. Find the mean lifetime.

Lifetime (in hours)	Number of lamps
$301 - 400$	$14$
$401 - 500$	$56$
$501 - 600$	$60$
$601 - 700$	$86$
$701 - 800$	$74$
$801 - 900$	$62$
$901 - 1000$	$48$

Answer 1

Hint: In this question, we are given the life time of neon lamps in hours and we have been asked the mean life time. We will use the direct method here. In this method, our first step is to convert the intervals into continuous intervals. After that, we will find the mid-points of intervals and multiply them with the frequency to find ${f_i}{x_i}$. Then we will put all the values in the formula and find the required mean.

Complete step-by-step solution:
We are given the life times of 400 neon lamps and we have been asked the mean lifetime. But as we can see that the intervals are discontinuous, our first step is to make them continuous.
To make the intervals continuous, subtract $0.5$ from the lower limit and add $0.5$ to the upper limit.
For example: our first interval is $301 - 400$, it can be made discontinuous in the following way-
$ \Rightarrow 301 - 0.5 = 300.5$
$ \Rightarrow 400 + 0.5 = 400.5$
Now, our interval has become $300.5 - 400.5$. After making the intervals, we will find the mid-points $({x_i})$ of those intervals, then we will multiply $({x_i})$ with $\left( {{f_i}} \right)$. Then, we will sum up the ${f_i}{x_i}$ and divide it by ${f_i}$.
Now, let us make the table.

Lifetime (in hours)	Number of lamps$\left( {{f_i}} \right)$	Mid-points $({x_i})$	${f_i}{x_i}$
$300.5 - 400.5$	$14$	$350.5$	$4907$
$400.5 - 500.5$	$56$	$450.5$	$25228$
$500.5 - 600.5$	$60$	$550.5$	$33030$
$600.5 - 700.5$	$86$	$650.5$	$55943$
$700.5 - 800.5$	$74$	$750.5$	$55537$
$800.5 - 900.5$	$62$	$850.5$	$52731$
$900.5 - 1000.5$	$48$	$950.5$	$45624$
	$\sum {{f_i} = 400} $		$\sum {{f_i}{x_i} = 2,73,000} $

Now, let us put the values in the formula-
Mean = $\dfrac{{\sum {{f_i}{x_i}} }}{{\sum {{f_i}} }}$
$ \Rightarrow \bar X = \dfrac{{\sum {{f_i}{x_i} = 2,73,000} }}{{\sum {{f_i} = 400} }}$
$ \Rightarrow \bar X = \dfrac{{273000}}{{400}}$
On simplifying we have,
$ \Rightarrow \bar X = 682.5$

Therefore, the mean lifetime of neon lamps is $682.5$ hours.

Note: The above method includes a lot of calculation. We can use the assumed mean method to find the mean. It will help us in reducing the calculation. In this method, the steps involving converting discontinuous into continuous and finding the mid-points remains the same. After this, one mid-point is selected as the assumed mean. Then using this mean, we find the deviations and multiply them with ${f_i}$.

Lifetime (in hours)	Number of lamps$\left( {{f_i}} \right)$	Mid-points $({x_i})$	${d_i} = {x_i} - A$$A = 650.5$	${f_i}{d_i}$
$300.5 - 400.5$	$14$	$350.5$	$ - 300$	$ - 4200$
$400.5 - 500.5$	$56$	$450.5$	$ - 200$	$ - 11200$
$500.5 - 600.5$	$60$	$550.5$	$ - 100$	$ - 6000$
$600.5 - 700.5$	$86$	$650.5$=A	$0$	$0$
$700.5 - 800.5$	$74$	$750.5$	$100$	$7400$
$800.5 - 900.5$	$62$	$850.5$	$200$	$12400$
$900.5 - 1000.5$	$48$	$950.5$	$300$	$14400$
	$\sum {{f_i} = 400} $			$\sum {{f_i}{d_i} = 12800} $

Now, we put all the values in the formula-
$ \Rightarrow \bar X = A + \dfrac{{\sum {{f_i}{d_i}} }}{{\sum {{f_i}} }}$
$ \Rightarrow \bar X = 650.5 + \dfrac{{12800}}{{400}}$
Simplifying,
$ \Rightarrow \bar X = 650.5 + 32$
$ \Rightarrow \bar X = 682.5$
Therefore, the mean lifetime of neon lamps is $682.5$ hours.