Question

Below is given frequency distribution of words in an essay:

Number of words	Number of candidates
600-800	14
800-1000	22
1000-1200	30
1200-1400	18
1400-1600	16

Find the mean number of the words written.

Answer 1

Hint: According to the question we have to find the mean number of the words written. So, first of all we have to find the class mark ${x_i}$ for each number of words between 600-800, 800-1000, 1000-1200, 1200-1400, and 1400-1600.
Now, as given the number of candidates which are the frequencies ${f_i}$ as 14, 22, 30, 18, and 16 so, with the help of the frequencies we have to obtain the value of ${f_i}{x_i}$
Now, to find the mean of the given number of words we have to use the formula to find mean as given below:

Formula used: Mean $\overline X = \dfrac{{\sum {{f_i}{x_i}} }}{{\sum {{f_i}} }}................(a)$

Complete step-by-step solution:
Step 1: First of all we have to find the class marks for the each number of words between 600-800, 800-1000, 1000-1200, 1200-1400, and 1400-1600 which can be obtained by dividing the sum of upper and lower limit of the given words by 2 hence,

Number of words	Number of candidates	Class marks ${x_i}$
600-800	14	$\dfrac{{600 + 800}}{2} = 700$
800-1000	22	$\dfrac{{800 + 1000}}{2} = 900$
1000-1200	30	$\dfrac{{1000 + 1200}}{2} = 1100$
1200-1400	18	$\dfrac{{1200 + 1400}}{2} = 1300$
1400-1600	16	$\dfrac{{1400 + 1600}}{2} = 1500$

Step 2: Now, we have to calculate ${f_i}{x_i}$ which can be obtained by multiplying the marks obtained as 700, 900, 1100, 1300, and 1500 with the frequencies ${f_i}$ as 14, 22, 30, 18, and 16 hence,

Number of words	Number of candidates ${f_i}$	Class marks ${x_i}$	${f_i}{x_i}$
600-800	14	700	$14 \times 700 = 9800$
800-1000	22	900	$22 \times 900 = 19800$
1000-1200	30	1100	$30 \times 1100 = 33000$
1200-1400	18	1300	$18 \times 1300 = 23400$
1400-1600	16	1500	$16 \times 1500 = 24000$

Step 3: Now, we have to find the value of $\sum {{f_i}{x_i}} $which can be obtained by the sum of all the number obtained for ${f_i}{x_i}$
$
   \Rightarrow \sum {{f_i}{x_i}} = 9800 + 19800 + 33000 + 23400 + 24000 \\
   \Rightarrow \sum {{f_i}{x_i}} = 110000................(1)
 $
Step 4: Now, we have to find the value of $\sum {{f_i}} $which can be obtained by the sum of all the frequencies obtained.
\[
   \Rightarrow \sum {{f_i} = 14 + 22 + 30 + 18 + 16} \\
   \Rightarrow \sum {{f_i} = 100} .............(2)
 \]
Step 5: Now, with the help of the formula (a) as mentioned in the solution hint to find the mean of the given words. Hence, on substituting (1) and (2) in formula (a),
Mean $ = $$\dfrac{{110000}}{{100}}$
Mean $ = $$1100$

Hence, with the help of formula (a) as mentioned in the solution hint we have obtained the mean number of the words written = 1100

Note: To find a mean from the frequency table we have to add up all the terms/numbers, then divide by the numbers how many there are.
If the frequency distribution is then it should be first converted into exclusive distribution.
Mean is one of the representative values of data and we can find the mean of observations by dividing the sum of all the observations by the total number of observations.