Find the median of the following distribution.

Steps to Solve

1. List Values and Frequencies Organize the data into two lists: the values of $x$ and their corresponding frequencies $f$.

   $x$	$f$
   10	10
   11	4
   12	7
   14	16
   15	18
   16	25
   17	22
   18	20
   19	18

2. Calculate Cumulative Frequency Compute the cumulative frequency for each value.

   • Cumulative frequencies:
     • For $x = 10$: $10$
     • For $x = 11$: $10 + 4 = 14$
     • For $x = 12$: $14 + 7 = 21$
     • For $x = 14$: $21 + 16 = 37$
     • For $x = 15$: $37 + 18 = 55$
     • For $x = 16$: $55 + 25 = 80$
     • For $x = 17$: $80 + 22 = 102$
     • For $x = 18$: $102 + 20 = 122$
     • For $x = 19$: $122 + 18 = 140$

3. Find Total Frequency (N) Sum all the frequencies to find the total frequency $N$.

   $$ N = 10 + 4 + 7 + 16 + 18 + 25 + 22 + 20 + 18 = 140 $$

4. Determine the Median Position Calculate the position of the median using the formula:

   $$ \text{Median position} = \frac{N}{2} = \frac{140}{2} = 70 $$

5. Identify Median Class Locate the cumulative frequency that first meets or exceeds the median position (70).

   • The cumulative frequency for $x = 16$ is 80, which is the first cumulative frequency greater than 70.

6. Calculate Median Apply the median formula for grouped data:

   $$ \text{Median} = L + \left(\frac{\frac{N}{2} - CF}{f}\right) \times h $$

   Where:

   • $L = 16$ (lower boundary of the median class)
   • $CF = 55$ (cumulative frequency of the class before the median class)
   • $f = 25$ (frequency of the median class)
   • $h = 1$ (class width, assuming intervals of 1)

   Substitute in the values:

   $$ \text{Median} = 16 + \left(\frac{70 - 55}{25}\right) \times 1 $$

   $$ = 16 + \left(\frac{15}{25}\right) $$

   $$ = 16 + 0.6 = 16.6 $$