Question

find the median for the data items in the given frequency distribution. the median is

(type an integer or a decimal.)

Explanation:

Step1: Calculate total frequency

First, we sum up all the frequencies. The frequencies are \(1, 2, 6, 4, 1, 4, 3, 1\). So, \(n=\sum f = 1 + 2 + 6 + 4 + 1 + 4 + 3 + 1\) \(= 22\).

Step2: Determine median position

The median position is \(\frac{n + 1}{2}=\frac{22+ 1}{2}=11.5\). This means the median is the average of the 11th and 12th values.

Step3: Calculate cumulative frequency

We calculate the cumulative frequency (cf) as follows:

• For \(x = 1\), \(cf = 1\)
• For \(x = 2\), \(cf = 1+2 = 3\)
• For \(x = 3\), \(cf = 3 + 6=9\)
• For \(x = 4\), \(cf = 9+4 = 13\)
• For \(x = 5\), \(cf = 13 + 1=14\)
• For \(x = 6\), \(cf = 14+4 = 18\)
• For \(x = 7\), \(cf = 18+3 = 21\)
• For \(x = 8\), \(cf = 21+1 = 22\)

Step4: Find the 11th and 12th values

Looking at the cumulative frequencies, the 9th value is the last value of \(x = 3\), and the 10th, 11th, 12th,... values are \(x = 4\) (since cumulative frequency for \(x = 4\) starts at 10 (9 + 1) and goes up to 13). So the 11th and 12th values are both 4.

Step5: Calculate the median

The median is the average of the 11th and 12th values. Since both are 4, the median is \(\frac{4 + 4}{2}=4\).

Answer:

4