Question

Find the mean deviation about the median for the data.

\(x_i\)	5	7	9	10	12	15
\(f_i\)	8	6	2	2	2	6

Answer 1

The given observations are already in ascending order. 

Adding a column corresponding to cumulative frequencies of the given data, we obtain the following table

\(x_i\)	\(f_i\)	\(c.f\)
5	8	8
7	6	14
9	2	16
10	2	18
12	2	20
15	6	26

Here, N = 26, which is even. 

Median is the mean of 13^th and 14t^h observations. Both of these observations lie in the cumulative frequency 14, for which the corresponding observation is 7.

Median,M= \(\frac{\frac13^{th}observation+14^{14}observation}{2}=\frac{7+7}{2}=7\)

The absolute values of the deviations from median, i.e \(|x_i-M|,\) are 

\(|x_i,M|\)	2	0	2	3	5	8
\(f_i\)	8	6	2	2	2	6
\(f_i|x_iM|\)	16	0	4	6	10	48

\(\sum_{I=1}^{6}f_i=26\) and   \(\sum_{I=1}^{6}f_i|x_i-M|=84\)

\(=MD(M)=\frac{1}{N}\sum_{i=1}^{6}f_i|x_i-M|=\frac{1}{26}X84=3.23\)

Answer 2

The given data is:
\(\begin{array}{|c|c|c|c|c|c|c|} \hline x_i & 5 & 7 & 9 & 10 & 12 & 15 \\ \hline f_i & 8 & 6 & 2 & 2 & 2 & 6 \\ \hline \end{array}\)

Step 1: Let's calculate the cumulative frequencies cf:
\(\begin{array}{|c|c|} \hline x_i & cf \\ \hline 5 & 8 \\ 7 & 8 + 6 = 14 \\ 9 & 14 + 2 = 16 \\ 10 & 16 + 2 = 18 \\ 12 & 18 + 2 = 20 \\ 15 & 20 + 6 = 26 \\ \hline \end{array}\)

Step 2: Find the Median
The median is the value that separates the higher half from the lower half of the data. To find the median, we use the cumulative frequency.
Total frequency N = 26
Median position = \(\frac{N + 1}{2} = \frac{26 + 1}{2} = 13.5\)

We look at the cumulative frequency to determine where the 13.5th value lies:
The 13.5th value lies in the interval of \(x_i = 7\), since the cumulative frequency just before 13.5 is 8 and the next cumulative frequency is 14.
Thus, the median M = 7.

Step 3: Calculate the Absolute Deviations from the Median
\(\begin{array}{|c|c|c|c|} \hline x_i & f_i & |x_i - M| & f_i \cdot |x_i - M| \\ \hline 5 & 8 & |5 - 7| = 2 & 8 \cdot 2 = 16 \\ 7 & 6 & |7 - 7| = 0 & 6 \cdot 0 = 0 \\ 9 & 2 & |9 - 7| = 2 & 2 \cdot 2 = 4 \\ 10 & 2 & |10 - 7| = 3 & 2 \cdot 3 = 6 \\ 12 & 2 & |12 - 7| = 5 & 2 \cdot 5 = 10 \\ 15 & 6 & |15 - 7| = 8 & 6 \cdot 8 = 48 \\ \hline & & & \sum f_i \cdot |x_i - M| = 84 \\ \hline \end{array}\)

Step 4: Find the Mean Deviation of the Median
The mean deviation about the median is given by:
\(\text{Mean Deviation} = \frac{\sum f_i \cdot |x_i - M|}{N}\)

Substitute the values:
\(\text{Mean Deviation} = \frac{84}{26} = 3.23\)

So, the answer is 3.23