Question

The mean deviation of first 8 composite numbers is

Answer 1

Hint: A complete No is a \[ + ve\] integer that can be formed by multiplying two smaller positive integer, equivalently it is a \[ + ve\] integer that has at least one divisor other then and itself every \[ + ve\]integer is composite prime or the Unit 1, So the composite no. are exactly the No. that are not prime and not a unit.
Every composite No. can be written as the product of two or more primes.
Ex:- The composite No.\[299\]can be written as \[13 \times 23\] and the composite No. \[360\] can be written as \[{2^3} \times {3^2} \times {5^2}\]. This representation is unique up to the order of the factor. This fact is called the fundamental He theorem of Arithmetic.

Complete Step-by-step solution
We know a composite number is an integer which is not a prime number i.e. it has more than 2 factors.
Our very first 8 Composite Numbers are \[4,6,8,9,12,14,15\]
By mean of the composite number we mean that:
Mean= sum of all numbers total numbers = \[\dfrac{{4 + 6 + 8 + 9 + 10 + 12 + 14 + 15}}{8}\]
\[Mean = \dfrac{{78}}{8} = 9.75\]
On subtracting all given values from mean value, we get:
The absolute difference between data value and mean.
\[\begin{gathered}
  \left| {9.75 - 4} \right| = 5.73, \\
  \left| {9.75 - 6} \right| = 3.75, \\
  \left| {9.75 - 8} \right| = 1.75, \\
  \left| {9.75 - 9} \right| = 0.75, \\
  \left| {9.75 - 10} \right| = 0.25, \\
  \left| {9.75 - 12} \right| = 2.25, \\
  \left| {9.5 - 14} \right| = 4.25, \\
  \left| {9.75 - 15} \right| = 5.25. \\
\end{gathered} \]
Therefore, the respective data values after facing the absolute differences are:
\[5.75,3.75,1.75,0.75,0.25,2.25,4.25and5.25\]
We know the mean deviation is given by:
sum of all absolute differences total numbers=\[\dfrac{{5.75 + 3.75 + 1.75 + 0.75 + 0.25 + 2.25 + 4.25 + 5.25}}{8}\]
\[ = \dfrac{{24}}{8} = 3\]
Therefore, the mean deviation of the first 8 composite numbers is 3.

Note: Mean deviation is also given by the formulae:
\[\dfrac{{\sum \left| {\mathop {xi}\limits^ \to - A} \right|}}{n}\]
\[A = \]Arithmetic Mean
\[\sum \to \]Summation
\[A = \dfrac{{Sum\,of\,Au\,Obs}}{{No.\,of\,obs}}\]
Mean deviation is Average absolute deviation, or mean absolute deviation of data. Set is the average of the absolute deviation from a central point. It is a summary statistic of statistical dispersion or variability