Average

Average

An average or an arithmetic mean of given data is the sum of the given observations divided by number of observations.

Average (A) \[=\frac{\text{Sum}\,\,\text{of}\,\,\text{given}\,\,\text{observations}\,\text{(S)}}{\text{Number}\,\,\text{of}\,\,\text{observations}\,\text{(N)}}\]

More About Average

·         Average of a given data is less than the greatest observation and greater than the smallest observation of the given data.

·         If the observations of given data are equal, then the average will also be the same as observations.

·         If '0' is one of the observations of a given data, then that 0 will also be included while calculating average.

·         If all the numbers get increased by 'a', then their average must be increased by 'a'.

·         If all the numbers get decreased by 'a', then their average must be decreased by 'a'.

·         If all the numbers get multiplied by 'a', then their average must be multiplied by 'a'.

·         If all the numbers get divided by 'a', then their average must be divided by 'a'.

Shortcut Method for Average of n Consecutive Natural/Odd/Even Numbers

Let the consecutive n odd numbers are (Here, x is odd/even and n is odd)

\[\begin{matrix}

   x & x+2 & x+4 & x+6 & x+8  \\

   \downarrow  & \downarrow  & \downarrow  & \downarrow  & \downarrow   \\

   O & O & [A] & O & O  \\

\end{matrix}\]

(i) Here, \[x+4\] is average number.

(ii) Highest number is \[x+8.\]

(iii) Lowest number is x

For example. Average of 5, 7, 9, 11, 13 is 9. Average of 3, 4, 5, 6, 7, 8, 9 is 6

Average of 6, 8, 10, 12, 14, 16, 18, 20, 22 is 14.

Average Speed

The average speed of a body is the total distance travelled divided by the total time taken to cover a distance

Average speed \[=\frac{\text{Total distance travelled}}{\text{Total time taken}}\]

Case I  If a person covers a certain distance at a speed of A km/h and again covers the same distance at a speed of B km/h, then the average speed during the whole journey will be \[\frac{2AB}{A+B}.\]

Case II If a person covers three equal distances at a speed of A km/h, B km/h, C km/h, then the average speed during the whole journey will be \[\frac{3ABC}{AB+BC+CA}.\]

Case III If distance ‘P’ is covered with speed x, distance

‘Q’ is covered with speed y and distance ‘R’ is covered with speed z, then for the whole journey.

Average speed \[=\frac{P+Q+R+....}{\frac{P}{x}+\frac{Q}{y}+\frac{R}{z}+...}\]

Case IV If a person covers ‘P’ part of his total distance

with speed of ‘x’, Q part of total distance with speed ‘y’ and R part of total distance with speed of ‘z’, then

Average speed \[=\frac{1}{\frac{P}{x}+\frac{Q}{y}+\frac{R}{z}+...}\]

Quicker One

Ø   Average of first n natural numbers \[=\left( \frac{n+1}{2} \right)\]

Ø   Average of first n even numbers \[=(n+1)\]

Ø   Average of first n odd numbers \[=n\]

Ø   Average of consecutive numbers\[=\frac{\text{First}\,\,\text{number+Last}\,\,\text{number}}{2}\]

Ø   Average of  1 to n odd numbers\[=\frac{\text{Last}\,\,\text{odd}\,\,\text{numbers+1}}{2}\]

Ø   Average of 1 to n even numbers \[=\frac{\text{Last}\,\,\text{even}\,\,\text{number+2}}{2}\]

Ø   Average of  squares of first n natural numbers\[=\frac{(n+1)(2n+1)}{6}\]

Ø   Average of the cubes of first n natural numbers\[=\frac{n\,\,{{(n+1)}^{2}}}{4}\]

Ø   Average of n multiples of nay number \[=\frac{\text{Number}\times (n+1)}{2}\]

Ø   If the average of \[{{n}_{1}}\] observations is \[{{a}_{1}},\] the average of \[{{n}_{2}}\] observations is \[{{a}_{2}}\] and so on, then average of all the observations \[=\frac{{{n}_{1}}{{a}_{1}}+{{n}_{2}}{{a}_{2}}+...}{{{n}_{1}}+{{n}_{2}}+...}\]

Ø   If the average of m observations is a and the average of n  observations taken out of m is b, then average of rest of the observation \[=\frac{ma-nb}{m-n}\]