Calculation of Harmonic Mean

\(HM=\frac { n }{ \sum { \left( \frac { f }{ X } \right) } } =\frac {120 }{ 5.975 } =20.08\)

If the problem is given to a layman he is most likely to compute the arithmetic mean of two speeds 
i.e., \(\overset { - }{ X } =\frac { 30km+20km }{ 2 } =25kmph\)
But this is not the correct average.
So harmonic mean would be mean suitable in this situation. Harmonic Mean of 30 and 20 is
\(HM=\frac { 2 }{ \left( \frac { 1 }{ 10 } \right) +\left( \frac { 1 }{ 30 } \right) } =\frac { 2 }{ \left( \frac { 5 }{ 60 } \right) } =\frac { 2\left( 60 \right) }{ 5 } \)
= 24 kmph

Marks are arranged in ascending order
12 15 20 28 30 40 50
n = number of observations =7
Q₁ = Size of \({ \left( \frac { \left( n+1 \right) }{ 4 } \right) }^{ th }\)value = Size of \({ \left( \frac { 7+1 }{ 4 } \right) }^{ th }\)value = Size of 2^nd value = 15
Hence Q₁ = 15
Q₃ = Size of \({ \left( \frac {3 \left( n+1 \right) }{ 4 } \right) }^{ th }\) value = Size of \({ \left( \frac { 3\times 8 }{ 4 } \right) }^{ th } \)value = Size of 6^th value = 40
Hence Q₃ = 40
QD =\(\quad \frac { 1 }{ 2 } \left( { Q }_{ 3 }-{ Q }_{ 1 } \right) \frac { 40-15 }{ 2 } =12.5\)
Coefficient of QD =\(\frac { { Q }_{ 3 }-{ Q }_{ 1 } }{ { Q }_{ 3 }+{ Q }_{ 1 } } \)=\(\frac { 40-15 }{ 40+15 } =\frac { 25 }{ 55 } =0.455\)
Hence coefficient of QD = 0.455

Q₁ = size of \({ \left( \frac { N+1 }{ 4 } \right) }^{ th }\)value
= size of \({ \left( \frac { 156 }{ 4 } \right) }^{ th }\) value
= size of 39^th value = 5
Q₃ = size of 3\({ \left( \frac { N+1 }{ 4 } \right) }^{ th }\) value
= size of 3(39)^th value
= size of 117^th value = 6.5 
D₈ = size of 8\({ \left( \frac { N+1 }{ 10 } \right) }^{ th }\) value
= size of 8\(\left( \frac { 156 }{ 10} \right) ^{ th }\) value
= size of 124.8^th value = 6.5
P₆₇ = size of 67\({ \left( \frac { (N+1) }{ 100 } \right) }^{ th }\) value 
= size of 104.52^th value
= 6

GM = Antilog\(\left( \frac { \sum { logx } }{ n } \right) =Anitlog\left( \frac { 6.3026 }{ 3 } \right) =Antilog(2.1009)\)
GM = 126.2
\(\therefore\) Average increase of the commodity from 2004 to 2007
= 126.1 - 100 = 26.1%