mpg cyl disp
Min. :14.30 Min. :4.0 Min. :108.0
1st Qu.:18.55 1st Qu.:5.5 1st Qu.:156.7
Median :21.00 Median :6.0 Median :192.5
Mean :20.21 Mean :6.0 Mean :222.2
3rd Qu.:21.75 3rd Qu.:6.5 3rd Qu.:283.5
Max. :24.40 Max. :8.0 Max. :360.0
“An admirable artifice which, by reducing to a few days the labour of many months, doubles the life of the astronomer, and spares him the errors and disgust inseparable from long calculations.”
Reciprocal of \(\frac{\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}}{n}\)
Thus, \(HM = \frac{n}{\sum \frac{1}{x_i}}\)
Calculate: 2, 4, 8
Answer
3.43
For grouped data
\(HM=\frac{\sum f}{\sum \frac{f}{x}}\)
For wighted data: \(\frac{\sum w}{\sum \frac{w}{x}}\)
Why and When HM
When there are rates associated, say speed, and numerator is fixed.
\(Speed, v = \frac{S}{t}\); HM if S is fixed
Example: A man travels 120 km the first day at 12 kph, the same distance at 10 kph on the 2nd day, and at 8 kph on the 3rd day. Find his average speed.
Wighted AM vs Weighted HM
Suppose, a bus travels 10 km at 10 kph, another 15 km at 20 kph, and another 20 km at 25 kph. What is the average speed.
For two non-zero positive numbers, \(AM \times HM =(GM)^2\)
Theorem 10
Mean and Median of first n natural numbers are \(\frac {n+1} 2\)
Theorem 11
If \(\bar x_1\) and \(\bar x_2\) are means of 2 data sets of sizes \(n_1\) and \(n_2\), respectively, the combined mean is \(\bar x_c=\frac{n_1 \bar x_1+n_2 \bar x_2}{n_1+n_2}\)
Theorem 12
If \(u=x+y, \bar u=\bar x + \bar y\); if \(n_1=n_2=n\)
Given \(u=x+y\)
\[\begin{eqnarray}
\bar u &=& \frac{\sum u}{n} \nonumber \\
&=& \frac{\sum (x+y)}{n} \nonumber \\
&=& \frac{\sum x}{n}+ \frac{\sum y}{n} \nonumber \\
&=& \bar x + \bar y \nonumber \\
\end{eqnarray}\]
Theorem 13
For equal number of observations, GM of two variables is equal to the product of their individual means.
Theorem 14
\(GM=Antilog(\frac{\sum \log x_i}{n})\) or \(Antilog(\frac{\sum f_i \log x_i}{\sum f_i})\)
Theorem 15
If \(y = a + bx, \bar y = a + b \bar x\)
Theorem 16
If \(z_i=ax_i+by_i, \bar z=a \bar x + b \bar y\)
Example Problems
Example Problem 01
AM and GM of two positive numbers are 25 and 15, respectively. Find HM and the numbers.
Solution (i)
We know, \(AM \times HM=(GM)^2\)
Thus, \(HM=\) 9
Solution (ii)
If the numbers are \(a, b; a>b\)\(\frac{a+b}{2}=25\) and \(\sqrt{ab}=15\)
The mean of 200 numbers was 50. Later it was revealed that two observations were incorrectly given as 92 and 8, instead of 192 and 88, respectively. Find the correct mean.
Solution
\(n=200, \bar x = 200\)
\(\therefore\) Incorrect total, \(\sum x = n \times \bar x=\) 10000
