Question

Following is the number of heads of 8 coins when tossed 205 times:

Number of Heads	0	1	2	3	4	5	6	7	8
Frequency	2	19	46	62	47	20	5	2	2

Calculate arithmetic mean (mean number of heads per toss), using Direct Method and Short-cut Method.

Answer 1

No. of Heads (X)	Frequency (f)	fX	Deviation d = X − A (A = 4)	Multiplication of deviation and frequency (fd)
0 1 2 3 4 (A) 5 6 7 8	2 19 46 62 47 20 5 2 2	0 19 92 186 188 100 30 14 16	−4 −3 −2 −1 0 1 2 3 4	−8 −57 −92 −62 0 20 10 6 8
	Σf = 205	Σfx = 645		Σfd = −219 + 44 = −175

Calculation of mean by direct method.
$$\begin{gathered} \overline{X}=\frac{\Sigma fX}{\Sigma f} \\ \mathrm{or},\overline{\mathit{X}}\mathit{}=\frac{645}{205} \\ \mathrm{or},\overline{X}=3.146 \\ \Rightarrow \overline{\mathit{X}}=3.15approx \end{gathered}$$

Calculation mean by short cut method
$$\begin{gathered} \overline{\mathit{X}}=\mathrm{A}+\frac{\Sigma fd}{\Sigma f} \\ \mathrm{or},\overline{X}=4+\frac{-175}{205} \\ \mathrm{or},\overline{X}=4-0.853 \\ \mathrm{or},\overline{X}=3.147 \\ \Rightarrow \overline{X}=3.15approx \end{gathered}$$

So, mean number of heads per toss is 3.15.