Question

Students of Class XI secured following marks in their Statistics paper.
Calculate arithmetic mean using Direct Method and Short-cut Method.
20, 44, 65, 28, 45, 67, 30, 50, 68, 39, 53, 70, 40, 60, 75

Answer 1

Sr. No.	Marks (X)	Deviation from the Assumed mean d = X − A where A = 50
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.	20 44 65 28 45 67 30 50 (A) 68 39 53 70 40 60 75	-30 -06 15 -22 -05 17 -20 0 18 -11 3 20 -10 10 25
N = 15	ΣX = 754	Σd = -104 + 108 = 4

where,
N= Number of items
$\Sigma X$= Total of monthly income of all the families
$\Sigma d$= net sum of deviations

Calculation of Mean by direct method
$$\begin{gathered} \overline{X}=\frac{\Sigma X}{N} \\ \mathrm{or},\overline{X}=\frac{754}{15} \\ \mathrm{or},\overline{X}=50.27 \\ \Rightarrow \overline{X}=50.27 \end{gathered}$$

Calculation of mean by short cut method
$$\begin{gathered} \overline{X}=\mathrm{A}+\frac{\Sigma d}{N} \\ \mathrm{or},\mathit{}\overline{X}=50+\frac{4}{15} \\ \mathrm{or},\overline{X}=50+0.27=50.27 \\ \Rightarrow \overline{X}=50.27 \end{gathered}$$

So, mean marks in statistics is 50.27.