Q.The marks obtained by 50 first-year Commerce students in an Economics test are grouped as follows: 0–10: 5 students, 10–20: 8 students, 20–30: 15 students, 30–40: 12 students, 40–50: 10 students. Calculate the arithmetic mean using the step-deviation method, and verify your answer using the direct method.

The arithmetic mean is the sum of all observations divided by their number. For an individual series it is computed directly as $\bar X=\Sigma X/N$, or by the short-cut (assumed mean) method $\bar X = A + \Sigma d/N$ using deviations $d=X-A$. For a discrete series the same two methods apply weighted by frequency: $\bar X=\Sigma fX/N$ or $\bar X=A+\Sigma fd/N$. For a continuous series, the step-deviation method is generally quickest: $\bar X = A + (\Sigma fd'/N)\times h$, with $d'=(m-A)/h$ computed from each class's mid-value $m$ and common class width $h$. All valid methods on the same data give the same final mean; the mean is based on every observation and suited to further algebraic use, but is pulled noticeably by extreme values.