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Q.

For grouped data, Arithmetic mean by direct method: -

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a

SfX / Sf

b

Sd / N

c

SX / N

d

None of these

answer is A.

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Detailed Solution

Here, we have to find the arithmetic mean by direct method, for grouped data.
Let us consider an example to find the formula for mean by direct method. Let us consider the following observations in a grouped form: -

Marks obtained	Xi	fi
0 – 10	X1 = 5	f1 = 1
10 – 20	X2 = 15	f2 = 3
20 – 30	X3 = 25	f3 = 2
30 – 40	X4 = 35	f4 = 5
40 – 50	X5 = 45	f5 = 2

The following table indicates the marks obtained by 18 students. Here, marks are classified in intervals. This is an example of grouped data.
Now, Xi is the mid – point of marks intervals and fi is the frequency or number of students in that interval of marks. So, here the mean is calculated by taking the ratio of ∑Xifi and ∑fi .
⇒ Mean =

\frac{\sum X_{i} f_{i}}{\sum f_{i}}

⇒ Mean =

\frac{f_{1} X_{1} + f_{2} X_{2} + f_{3} X_{3} + f_{4} X_{4} + f_{5} X_{5}}{f_{1} + f_{2} + f_{3} + f_{4} + f_{5}}

⇒ Mean =

\frac{(5 \times 1) + (15 \times 3) + (25 \times 2) + (35 \times 5) + (45 \times 2)}{1 + 3 + 2 + 5 + 2}

⇒ Mean =

\frac{5 + 45 + 50 + 175 + 90}{13}

⇒ Mean =

\frac{365}{13}

≈ 28.08
So, the formula for mean using direct method, for grouped data is

\frac{\sum X_{i} f_{i}}{\sum X_{i}}

=

\frac{\sum fX}{\sum f}

(written without indices)
So, the correct answer is “Option 1”.

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