Question

Calculate the average marks from the following data by using short-cut method:
Marks:
6-10
11-15
16-20
21-25
26-30
No. of students:
20
30
50
40
10
[Ans. X = 17.67]
Find out the arithmetic mean from the following data:​

Answer 1

Calculate the average marks from the following data by using the short-cut method:

$\begin{tabular}{|c|c|} \underline{Marks} & \underline{No. of students}\\6-10& 20\\ 11-15 & 30 \\ 16-20 & 50 \\ 21-25 & 40\\ 26-30 & 10\end{tabular}$

Answer

The average marks is 17.67 marks (approx.)

Given

Frequency table

To Find

Average Marks using the Short-cut method

Solution

First, we need to convert the classes to continuous form.

The formula is:-

new lower limit

= current lower limit - (current lower limit - the upper limit of the previous class)/2

new upper limit

= current upper limit + (the lower limit of the next class - current upper limit)/2

Hence we get

$\begin{tabular}{|c|} \underline{Marks}\\6-10\\11-15\\16-20\\21-25\\26-30 \\\end{tabular}$$\begin{tabular}{|c|} \underline{Marks(continuous)}\\5.5-10.5\\10.5-15.5\\15.5-20.5\\20.5-25.5\\25.5-30.5 \\\end{tabular}$$\begin{tabular}{|c|} \underline{No. of Student}\\ 20\\30\\50\\40\\10\\\end{tabular}$

The formula short-cut method is

A + ∑df/∑f

where,

A = A common term between X

d = X - A

f = frequency

X = class mark

The formula for class mark is

(lower limit + upper limit)/2

Hence we get the following

$\begin{tabular}{|c|} \underline{Marks(continuous)}\\5.5-10.5\\10.5-15.5\\15.5-20.5\\20.5-25.5\\25.5-30.5 \\\end{tabular}$$\begin{tabular}{|c|} \underline{Class Mark(X)}\\ 8\\13\\18\\23\\28\\\end{tabular}$$\begin{tabular}{|c|} \underline{No. of Students (f)}\\ 20\\30\\50\\40\\10\\\end{tabular}$

Now we will select a common term A

let the common term be 18

Therefore now we will calculate

d = X - A and df

$\begin{tabular}{|c|} \underline{d = X - A, (A = 18)}\\ -10\\-5\\0\\5\\10\\\\\end{tabular}$$\begin{tabular}{|c|} \underline{No. of Students (f)}\\ 20\\30\\50\\40\\10\\ \overline{\sum f = 150}\\\end{tabular}$$\begin{tabular}{|c|} \underline{df}\\ -200\\-150\\0\\200\\100\\ \overline{\sum df = -50}\\\end{tabular}$

Therefore, Average marks is

= 18 + (-50)/(150) marks

= 18 - 1/3 marks

= 17 2/3 marks

= 17.67 marks (Approx)

Therefore, the average marks is 17.67 marks (approx.)

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