Question

The arithmetic mean of consumption expenditures for \(5\) members is Rupees \(40\), with standard deviation of Rupees \(7\). Three new members are added in the group, and their consumption expenditures (in Rupees) are \(30\), \(35\), and \(40\). Then, the standard deviation of consumption expenditures of these \(8\) members is (rounded off to two decimal places).

Hint:

When combining two groups, calculate the new mean and new sum of squares separately before applying the variance formula. Direct averaging of standard deviations is incorrect.

Answer 1

Correct Answer: 6.54

Step 1: Use the variance shortcut.
For any group, $\sigma^2=\dfrac{\sum x^2}{n}-\bar{x}^2$. For the first $5$ members, $\bar{x}=40$ and $\sigma=7$, so the variance is $49$.

Step 2: Recover their sum of squares.
From $49=\dfrac{\sum x^2}{5}-1600$ we get $\sum x^2=5(1649)=8245$. Their total is $5\times40=200$.

Step 3: Bring in the three new members.
The new values $30,35,40$ add $105$ to the total and $900+1225+1600=3725$ to the sum of squares.

Step 4: Find the new mean.
Now $\bar{x}=\dfrac{200+105}{8}=\dfrac{305}{8}=38.125$.

Step 5: Find the new variance.
With total sum of squares $8245+3725=11970$,
\[ \sigma^2=\frac{11970}{8}-(38.125)^2=1496.25-1453.5156=42.7344 \]

Step 6: Take the root.
\[ \sigma=\sqrt{42.7344}\approx6.54 \]
\[ \boxed{\text{Rupees }6.54} \]