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: Recall the formula relating variance, mean, and sum of squares.
For a dataset with \(n\) observations, mean \(\bar{x}\), and standard deviation \(\sigma\), we use
\[ \sigma^2=\frac{\sum x_i^2}{n}-\bar{x}^2 \]
We are given for the first \(5\) members:
\[ n_1=5 \] \[ \bar{x}_1=40 \] \[ \sigma_1=7 \]
Therefore, variance is
\[ \sigma_1^2=7^2=49 \]

Step 2: Find the sum of squares for the first \(5\) members.
Using the variance formula,
\[ 49=\frac{\sum x_i^2}{5}-40^2 \]
Since
\[ 40^2=1600 \] we get
\[ 49=\frac{\sum x_i^2}{5}-1600 \]
Adding \(1600\) on both sides,
\[ 1649=\frac{\sum x_i^2}{5} \]
Multiplying by \(5\),
\[ \sum x_i^2=8245 \]
Also, total expenditure of the first \(5\) members is
\[ \sum x_i = 5 \times 40 \] \[ \sum x_i = 200 \]

Step 3: Add the expenditures of the three new members.
The expenditures of the new members are
\[ 30,\ 35,\ 40 \]
Their total is
\[ 30+35+40=105 \]
Their sum of squares is
\[ 30^2+35^2+40^2 \] \[ =900+1225+1600 \] \[ =3725 \]

Step 4: Find the combined mean for \(8\) members.
Total expenditure of all \(8\) members is
\[ 200+105=305 \]
Thus, the new mean is
\[ \bar{x}=\frac{305}{8} \] \[ \bar{x}=38.125 \]

Step 5: Find the combined sum of squares.
Total sum of squares becomes
\[ 8245+3725 \] \[ =11970 \]
Now apply the variance formula for all \(8\) members:
\[ \sigma^2 = \frac{11970}{8}-(38.125)^2 \]
First, calculate
\[ \frac{11970}{8}=1496.25 \]
Next,
\[ (38.125)^2=1453.515625 \]
Therefore,
\[ \sigma^2 = 1496.25-1453.515625 \] \[ \sigma^2 = 42.734375 \]

Step 6: Calculate the standard deviation.
\[ \sigma = \sqrt{42.734375} \] \[ \sigma \approx 6.536 \]
Rounded off to two decimal places,
\[ \sigma \approx 6.54 \]

Step 7: Final conclusion.
Hence, the standard deviation of consumption expenditures of the \(8\) members is
\[ \boxed{6.54} \]