Find standard deviation. Expenditure 10 30 30 50 50 70 70 90 90 110 : : : : : : Individual 3 2 4 3 3

Solution to Find Standard Deviation

Step 1: Find the Midpoint of Each Class Interval

The given data is grouped, so we first calculate the midpoint (class mark) of each interval using the formula:

$$\text{Class mark}=\frac{\text{Lower limit}+\text{Upper limit}}{2}$$

Step 2: Calculate Mean ($\bar{x}$)

$$\bar{x}=\frac{\sum (f\times x)}{\sum f}$$

$$\sum f=3+2+4+3+3=15$$

$$\sum (f\times x)=60+80+240+240+300=920$$

$$\bar{x}=\frac{920}{15}\approx 61.33$$

Step 3: Calculate Variance (${\sigma }^2$)

$${\sigma }^2=\frac{\sum (f\times x^2)}{N}-{\left(\frac{\sum (f\times x)}{N}\right)}^2$$

Where $N=\sum f=15$.

$$\sum (f\times x^2)=1200+3200+14400+19200+30000=68000$$

$${\sigma }^2=\frac{68000}{15}-(61.33{)}^2$$

$${\sigma }^2=4533.33-3761.69=771.64$$

Step 4: Calculate Standard Deviation ($\sigma$)

$$\sigma =\sqrt{{\sigma }^2}=\sqrt{771.64}\approx 27.78$$

Final Answer: The standard deviation of the given expenditure data is 27.78.