Following are the ages (in years) of 6 people in a group: 25, 30, 35, 40, 45 and 50. What is the standard deviation of their ages (rounded to two decimal places)?

Given:

Ages of the group: 25, 30, 35, 40, 45, 50

n = 6

Formula used:

Standard Deviation (σ) = \(\sqrt{\dfrac{\sum(x_i - \mu)^2}{n}}\)

Where:

n = number of data points

\(\mu\) = mean of the data points

\((x_i)\) = each data point

Calculation:

Step 1: Calculate the mean:

\(\mu = \dfrac{\text{Sum of ages}}{n}\)

⇒ \(\mu = \dfrac{25+30+35+40+45+50}{6}\)

⇒ \(\mu = \dfrac{225}{6} = 37.5\)

Step 2: Calculate the squared differences from the mean:

\((x_i - \mu)^2\) for each data point:

For 25: \((25 - 37.5)^2 = (-12.5)^2 = 156.25\)

For 30: \((30 - 37.5)^2 = (-7.5)^2 = 56.25\)

For 35: \((35 - 37.5)^2 = (-2.5)^2 = 6.25\)

For 40: \((40 - 37.5)^2 = (2.5)^2 = 6.25\)

For 45: \((45 - 37.5)^2 = (7.5)^2 = 56.25\)

For 50: \((50 - 37.5)^2 = (12.5)^2 = 156.25\)

Step 3: Calculate the variance:

Variance = \(\dfrac{\text{Sum of squared differences}}{n}\)

⇒ Variance = \(\dfrac{156.25+56.25+6.25+6.25+56.25+156.25}{6}\)

⇒ Variance = \(\dfrac{437.5}{6} = 72.92\)

Step 4: Calculate the standard deviation:

σ = \(\sqrt{72.92}\)

⇒ σ = 8.54

∴ The correct answer is option (1).