Understanding Standard Deviation Calculation for Ages

This solution explains the step-by-step process to calculate the standard deviation for a given set of ages: 25, 30, 35, 40, 45, and 50 years. Standard deviation measures the amount of variation or dispersion of a set of values.

Step 1: Calculate the Mean (Average) of the Ages

First, we find the mean ($\mu$) of the given ages. The mean is the sum of all values divided by the total number of values.

The ages are: 25, 30, 35, 40, 45, 50.

Number of observations ($n$) = 6.

Sum of ages = $25 + 30 + 35 + 40 + 45 + 50 = 225$ years.

The formula for the mean is:

$$ \mu = \frac{\sum x_i}{n} $$

Calculating the mean:

$$ \mu = \frac{225}{6} = 37.5 \text{ years} $$

Step 2: Calculate the Deviations from the Mean

Next, we find the difference between each age and the mean ($\mu$).

Step 3: Square the Deviations

Now, we square each of the deviations calculated in the previous step.

Step 4: Calculate the Variance

The variance ($\sigma^2$) is the average of the squared deviations. Since we are given a specific group of 6 people and asked for the standard deviation of *their* ages, we treat this as a population. Therefore, we divide the sum of squared deviations by the number of observations ($n$).

Sum of Squared Deviations = $156.25 + 56.25 + 6.25 + 6.25 + 56.25 + 156.25 = 437.5$

The formula for population variance is:

$$ \sigma^2 = \frac{\sum (x_i - \mu)^2}{n} $$

Calculating the variance:

$$ \sigma^2 = \frac{437.5}{6} \approx 72.9167 $$

Step 5: Calculate the Standard Deviation

The standard deviation ($\sigma$) is the square root of the variance.

The formula for population standard deviation is:

$$ \sigma = \sqrt{\sigma^2} $$

Calculating the standard deviation:

$$ \sigma = \sqrt{72.9167} \approx 8.5404 $$

Step 6: Round the Result

We need to round the standard deviation to two decimal places.

$$ \sigma \approx 8.54 $$

Therefore, the standard deviation of the ages is approximately 8.54 years.