Final answer:
To find the standard deviation, calculate the mean of the length of surnames, then find the variance and finally take the square root of the variance to get the standard deviation.
Step-by-step explanation:
To find the standard deviation, we first need to calculate the mean of the length of surnames. To calculate the mean, we multiply each length by its frequency and sum up the results. Then we divide the sum by the total number of people in the sample, which is 400 in this case. The formula for mean is:
Mean = (Sum of (Length x Frequency)) / Total number of people
Next, we calculate the variance by summing up the squared differences between each length and the mean, multiplied by their respective frequencies. The formula for variance is:
Variance = Sum of ((Length - Mean)^2 x Frequency) / Total number of people
The standard deviation is the square root of the variance. So we take the square root of the calculated variance to get the standard deviation. Using the formula:
Standard Deviation = Square Root of Variance
Plugging in the values from the given data, we can calculate the standard deviation, which is approximately 2.71. Therefore, the correct answer is b) 2.71.