Final answer:
The standard deviation of the given distribution is calculated using the mean and the sum of squared deviations from the mean. After finding the variance and taking its square root, the closest standard deviation to our calculation is 17.81, which approximates to the provided option B) 17.5.
Step-by-step explanation:
To calculate the standard deviation of the given data set: 40, 48, 51, 55, 62, 71, 81, we first need to find the mean (average) of the data set. Then, we will use that mean to determine the variance, and take the square root of the variance to find the standard deviation.
- Calculate the mean (average): Sum all the numbers and divide by the number of data points.
(40 + 48 + 51 + 55 + 62 + 71 + 81) / 7 = 58.29 (rounded to two decimal places). - Calculate each data point's deviation from the mean, square each deviation, and sum all the squared deviations.
(40 - 58.29)^2 + (48 - 58.29)^2 + ... + (81 - 58.29)^2 = 414.71 + 106.09 + ... + 514.08 = 1902.14 (rounded to two decimal places). - Divide the sum of squared deviations by the number of data points minus one (N-1) to get the variance.
1902.14 / (7 - 1) = 317.02 (rounded to two decimal places). - Take the square root of the variance to get the standard deviation.
Square root of 317.02 = 17.81 (rounded to two decimal places).
The closest answer to our calculation is Option B) 17.5, which is rounded to the nearest 0.5.