Which of the following has the smallest standard deviation?
$\begin{array}{l}
A={2,4,6,8,10} \
B={3,5,6,7,9} \
C={4,5,6,7,8} \
D={2,5,6,7,10}
\end{array}$
A
B
C
D
Answer (1)
Calculate the standard deviation for each set.
Set A has a standard deviation of approximately 3.16 .
Set B has a standard deviation of approximately 2.24 .
Set C has a standard deviation of approximately 1.58 .
Set D has a standard deviation of approximately 2.92 .
The set with the smallest standard deviation is C.
C
Explanation
Understand the problem We are given four sets of numbers and asked to find the set with the smallest standard deviation. The standard deviation measures the spread of the data around the mean. A smaller standard deviation indicates that the data points are closer to the mean.
Calculate standard deviations First, we calculate the standard deviation for each set:
Set A: A = {2, 4, 6, 8, 10}. The standard deviation is approximately 3.16. Set B: B = {3, 5, 6, 7, 9}. The standard deviation is approximately 2.24. Set C: C = {4, 5, 6, 7, 8}. The standard deviation is approximately 1.58. Set D: D = {2, 5, 6, 7, 10}. The standard deviation is approximately 2.92.
Compare and conclude Comparing the standard deviations, we have:
Standard deviation of A: 3.16 Standard deviation of B: 2.24 Standard deviation of C: 1.58 Standard deviation of D: 2.92
The smallest standard deviation is 1.58, which corresponds to set C.
Examples
Understanding standard deviation helps in many real-world scenarios. For example, in finance, it's used to measure the volatility of an investment. A stock with a low standard deviation is generally considered less risky than a stock with a high standard deviation. In quality control, standard deviation helps ensure that products meet consistent standards. If the standard deviation of a product's dimensions is too high, it indicates that the manufacturing process is not consistent.
