Which of the following has the largest 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 mean of each set: all means are 6.
Calculate the variance of each set: A has variance 8, B has variance 4, C has variance 2, and D has variance 6.8.
Calculate the standard deviation of each set by taking the square root of the variance: A ≈ 2.83, B = 2, C ≈ 1.41, and D ≈ 2.61.
The set with the largest standard deviation is A.
A
Explanation
Analyze the problem and data We are given four sets of numbers and asked to determine which has the largest standard deviation. The sets are:
A = { 2 , 4 , 6 , 8 , 10 } B = { 3 , 5 , 6 , 7 , 9 } C = { 4 , 5 , 6 , 7 , 8 } D = { 2 , 5 , 6 , 7 , 10 }
Explain the concept of standard deviation To find the set with the largest standard deviation, we need to calculate the standard deviation for each set. The standard deviation measures the spread of the data around the mean. The larger the standard deviation, the more spread out the data is.
Calculate the mean for each set First, we calculate the mean for each set:
For set A: 5 2 + 4 + 6 + 8 + 10 = 5 30 = 6 For set B: 5 3 + 5 + 6 + 7 + 9 = 5 30 = 6 For set C: 5 4 + 5 + 6 + 7 + 8 = 5 30 = 6 For set D: 5 2 + 5 + 6 + 7 + 10 = 5 30 = 6
All sets have a mean of 6.
Calculate the variance for each set Next, we calculate the variance for each set. The variance is the average of the squared differences from the mean.
For set A: 5 ( 2 − 6 ) 2 + ( 4 − 6 ) 2 + ( 6 − 6 ) 2 + ( 8 − 6 ) 2 + ( 10 − 6 ) 2 = 5 16 + 4 + 0 + 4 + 16 = 5 40 = 8 For set B: 5 ( 3 − 6 ) 2 + ( 5 − 6 ) 2 + ( 6 − 6 ) 2 + ( 7 − 6 ) 2 + ( 9 − 6 ) 2 = 5 9 + 1 + 0 + 1 + 9 = 5 20 = 4 For set C: 5 ( 4 − 6 ) 2 + ( 5 − 6 ) 2 + ( 6 − 6 ) 2 + ( 7 − 6 ) 2 + ( 8 − 6 ) 2 = 5 4 + 1 + 0 + 1 + 4 = 5 10 = 2 For set D: 5 ( 2 − 6 ) 2 + ( 5 − 6 ) 2 + ( 6 − 6 ) 2 + ( 7 − 6 ) 2 + ( 10 − 6 ) 2 = 5 16 + 1 + 0 + 1 + 16 = 5 34 = 6.8
Calculate the standard deviation for each set Finally, we calculate the standard deviation for each set by taking the square root of the variance.
For set A: 8 ≈ 2.83 For set B: 4 = 2 For set C: 2 ≈ 1.41 For set D: 6.8 ≈ 2.61
Comparing the standard deviations, we see that set A has the largest standard deviation.
Determine the set with the largest standard deviation The set with the largest standard deviation is A.
Examples
Understanding standard deviation helps in many real-world scenarios. For example, in finance, it's used to measure the risk of an investment; a higher standard deviation means the investment's returns are more volatile. In quality control, it helps ensure product consistency; a lower standard deviation indicates more uniform products. In sports, it can show how consistently a player performs; a lower standard deviation means the player is more reliable.
