You are allowed to choose four whole numbers from 1 to 10 (including 1 and
10). You can only choose each number once though. Which of the following
statements is FALSE?
A. The numbers 4, 5, 6, and 7 have the smallest possible standard
deviation.
B. The numbers 1, 2, 3, and 4 have the smallest possible standard
deviation.
C. The numbers 1, 5, 6, and 10 have the largest possible standard
deviation.
D. The numbers 1, 2, 9, and 10 have the largest possible standard
deviation
To determine which statement is false, we need to calculate the standard deviation for each set of numbers and compare them.
Let's calculate the standard deviation for each option step-by-step:
A. The numbers 4, 5, 6, and 7.
- Calculate the mean (average) of the numbers: (4 + 5 + 6 + 7) / 4 = 5.5
- Calculate the deviation of each number from the mean: (4 - 5.5), (5 - 5.5), (6 - 5.5), (7 - 5.5)
- Square each deviation: (-1.5)^2, (-0.5)^2, (0.5)^2, (1.5)^2
- Calculate the sum of squared deviations: 2.25 + 0.25 + 0.25 + 2.25 = 5
- Calculate the variance: 5 / 4 ≈ 1.25
- Calculate the standard deviation: √1.25 ≈ 1.12
B. The numbers 1, 2, 3, and 4.
- Calculate the mean (average) of the numbers: (1 + 2 + 3 + 4) / 4 = 2.5
- Calculate the deviation of each number from the mean: (1 - 2.5), (2 - 2.5), (3 - 2.5), (4 - 2.5)
- Square each deviation: (-1.5)^2, (-0.5)^2, (0.5)^2, (1.5)^2
- Calculate the sum of squared deviations: 2.25 + 0.25 + 0.25 + 2.25 = 5
- Calculate the variance: 5 / 4 ≈ 1.25
- Calculate the standard deviation: √1.25 ≈ 1.12
C. The numbers 1, 5, 6, and 10.
- Calculate the mean (average) of the numbers: (1 + 5 + 6 + 10) / 4 = 5.5
- Calculate the deviation of each number from the mean: (1 - 5.5), (5 - 5.5), (6 - 5.5), (10 - 5.5)
- Square each deviation: (-4.5)^2, (-0.5)^2, (0.5)^2, (4.5)^2
- Calculate the sum of squared deviations: 20.25 + 0.25 + 0.25 + 20.25 = 41
- Calculate the variance: 41 / 4 = 10.25
- Calculate the standard deviation: √10.25 ≈ 3.21
D. The numbers 1, 2, 9, and 10.
- Calculate the mean (average) of the numbers: (1 + 2 + 9 + 10) / 4 = 5.5
- Calculate the deviation of each number from the mean: (1 - 5.5), (2 - 5.5), (9 - 5.5), (10 - 5.5)
- Square each deviation: (-4.5)^2, (-3.5)^2, (3.5)^2, (4.5)^2
- Calculate the sum of squared deviations: 20.25 + 12.25 + 12.25 + 20.25 = 65
- Calculate the variance: 65 / 4 = 16.25
- Calculate the standard deviation: √16.25 ≈ 4.04
Comparing the standard deviations:
A. 1.12
B. 1.12
C. 3.21
D. 4.04
Based on the calculations, the FALSE statement is:
D. The numbers 1, 2, 9, and 10 have the largest possible standard deviation.
To determine which statement is false, we need to calculate the standard deviation for each set of four numbers mentioned in the options and compare them.
The standard deviation measures the spread or dispersion of a set of values. A smaller standard deviation indicates that the numbers are closer together, while a larger standard deviation implies that the numbers are more spread out.
To calculate the standard deviation of a set of numbers, follow these steps:
1. Calculate the mean (average) of the numbers.
2. Find the difference between each number and the mean.
3. Square each difference.
4. Calculate the average (mean) of the squared differences.
5. Take the square root of the average.
Let's calculate the standard deviation for each set of numbers:
A. Numbers 4, 5, 6, and 7:
Mean: (4 + 5 + 6 + 7) / 4 = 5.5
Differences: (-1.5, -0.5, 0.5, 1.5)
Squared differences: (2.25, 0.25, 0.25, 2.25)
Average of squared differences: (2.25 + 0.25 + 0.25 + 2.25) / 4 = 1.25
Standard deviation: √1.25 ≈ 1.12
B. Numbers 1, 2, 3, and 4:
Mean: (1 + 2 + 3 + 4) / 4 = 2.5
Differences: (-1.5, -0.5, 0.5, 1.5)
Squared differences: (2.25, 0.25, 0.25, 2.25)
Average of squared differences: (2.25 + 0.25 + 0.25 + 2.25) / 4 = 1.25
Standard deviation: √1.25 ≈ 1.12
C. Numbers 1, 5, 6, and 10:
Mean: (1 + 5 + 6 + 10) / 4 = 5.5
Differences: (-4.5, -0.5, 0.5, 4.5)
Squared differences: (20.25, 0.25, 0.25, 20.25)
Average of squared differences: (20.25 + 0.25 + 0.25 + 20.25) / 4 = 10
Standard deviation: √10 ≈ 3.16
D. Numbers 1, 2, 9, and 10:
Mean: (1 + 2 + 9 + 10) / 4 = 5.5
Differences: (-4.5, -3.5, 3.5, 4.5)
Squared differences: (20.25, 12.25, 12.25, 20.25)
Average of squared differences: (20.25 + 12.25 + 12.25 + 20.25) / 4 = 16.25
Standard deviation: √16.25 ≈ 4.03
Comparing the standard deviations:
A. Standard deviation ≈ 1.12
B. Standard deviation ≈ 1.12
C. Standard deviation ≈ 3.16
D. Standard deviation ≈ 4.03
Based on the calculations, the FALSE statement is option D. The numbers 1, 2, 9, and 10 do not have the largest possible standard deviation.
Therefore, the correct answer is D. The numbers 1, 2, 9, and 10 have the largest possible standard deviation.