Question:
A dataset contains the numbers 5, 7, 9, 11, 13, and 15. What is the standard deviation?
A dataset contains the numbers 5, 7, 9, 11, 13, and 15. What is the standard deviation?
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Standard deviation measures the spread of data around the mean; calculate variance first, then take the square root.
Updated On: Oct 4, 2025
- 2.58
- 3.74
- 4.20
- 5.00
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The Correct Option is B
Solution and Explanation
Step 1: Calculate the mean.
[6pt] The mean is the sum of all numbers divided by the number of numbers. Here, we have the numbers 5, 7, 9, 11, 13, and 15. The mean is calculated as: \[ \text{Mean} = \frac{5 + 7 + 9 + 11 + 13 + 15}{6} = \frac{60}{6} = 10. \] Step 2: Calculate the squared differences from the mean.
[6pt] Next, we subtract the mean from each number, square the result, and then sum the squared differences: \[ (5 - 10)^2 = (-5)^2 = 25,
(7 - 10)^2 = (-3)^2 = 9,
(9 - 10)^2 = (-1)^2 = 1, \] \[ (11 - 10)^2 = 1^2 = 1,
(13 - 10)^2 = 3^2 = 9,
(15 - 10)^2 = 5^2 = 25. \] Sum of the squared differences: \[ 25 + 9 + 1 + 1 + 9 + 25 = 70. \] Step 3: Calculate the variance.
[6pt] The variance is the sum of squared differences divided by \(n-1\), where \(n\) is the number of data points. Here, \(n = 6\), so we divide by \(5\): \[ \text{Variance} = \frac{70}{5} = 14. \] Step 4: Calculate the standard deviation.
[6pt] The standard deviation is the square root of the variance: \[ \sigma = \sqrt{14} \approx 3.74. \] Thus, the standard deviation is approximately \( 3.74 \).
[6pt] The mean is the sum of all numbers divided by the number of numbers. Here, we have the numbers 5, 7, 9, 11, 13, and 15. The mean is calculated as: \[ \text{Mean} = \frac{5 + 7 + 9 + 11 + 13 + 15}{6} = \frac{60}{6} = 10. \] Step 2: Calculate the squared differences from the mean.
[6pt] Next, we subtract the mean from each number, square the result, and then sum the squared differences: \[ (5 - 10)^2 = (-5)^2 = 25,
(7 - 10)^2 = (-3)^2 = 9,
(9 - 10)^2 = (-1)^2 = 1, \] \[ (11 - 10)^2 = 1^2 = 1,
(13 - 10)^2 = 3^2 = 9,
(15 - 10)^2 = 5^2 = 25. \] Sum of the squared differences: \[ 25 + 9 + 1 + 1 + 9 + 25 = 70. \] Step 3: Calculate the variance.
[6pt] The variance is the sum of squared differences divided by \(n-1\), where \(n\) is the number of data points. Here, \(n = 6\), so we divide by \(5\): \[ \text{Variance} = \frac{70}{5} = 14. \] Step 4: Calculate the standard deviation.
[6pt] The standard deviation is the square root of the variance: \[ \sigma = \sqrt{14} \approx 3.74. \] Thus, the standard deviation is approximately \( 3.74 \).
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