What is the standard deviation of the data set {10, 12, 23, 23, 16, 23, 21}?

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Q: What is the standard deviation of the data set {10, 12, 23, 23, 16, 23, 21}?

Solution: Mean = 19.14, Variance = 20.25, Standard Deviation = √20.25 = 4.5.

Steps: 6

Step 1: Find the mean (average) of the data set. Add all the numbers together: 10 + 12 + 23 + 23 + 16 + 23 + 21 = 138. Then divide by the number of values (7): 138 / 7 = 19.14.
Step 2: Find the variance. Subtract the mean from each number to find the difference: (10 - 19.14), (12 - 19.14), (23 - 19.14), (23 - 19.14), (16 - 19.14), (23 - 19.14), (21 - 19.14). This gives you: -9.14, -7.14, 3.86, 3.86, -3.14, 3.86, 1.86.
Step 3: Square each of these differences: (-9.14)^2, (-7.14)^2, (3.86)^2, (3.86)^2, (-3.14)^2, (3.86)^2, (1.86)^2. This results in: 83.58, 50.98, 14.82, 14.82, 9.86, 14.82, 3.46.
Step 4: Add all the squared differences together: 83.58 + 50.98 + 14.82 + 14.82 + 9.86 + 14.82 + 3.46 = 192.34.
Step 5: Divide the total by the number of values (7) to find the variance: 192.34 / 7 = 27.48.
Step 6: Find the standard deviation by taking the square root of the variance: √27.48 = 5.24.