If the data set {1, 2, 3, 4, 5} is transformed to {2, 3, 4, 5, 6}, what happens to the standard deviation?

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Increases
Decreases
Remains the same
Cannot be determined

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Q: If the data set {1, 2, 3, 4, 5} is transformed to {2, 3, 4, 5, 6}, what happens to the standard deviation?

Solution: The standard deviation remains the same because the transformation is a shift.

Steps: 15

Step 1: Understand what standard deviation is. It measures how spread out the numbers are in a data set.
Step 2: Look at the original data set: {1, 2, 3, 4, 5}.
Step 3: Calculate the mean (average) of the original data set: (1 + 2 + 3 + 4 + 5) / 5 = 3.
Step 4: Find the deviations from the mean for each number in the original data set: (1-3), (2-3), (3-3), (4-3), (5-3) which gives us: -2, -1, 0, 1, 2.
Step 5: Square each of these deviations: (-2)^2, (-1)^2, (0)^2, (1)^2, (2)^2 which gives us: 4, 1, 0, 1, 4.
Step 6: Calculate the average of these squared deviations to find the variance: (4 + 1 + 0 + 1 + 4) / 5 = 2.
Step 7: Take the square root of the variance to find the standard deviation: sqrt(2).
Step 8: Now look at the transformed data set: {2, 3, 4, 5, 6}.
Step 9: Calculate the mean of the transformed data set: (2 + 3 + 4 + 5 + 6) / 5 = 4.
Step 10: Find the deviations from the mean for the transformed data set: (2-4), (3-4), (4-4), (5-4), (6-4) which gives us: -2, -1, 0, 1, 2.
Step 11: Notice that the deviations are the same as in the original data set.
Step 12: Square these deviations: (-2)^2, (-1)^2, (0)^2, (1)^2, (2)^2 which gives us: 4, 1, 0, 1, 4.
Step 13: Calculate the average of these squared deviations: (4 + 1 + 0 + 1 + 4) / 5 = 2, which is the same as before.
Step 14: Take the square root of the variance to find the standard deviation: sqrt(2), which is also the same as before.
Step 15: Conclude that the standard deviation remains the same because the transformation was just a shift (adding 1 to each number).

If the data set {1, 2, 3, 4, 5} is transformed to {2, 3, 4, 5, 6}, what happens to the standard deviation?

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