If the data set {10, 20, 30, 40, 50} is transformed to {x + 5}, what happens to

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Question: If the data set {10, 20, 30, 40, 50} is transformed to {x + 5}, what happens to the standard deviation?

Options:

Correct Answer: Remains the same

Solution:

Standard deviation remains the same as adding a constant does not affect dispersion.

If the data set {10, 20, 30, 40, 50} is transformed to {x + 5}, what happens to the standard deviation?

If the data set {10, 20, 30, 40, 50} is transformed to {x + 5}, what happens to the standard deviation?

Step 1: Understand what standard deviation measures. It measures how spread out the numbers are in a data set.
Step 2: Look at the original data set: {10, 20, 30, 40, 50}.
Step 3: Calculate the mean (average) of the original data set. (10 + 20 + 30 + 40 + 50) / 5 = 30.
Step 4: Find the differences between each number and the mean: (10-30), (20-30), (30-30), (40-30), (50-30). This gives us: -20, -10, 0, 10, 20.
Step 5: Square each of these differences: 400, 100, 0, 100, 400.
Step 6: Find the average of these squared differences. (400 + 100 + 0 + 100 + 400) / 5 = 200.
Step 7: Take the square root of this average to find the standard deviation. √200 ≈ 14.14.
Step 8: Now, transform the data set by adding 5 to each number: {15, 25, 35, 45, 55}.
Step 9: Calculate the mean of the new data set: (15 + 25 + 35 + 45 + 55) / 5 = 35.
Step 10: Find the differences between each number in the new set and the new mean: (15-35), (25-35), (35-35), (45-35), (55-35). This gives us: -20, -10, 0, 10, 20.
Step 11: Notice that the differences are the same as before. The squared differences will also be the same: 400, 100, 0, 100, 400.
Step 12: The average of these squared differences is still 200, and the standard deviation remains √200 ≈ 14.14.
Step 13: Conclusion: Adding a constant (like 5) to each number does not change how spread out the numbers are, so the standard deviation stays the same.

Standard Deviation – A measure of the amount of variation or dispersion in a set of values.
Effect of Adding a Constant – Adding a constant to each data point in a dataset shifts the entire dataset without affecting its spread or dispersion.