Question

"Standard deviation II. Consider these six data values: 10, 16, 18, 20, 22, 28. a) Find the mean and standard deviation by hand. b) Now look at these data values: 110, 116, 118, 120, 122, 128. Without doing any calculations, explain why the standard deviation will be the same even though the mean is obviously different."

Answer 1

The mean of the first set of values is 19, and the standard deviation is approximately 6.037. The standard deviation for the second set of values will be the same because it has the same spread, despite the mean being higher due to an addition of 100 to each data point.

Step-by-step explanation:

Calculating Mean and Standard Deviation by Hand

To find the mean of the first set of data values (10, 16, 18, 20, 22, 28), you add up all the numbers and divide by the count of numbers:

Mean = (10 + 16 + 18 + 20 + 22 + 28) / 6 = 114 / 6 = 19

Next, to find the standard deviation by hand, follow these steps:

1. Calculate the mean (average).
2. Subtract the mean from each of the numbers to find the deviations.
3. Square each deviation.
4. Find the sum of the squares.
5. Divide by the count of the numbers minus one for a sample (n-1).
6. Take the square root of the result.
7. For our set,

• Squared deviations: (10-19)^2 + (16-19)^2 + (18-19)^2 + (20-19)^2 + (22-19)^2 + (28-19)^2 = 81 + 9 + 1 + 1 + 9 + 81 = 182
• Divide by count-1: 182 / (6-1) = 182 / 5 = 36.4
• Square root: √36.4 ≈ 6.037

The standard deviation is approximately 6.037.

Comparing Standard Deviation of Different Means

For the second set of numbers (110, 116, 118, 120, 122, 128), although the mean is different, the standard deviation will be the same because the spread of the numbers is the same. The only difference in the datasets is an addition of 100 to each data point which shifts the mean but does not affect the variation or the distances between each data point.