Use This Data For The Following Question

Use This Data For The Following Question1020304050 If You Call

Use this data for the following question {10,20,30,40,50}. If you call this data set x, the Stata command (generate xnew=5*x) would generate a new variable which is 5 times each element of the original data set. a) Find the standard deviation and the mean. b) Add 5 to each value, and then find the standard deviation and the mean. c) Subtract 5 from each value and find the standard deviation and mean. d) Multiply each value by 5 and find the standard deviation and mean. e) Divide each value by 5 and find the standard deviation and the mean. f) Generalize the results of parts b through e.

Paper For Above instruction

The dataset provided consists of five values: 10, 20, 30, 40, and 50. These values serve as a fundamental example for understanding how various transformations affect statistical measures such as the mean and standard deviation. This paper explores these effects thoroughly, demonstrating the mathematical principles behind each transformation and their impact on the distribution of data.

Initially, the original dataset's mean and standard deviation are computed. The mean is obtained by summing all values and dividing by the number of data points, while the standard deviation measures the dispersion of data around the mean. This baseline provides a reference point for subsequent transformations.

Calculations for the original dataset:

- Mean (μ) = (10 + 20 + 30 + 40 + 50) / 5 = 150 / 5 = 30

- Standard deviation (σ): To compute this, find the squared differences from the mean, sum those, divide by the degrees of freedom (since this is a sample), and take the square root. The squared differences are: (10-30)^2=400, (20-30)^2=100, (30-30)^2=0, (40-30)^2=100, (50-30)^2=400. Sum = 1000. Dividing by n-1=4 gives 250, thus, σ = √250 ≈ 15.81.

Now, consider each transformation:

a) The original data's mean and standard deviation serve as the baseline with values 30 and approximately 15.81, respectively.

b) Adding 5 to each value results in the dataset {15, 25, 35, 45, 55}. The mean increases by 5, so the new mean = 30 + 5 = 35. The standard deviation, a measure of spread, remains unchanged because adding a constant shifts the entire distribution but does not affect the spread. Hence, σ ≈ 15.81. This stems from the fact that standard deviation is invariant under additive transformations.

c) Subtracting 5 from each value produces {5, 15, 25, 35, 45}. The mean decreases by 5, so the new mean = 30 - 5 = 25. The standard deviation remains unchanged at approximately 15.81, consistent with the properties of additive transformations.

d) Multiplying each value by 5 yields {50, 100, 150, 200, 250}. The new mean is the original mean multiplied by 5: 30 × 5 = 150. Concerning the standard deviation, multiplying all data points by a scalar multiplies the standard deviation by the same scalar. Therefore, σ_new = 15.81 × 5 ≈ 79.05.

e) Dividing each value by 5 results in {2, 4, 6, 8, 10}. The new mean becomes the original mean divided by 5: 30 / 5 = 6. The standard deviation is likewise divided by 5, so σ_new ≈ 15.81 / 5 ≈ 3.16.

f) Generalizing the effects of these transformations, we observe the following principles: Adding a constant c to each data point shifts the mean by c without changing the standard deviation. Multiplying each data point by a constant k scales both the mean and the standard deviation by k. Dividing by a positive constant c is equivalent to multiplying by 1/c, thus dividing both the mean and standard deviation by c. These properties exemplify the linearity of mean and the scaling behavior of standard deviation under affine transformations, which are foundational concepts in statistical analysis.

In summary, transformations involving addition/subtraction influence only the location of the distribution (mean), leaving the spread unchanged, whereas scaling transformations modify both the location and spread proportionally. Understanding these properties is essential for interpreting data transformations in statistical practice and highlights the importance of ratios and proportionality in variability measures like standard deviation. These insights are widely applicable across statistical analyses, from basic descriptive statistics to more advanced inferential techniques.

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