Math Problem Statement

Compute the (sample) variance and standard deviation of the data sample. (Round your answers to two decimal places.) 4.4, −4.8, 2.9, −0.4, −0.4 variance Incorrect: Your answer is incorrect. standard deviation

Solution

Let's calculate the sample variance and sample standard deviation for the data sample provided: 4.4,4.8,2.9,0.4,0.44.4, -4.8, 2.9, -0.4, -0.4.

Step-by-Step Solution

  1. Calculate the Mean: Mean=4.4+(4.8)+2.9+(0.4)+(0.4)5=2.55=0.5\text{Mean} = \frac{4.4 + (-4.8) + 2.9 + (-0.4) + (-0.4)}{5} = \frac{2.5}{5} = 0.5

  2. Calculate the Differences from the Mean and Square Them: (4.40.5)2=15.21(4.4 - 0.5)^2 = 15.21 (4.80.5)2=28.09(-4.8 - 0.5)^2 = 28.09 (2.90.5)2=5.76(2.9 - 0.5)^2 = 5.76 (0.40.5)2=0.81(-0.4 - 0.5)^2 = 0.81 (0.40.5)2=0.81(-0.4 - 0.5)^2 = 0.81

  3. Calculate the Sum of Squared Differences: Sum of Squares=15.21+28.09+5.76+0.81+0.81=50.68\text{Sum of Squares} = 15.21 + 28.09 + 5.76 + 0.81 + 0.81 = 50.68

  4. Calculate the Sample Variance: Since this is a sample, divide by n1n - 1 (where n=5n = 5): Variance=50.6851=50.684=12.67\text{Variance} = \frac{50.68}{5 - 1} = \frac{50.68}{4} = 12.67

  5. Calculate the Sample Standard Deviation: Standard Deviation=12.673.56\text{Standard Deviation} = \sqrt{12.67} \approx 3.56

Final Answers

  • Variance: 12.6712.67
  • Standard Deviation: 3.563.56

Would you like further details on these calculations or explanations?

Here are five related questions to deepen understanding:

  1. What is the difference between sample variance and population variance?
  2. How would the calculations change if we were finding the population variance instead?
  3. How does rounding impact the accuracy of variance and standard deviation?
  4. What are practical interpretations of variance and standard deviation in a data sample?
  5. How does variance behave when all data points are shifted by the same constant?

Tip: Always check whether you're working with a sample or population, as this affects whether you divide by n1n - 1 or nn.

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Math Problem Analysis

Mathematical Concepts

Statistics
Variance
Standard Deviation

Formulas

Sample variance formula: σ² = Σ(xᵢ - x̄)² / (n - 1)
Standard deviation formula: σ = √(variance)

Theorems

Formula for sample variance
Formula for sample standard deviation

Suitable Grade Level

Grades 9-12