Math Problem Statement

Compute the sample variance and sample standard deviation (to 2 decimals).

Solution

To compute the sample variance and sample standard deviation, we'll need a dataset. If you provide a list of numbers, I can calculate them for you.

Here’s the general approach:

Steps to Compute Sample Variance (s2s^2):

  1. Find the mean xˉ\bar{x} of the sample dataTo compute the sample variance and sample standard deviation, we'll need a dataset. If you provide a list of numbers, I can calculate them for you.

Here’s the general approach:

Steps to Compute Sample Variance (s2s^2):

  1. Find the mean xˉ\bar{x} of the sample xˉ=xin\bar{x} = \frac{\sum x_i}{n} where xix_i represents each value in the dataset and nn is the sample size.

  2. Subtract the mean from each data point and square the result (these are the squared deviations from the mean): (xixˉ)2(x_i - \bar{x})^2

  3. Sum up all the squared deviations: (xixˉ)2\sum (x_i - \bar{x})^2

  4. Divide by n1n - 1 (degrees of freedom): s2=(xixˉ)2n1s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1} This gives the sample variance.

Steps to Compute Sample Standard Deviation (ss):

  1. Take the square root of the sample variance: s=s2s = \sqrt{s^2}

Once you provide the data, I can carry out the calculations and give the results to 2 decimal places.

Do you have any questions or details you'd like to add? Here are some related questions you might find helpful:

  1. What’s the difference between sample variance and population variance?
  2. How do degrees of freedom affect the sample variance?
  3. Can variance be negative, and why or why not?
  4. What are some real-world applications of variance and standard deviation?
  5. What’s the relationship between standard deviation and normal distribution?

Tip: For small datasets, using the sample variance formula is important because it accounts for the potential variability that could be underestimated if using population variance formulas.

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Math Problem Analysis

Mathematical Concepts

Statistics
Sample Variance
Sample Standard Deviation

Formulas

Sample Mean: x̄ = (Σxi) / n
Sample Variance: s^2 = (Σ(xi - x̄)^2) / (n - 1)
Sample Standard Deviation: s = √s^2

Theorems

Degrees of Freedom in Sample Variance

Suitable Grade Level

Grades 10-12