Math Problem Statement

Compute the sample standard deviation of the data sample: 2, 5, 6, 7, -1, -1.

Solution

The problem asks to compute the sample standard deviation of the data sample: 2,5,6,7,1,12, 5, 6, 7, -1, -1, rounded to two decimal places.

Step-by-step solution:

Step 1: Calculate the sample mean.

The formula for the sample mean xˉ\bar{x} is: xˉ=xin\bar{x} = \frac{\sum x_i}{n} Where nn is the number of data points.

For the data 2,5,6,7,1,12, 5, 6, 7, -1, -1: xˉ=2+5+6+7+(1)+(1)6=186=3\bar{x} = \frac{2 + 5 + 6 + 7 + (-1) + (-1)}{6} = \frac{18}{6} = 3

Step 2: Calculate the squared deviations from the mean.

Next, subtract the mean from each data point and square the result: (23)2=1(2 - 3)^2 = 1 (53)2=4(5 - 3)^2 = 4 (63)2=9(6 - 3)^2 = 9 (73)2=16(7 - 3)^2 = 16 (13)2=16(-1 - 3)^2 = 16 (13)2=16(-1 - 3)^2 = 16

Step 3: Calculate the sample variance.

The formula for sample variance s2s^2 is: s2=(xixˉ)2n1s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1} Substituting the values: s2=1+4+9+16+16+165=625=12.4s^2 = \frac{1 + 4 + 9 + 16 + 16 + 16}{5} = \frac{62}{5} = 12.4

Step 4: Calculate the sample standard deviation.

The sample standard deviation is the square root of the sample variance: s=12.43.52s = \sqrt{12.4} \approx 3.52

Final Answer:

The sample standard deviation is approximately 3.52 (rounded to two decimal places).

Would you like further details or have any questions?


Here are five related questions to consider:

  1. How is the sample standard deviation different from population standard deviation?
  2. What is the significance of using n1n-1 in the sample variance formula?
  3. How would the standard deviation change if we added an additional data point to this sample?
  4. What is the relationship between variance and standard deviation?
  5. How does standard deviation help in understanding the spread of data?

Tip: Remember that standard deviation helps quantify how much data points in a sample deviate from the mean. The larger the standard deviation, the more spread out the data.

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Math Problem Analysis

Mathematical Concepts

Statistics
Standard Deviation
Sample Mean
Variance

Formulas

Sample mean: \( \bar{x} = \frac{\sum x_i}{n} \)
Sample variance: \( s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1} \)
Sample standard deviation: \( s = \sqrt{s^2} \)

Theorems

Sample standard deviation formula

Suitable Grade Level

Grades 9-12