Math Problem Statement

Find the standard deviation for the group of data items. 11​, 11​, 14​, 14​, 14​, 14

Solution

To calculate the standard deviation of the given data set:

Data: 11, 11, 14, 14, 14, 14

Steps:

  1. Find the Mean (μ): μ=xin=11+11+14+14+14+146=786=13\mu = \frac{\sum x_i}{n} = \frac{11 + 11 + 14 + 14 + 14 + 14}{6} = \frac{78}{6} = 13

  2. Find the squared deviations from the mean:

    • For 11: (1113)2=(2)2=4(11 - 13)^2 = (-2)^2 = 4
    • For 11: (1113)2=(2)2=4(11 - 13)^2 = (-2)^2 = 4
    • For 14: (1413)2=(1)2=1(14 - 13)^2 = (1)^2 = 1
    • For 14: (1413)2=(1)2=1(14 - 13)^2 = (1)^2 = 1
    • For 14: (1413)2=(1)2=1(14 - 13)^2 = (1)^2 = 1
    • For 14: (1413)2=(1)2=1(14 - 13)^2 = (1)^2 = 1
  3. Sum of the squared deviations: (xiμ)2=4+4+1+1+1+1=12\sum (x_i - \mu)^2 = 4 + 4 + 1 + 1 + 1 + 1 = 12

  4. Variance (σ²): Since this is a sample, use the formula for sample variance: σ2=(xiμ)2n1=125=2.4\sigma^2 = \frac{\sum (x_i - \mu)^2}{n-1} = \frac{12}{5} = 2.4

  5. Standard Deviation (σ): σ=2.41.55\sigma = \sqrt{2.4} \approx 1.55

So, the standard deviation is approximately 1.55.

Would you like more details or have any other questions?

Related questions:

  1. How do you calculate the variance from raw data?
  2. What is the difference between sample and population standard deviation?
  3. Can you explain how to find the mean for a larger data set?
  4. What is the importance of standard deviation in data analysis?
  5. How do outliers affect the standard deviation?

Tip: The standard deviation measures how spread out the data is around the mean.

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Math Problem Analysis

Mathematical Concepts

Statistics
Mean
Variance
Standard Deviation

Formulas

Mean (μ) = Σx_i / n
Variance (σ²) = Σ(x_i - μ)² / (n - 1) (for sample data)
Standard Deviation (σ) = √(Variance)

Theorems

Sample Variance and Standard Deviation

Suitable Grade Level

Grades 10-12