Math Problem Statement

Find the SD of the given data set - 7, 2, 8, 11, 6, 13, 16

Solution

To calculate the standard deviation (SD) of a given data set, follow these steps:

Data Set:

7, 2, 8, 11, 6, 13, 16

Step 1: Find the Mean (Average)

The formula for the mean μ\mu is:

μ=xin\mu = \frac{\sum x_i}{n}

where xix_i represents each value in the data set, and nn is the number of data points.

μ=7+2+8+11+6+13+167=637=9\mu = \frac{7 + 2 + 8 + 11 + 6 + 13 + 16}{7} = \frac{63}{7} = 9

Step 2: Find the Squared Differences from the Mean

For each data point, subtract the mean and then square the result.

(79)2=(2)2=4(7 - 9)^2 = (-2)^2 = 4 (29)2=(7)2=49(2 - 9)^2 = (-7)^2 = 49 (89)2=(1)2=1(8 - 9)^2 = (-1)^2 = 1 (119)2=(2)2=4(11 - 9)^2 = (2)^2 = 4 (69)2=(3)2=9(6 - 9)^2 = (-3)^2 = 9 (139)2=(4)2=16(13 - 9)^2 = (4)^2 = 16 (169)2=(7)2=49(16 - 9)^2 = (7)^2 = 49

Step 3: Find the Variance

The variance σ2\sigma^2 is the average of these squared differences. The formula for variance is:

σ2=(xiμ)2n\sigma^2 = \frac{\sum (x_i - \mu)^2}{n}

σ2=4+49+1+4+9+16+497=132718.86\sigma^2 = \frac{4 + 49 + 1 + 4 + 9 + 16 + 49}{7} = \frac{132}{7} \approx 18.86

Step 4: Find the Standard Deviation

The standard deviation σ\sigma is the square root of the variance:

σ=18.864.34\sigma = \sqrt{18.86} \approx 4.34

Conclusion:

The standard deviation of the data set is approximately 4.34.


Do you want more details or have any questions about the process?

Here are 5 related questions:

  1. How does the standard deviation change if the dataset values increase by 5 each?
  2. What is the variance of the data set?
  3. How do outliers in a data set affect the standard deviation?
  4. Can you explain the difference between sample standard deviation and population standard deviation?
  5. How would removing the largest number from the data set affect the standard deviation?

Tip: Standard deviation provides insight into how much data points deviate from the mean, with a larger SD indicating more spread in the data.

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Math Problem Analysis

Mathematical Concepts

Statistics
Standard Deviation
Variance

Formulas

Mean (μ) = (Σx_i) / n
Variance (σ^2) = Σ(x_i - μ)^2 / n
Standard Deviation (σ) = √σ^2

Theorems

None specifically required for basic SD calculation

Suitable Grade Level

Grades 9-12