Math Problem Statement

Find the standard deviation for 2 , 17 , 12 , 5 , 7 , 10 , 5 , 8 , 6

Solution

To calculate the standard deviation for the data set 2,17,12,5,7,10,5,8,62, 17, 12, 5, 7, 10, 5, 8, 6, follow these steps:

1. Find the mean:

The mean μ\mu is the sum of all numbers divided by the number of data points.

μ=2+17+12+5+7+10+5+8+69\mu = \frac{2 + 17 + 12 + 5 + 7 + 10 + 5 + 8 + 6}{9}

μ=729=8\mu = \frac{72}{9} = 8

2. Find the squared differences from the mean:

For each number in the data set, subtract the mean and square the result.

(28)2=(6)2=36(2 - 8)^2 = (-6)^2 = 36 (178)2=(9)2=81(17 - 8)^2 = (9)^2 = 81 (128)2=(4)2=16(12 - 8)^2 = (4)^2 = 16 (58)2=(3)2=9(5 - 8)^2 = (-3)^2 = 9 (78)2=(1)2=1(7 - 8)^2 = (-1)^2 = 1 (108)2=(2)2=4(10 - 8)^2 = (2)^2 = 4 (58)2=(3)2=9(5 - 8)^2 = (-3)^2 = 9 (88)2=(0)2=0(8 - 8)^2 = (0)^2 = 0 (68)2=(2)2=4(6 - 8)^2 = (-2)^2 = 4

3. Find the average of the squared differences:

Variance=36+81+16+9+1+4+9+0+49\text{Variance} = \frac{36 + 81 + 16 + 9 + 1 + 4 + 9 + 0 + 4}{9} Variance=160917.78\text{Variance} = \frac{160}{9} \approx 17.78

4. Take the square root of the variance to find the standard deviation:

σ=17.784.21\sigma = \sqrt{17.78} \approx 4.21

Conclusion:

The standard deviation is approximately 4.214.21.

Do you need further details or have any questions?

Here are 5 related questions:

  1. What is the difference between standard deviation and variance?
  2. How does changing a single value affect the standard deviation?
  3. What is the significance of a low standard deviation versus a high one?
  4. How would you calculate the standard deviation of a larger dataset?
  5. Can you calculate standard deviation using other measures of central tendency?

Tip: Standard deviation gives a measure of how spread out the numbers in your data set are from the mean.

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Math Problem Analysis

Mathematical Concepts

Statistics
Standard Deviation
Mean
Variance

Formulas

Mean: μ = (Σx) / N
Variance: σ² = Σ((x - μ)²) / N
Standard Deviation: σ = √σ²

Theorems

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Suitable Grade Level

Grades 9-12