Observation 4,8,11,17,20,24,32 find standard deviation

To find the standard deviation of the data set \( \{4, 8, 11, 17, 20, 24, 32\} \), we will follow these steps:

### Step 1: Calculate the Mean
The mean (average) of the data set is calculated as:

\[
\text{Mean} = \frac{\sum x_i}{n}
\]

Where:
- \( x_i \) are the observations,
- \( n \) is the number of observations.

\[
\text{Mean} = \frac{4 + 8 + 11 + 17 + 20 + 24 + 32}{7} = \frac{116}{7} \approx 16.57
\]

### Step 2: Calculate Each Deviation from the Mean
Subtract the mean from each observation:

\[
4 - 16.57 \approx -12.57,\quad 8 - 16.57 \approx -8.57,\quad 11 - 16.57 \approx -5.57
\]
\[
17 - 16.57 \approx 0.43,\quad 20 - 16.57 \approx 3.43,\quad 24 - 16.57 \approx 7.43,\quad 32 - 16.57 \approx 15.43
\]

### Step 3: Square Each Deviation
Square each of the deviations:

\[
(-12.57)^2 \approx 158.04,\quad (-8.57)^2 \approx 73.43,\quad (-5.57)^2 \approx 31.04
\]
\[
(0.43)^2 \approx 0.18,\quad (3.43)^2 \approx 11.76,\quad (7.43)^2 \approx 55.22,\quad (15.43)^2 \approx 238.06
\]

### Step 4: Calculate the Variance
The variance is the mean of the squared deviations:

\[
\text{Variance} = \frac{\sum (x_i - \text{Mean})^2}{n}
\]

\[
\text{Variance} = \frac{158.04 + 73.43 + 31.04 + 0.18 + 11.76 + 55.22 + 238.06}{7} \approx \frac{567.73}{7} \approx 81.10
\]

### Step 5: Calculate the Standard Deviation
The standard deviation is the square root of the variance:

\[
\text{Standard Deviation} = \sqrt{81.10} \approx 9.00
\]

So, the standard deviation of the data set \( \{4, 8, 11, 17, 20, 24, 32\} \) is approximately **9.00**.

Would you like further details or have any questions? 

Here are 5 related questions to explore further:

1. How does standard deviation help in understanding data distribution?
2. What is the difference between variance and standard deviation?
3. How do you calculate the standard deviation for a larger data set?
4. How would removing an outlier affect the standard deviation of a data set?
5. What is the significance of a low versus high standard deviation in a data set?

**Tip:** The standard deviation is a measure of the amount of variation or dispersion in a set of values. A lower standard deviation indicates that the values tend to be close to the mean, while a higher standard deviation indicates a wider range of values.

Learn how to calculate the standard deviation for the data set {4, 8, 11, 17, 20, 24, 32} with a detailed step-by-step solution. This problem covers key statistical concepts, including mean calculation, variance, and standard deviation formula applications. Suitable for students in Grades 11-12.

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