using this formula σ=∑(xi−μ)2N*σ*=*N*∑(*xi*​−*μ*)2​​  Find the standard deviation of the data set : 390, 394, 403, 409, 428, 437, 443, 461, 461, 461 SHOW ALL STOPS DONT ROUND ANY NUMBERS

We are asked to find the standard deviation of the data set:  
**390, 394, 403, 409, 428, 437, 443, 461, 461, 461**  
without rounding any numbers and showing all steps.

### Step 1: Find the Mean (μ)
The formula for the mean is:

\[
\mu = \frac{\sum x_i}{N}
\]

Where:
- \(x_i\) is each individual data point
- \(N\) is the total number of data points

**Data points:**
\[
\{390, 394, 403, 409, 428, 437, 443, 461, 461, 461\}
\]

**Number of data points:**
\[
N = 10
\]

**Sum of data points:**
\[
\sum x_i = 390 + 394 + 403 + 409 + 428 + 437 + 443 + 461 + 461 + 461 = 4247
\]

Now calculate the mean:
\[
\mu = \frac{4247}{10} = 424.7
\]

### Step 2: Calculate the squared deviations
For each data point, we will subtract the mean and square the result:

\[
(x_i - \mu)^2
\]

**For each data point:**

1. \( (390 - 424.7)^2 = (-34.7)^2 = 1204.09 \)
2. \( (394 - 424.7)^2 = (-30.7)^2 = 941.49 \)
3. \( (403 - 424.7)^2 = (-21.7)^2 = 470.89 \)
4. \( (409 - 424.7)^2 = (-15.7)^2 = 246.49 \)
5. \( (428 - 424.7)^2 = (3.3)^2 = 10.89 \)
6. \( (437 - 424.7)^2 = (12.3)^2 = 151.29 \)
7. \( (443 - 424.7)^2 = (18.3)^2 = 334.89 \)
8. \( (461 - 424.7)^2 = (36.3)^2 = 1317.69 \)
9. \( (461 - 424.7)^2 = (36.3)^2 = 1317.69 \)
10. \( (461 - 424.7)^2 = (36.3)^2 = 1317.69 \)

### Step 3: Calculate the sum of the squared deviations
\[
\sum (x_i - \mu)^2 = 1204.09 + 941.49 + 470.89 + 246.49 + 10.89 + 151.29 + 334.89 + 1317.69 + 1317.69 + 1317.69 = 7313.10
\]

### Step 4: Calculate the variance
The formula for the variance (σ²) is:

\[
\sigma^2 = \frac{\sum (x_i - \mu)^2}{N}
\]

Substituting the values we found:

\[
\sigma^2 = \frac{7313.10}{10} = 731.31
\]

### Step 5: Calculate the standard deviation
The standard deviation (σ) is the square root of the variance:

\[
\sigma = \sqrt{731.31} = 27.03629058207
\]

### Final Answer:
The standard deviation of the data set is approximately **27.03629058207**.

Would you like further details or clarifications?

#### Related questions:
1. How does changing one number in the data set affect the standard deviation?
2. What is the significance of using the population standard deviation formula?
3. How do we compute the sample standard deviation using the same data?
4. How do outliers impact the standard deviation of a data set?
5. What are the advantages of calculating the standard deviation over the variance?

**Tip:** Always ensure you know whether you're calculating the population or sample standard deviation, as they differ slightly in formula!

Using this formula σ=∑(xi−μ)²/N, find the standard deviation of the data set: 390, 394, 403, 409, 428, 437, 443, 461, 461, 461. SHOW ALL STEPS, DON'T ROUND ANY NUMBERS.

Calculate the standard deviation of the data set 390, 394, 403, 409, 428, 437, 443, 461, 461, 461 step by step, without rounding any numbers. Learn key statistical concepts like mean, variance, and standard deviation in this detailed guide. Suitable for students in Grades 9-12.

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