Find the Standard Deviation for below numbers:  
4,9,7,12,13,10,12

To find the standard deviation for the given numbers: \(4, 9, 7, 12, 13, 10, 12\), follow these steps:

### Step 1: Find the mean (average)
The mean is calculated by adding all the numbers and then dividing by the count of numbers.

\[
\text{Mean} = \frac{4 + 9 + 7 + 12 + 13 + 10 + 12}{7} = \frac{67}{7} = 9.57
\]

### Step 2: Calculate each number's deviation from the mean
Subtract the mean from each number:

\[
4 - 9.57 = -5.57, \quad 9 - 9.57 = -0.57, \quad 7 - 9.57 = -2.57
\]
\[
12 - 9.57 = 2.43, \quad 13 - 9.57 = 3.43, \quad 10 - 9.57 = 0.43, \quad 12 - 9.57 = 2.43
\]

### Step 3: Square each deviation
Square the results from step 2:

\[
(-5.57)^2 = 31.02, \quad (-0.57)^2 = 0.32, \quad (-2.57)^2 = 6.60
\]
\[
(2.43)^2 = 5.90, \quad (3.43)^2 = 11.76, \quad (0.43)^2 = 0.18, \quad (2.43)^2 = 5.90
\]

### Step 4: Find the variance
The variance is the average of these squared deviations. Since we are calculating the sample standard deviation, divide the sum of squared deviations by \(n-1\), where \(n\) is the total number of data points (7 in this case).

\[
\text{Variance} = \frac{31.02 + 0.32 + 6.60 + 5.90 + 11.76 + 0.18 + 5.90}{6} = \frac{61.68}{6} = 10.28
\]

### Step 5: Find the standard deviation
The standard deviation is the square root of the variance:

\[
\text{Standard Deviation} = \sqrt{10.28} \approx 3.21
\]

So, the standard deviation is approximately **3.21**.

Would you like further details on any step or have any questions?

Here are 5 related questions:
1. How is population standard deviation different from sample standard deviation?
2. What is the significance of standard deviation in statistics?
3. Can you calculate the variance from the numbers provided?
4. How do outliers affect the standard deviation?
5. Can you provide an example where a lower standard deviation is more favorable?

**Tip:** Standard deviation gives a sense of how spread out the values in a data set are. A smaller value means they are closer to the mean, while a larger value means more spread.

Find the Standard Deviation for below numbers: 4, 9, 7, 12, 13, 10, 12

Learn how to calculate the standard deviation for the set of numbers: 4, 9, 7, 12, 13, 10, 12. This step-by-step guide explains the process, including calculating the mean, variance, and standard deviation, with a result of approximately 3.21. Suitable for students in Grades 9-12.

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