Consider the following sample. 12 12 12 12 12 13 12. The standard deviation in this case is:

The standard deviation is a measure of how spread out the numbers in a data set are from their mean. To calculate the standard deviation for the sample \( 12, 12, 12, 12, 12, 13, 12 \), follow these steps:

### Step 1: Calculate the Mean
The mean (\(\mu\)) is the sum of the values divided by the number of values.

\[
\mu = \frac{12 + 12 + 12 + 12 + 12 + 13 + 12}{7} = \frac{85}{7} = 12.14
\]

### Step 2: Calculate the Variance
The variance is the average of the squared differences from the mean. First, we find the squared differences from the mean for each value:

\[
(12 - 12.14)^2 = (-0.14)^2 = 0.0196
\]
\[
(12 - 12.14)^2 = (-0.14)^2 = 0.0196
\]
\[
(12 - 12.14)^2 = (-0.14)^2 = 0.0196
\]
\[
(12 - 12.14)^2 = (-0.14)^2 = 0.0196
\]
\[
(12 - 12.14)^2 = (-0.14)^2 = 0.0196
\]
\[
(13 - 12.14)^2 = (0.86)^2 = 0.7396
\]
\[
(12 - 12.14)^2 = (-0.14)^2 = 0.0196
\]

Next, sum these squared differences:

\[
0.0196 + 0.0196 + 0.0196 + 0.0196 + 0.0196 + 0.7396 + 0.0196 = 0.8572
\]

### Step 3: Find the Sample Variance
For a sample, the variance is calculated by dividing the sum of squared differences by \(n-1\) (where \(n\) is the number of data points):

\[
\text{Variance} = \frac{0.8572}{7-1} = \frac{0.8572}{6} = 0.14287
\]

### Step 4: Calculate the Standard Deviation
The standard deviation is the square root of the variance:

\[
\text{Standard Deviation} = \sqrt{0.14287} \approx 0.378
\]

Thus, the standard deviation of the sample is approximately **0.378**.

Would you like more details or have any questions?

---

Here are some related questions to explore:
1. What happens to the standard deviation if the value 13 is replaced by 12?
2. How is the sample variance different from the population variance?
3. Can standard deviation ever be negative?
4. What is the relationship between the mean and standard deviation in a normal distribution?
5. How would adding a constant to each value in the sample affect the standard deviation?

**Tip:** Always double-check whether you're dealing with a sample or a population when calculating variance and standard deviation.

Learn how to calculate the standard deviation for the sample 12, 12, 12, 12, 12, 13, 12. This detailed solution covers the key statistical concepts of mean, variance, and standard deviation, providing step-by-step instructions. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation