3,5,8,9,7 , 10 . Find square or standard deviation 

To find the standard deviation of the given set of observations \(\{3, 5, 8, 9, 7, 10\}\), we first need to calculate the mean, variance, and then take the square root of the variance to find the standard deviation.

### Step 1: Calculate the Mean
The mean (\(\bar{x}\)) is calculated by summing all the observations and dividing by the number of observations (\(n\)).

\[ \bar{x} = \frac{3 + 5 + 8 + 9 + 7 + 10}{6} = \frac{42}{6} = 7 \]

### Step 2: Calculate the Variance
The variance (VAR) is calculated using the formula:

\[ \text{VAR} = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n} \]

First, we find the deviation of each observation from the mean and then square each deviation:

\[
\begin{align*}
(3 - 7)^2 &= (-4)^2 = 16 \\
(5 - 7)^2 &= (-2)^2 = 4 \\
(8 - 7)^2 &= (1)^2 = 1 \\
(9 - 7)^2 &= (2)^2 = 4 \\
(7 - 7)^2 &= (0)^2 = 0 \\
(10 - 7)^2 &= (3)^2 = 9 \\
\end{align*}
\]

Sum of squared deviations:

\[ 16 + 4 + 1 + 4 + 0 + 9 = 34 \]

Now, calculate the variance:

\[ \text{VAR} = \frac{34}{6} \approx 5.67 \]

### Step 3: Calculate the Sample Variance (Corrected Estimate)
The sample variance (VARA) is calculated by dividing the sum of squared deviations by \(n - 1\):

\[ \text{VARA} = \frac{34}{6 - 1} = \frac{34}{5} = 6.8 \]

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

\[ \text{SD} = \sqrt{\text{VAR}} = \sqrt{5.67} \approx 2.38 \]

The corrected standard deviation (Sample Standard Deviation) is:

\[ \text{SD}_{\text{sample}} = \sqrt{\text{VARA}} = \sqrt{6.8} \approx 2.61 \]

So, the standard deviation of the given set of observations is approximately \(2.38\) and the corrected standard deviation is approximately \(2.61\).

Would you like more details or have any questions?

Here are some more questions you can ask:
1. What is the difference between standard deviation and variance?
2. How do you interpret the value of standard deviation in a dataset?
3. Can you show how to calculate the mean absolute deviation?
4. What is the significance of using \(n-1\) in sample variance calculation?
5. How do you handle standard deviation for grouped data?

**Tip:** Always ensure you distinguish between population and sample statistics, especially in variance and standard deviation calculations, to apply the correct formula for accurate results.

Learn how to calculate the standard deviation and variance for a set of observations (3, 5, 8, 9, 7, 10) with a detailed step-by-step guide. This problem covers key statistical concepts, including mean calculation, variance formula, and standard deviation interpretation. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation