For the following set of data, find the population standard deviation, to the nearest thousandth.
44, 46, 44, 46, 45, 36, 47


To find the population standard deviation, we follow these steps:

### 1. **Find the Mean (μ)**:
First, calculate the mean of the data set.

\[
\mu = \frac{\text{Sum of all data values}}{\text{Number of data values}} = \frac{44 + 46 + 44 + 46 + 45 + 36 + 47}{7} = \frac{308}{7} = 44
\]

### 2. **Calculate Each Deviation from the Mean and Square It**:
Now, subtract the mean from each data value and square the result.

\[
(44 - 44)^2 = 0^2 = 0
\]
\[
(46 - 44)^2 = 2^2 = 4
\]
\[
(44 - 44)^2 = 0^2 = 0
\]
\[
(46 - 44)^2 = 2^2 = 4
\]
\[
(45 - 44)^2 = 1^2 = 1
\]
\[
(36 - 44)^2 = (-8)^2 = 64
\]
\[
(47 - 44)^2 = 3^2 = 9
\]

### 3. **Find the Average of These Squared Deviations**:
Since this is the population standard deviation, we divide by the total number of data points \( N = 7 \).

\[
\text{Variance} = \frac{0 + 4 + 0 + 4 + 1 + 64 + 9}{7} = \frac{82}{7} = 11.714
\]

### 4. **Take the Square Root of the Variance**:
The standard deviation is the square root of the variance.

\[
\sigma = \sqrt{11.714} \approx 3.423
\]

So, the population standard deviation is approximately **3.423** (to the nearest thousandth).

Do you want further details or explanations?

Here are 5 questions to expand your understanding:
1. How is the population standard deviation different from sample standard deviation?
2. Can you explain the difference between variance and standard deviation?
3. What does standard deviation indicate about a data set?
4. How would you calculate the sample standard deviation for the same data set?
5. What effect does a single outlier have on the standard deviation?

**Tip**: Always verify whether you are working with a sample or the entire population when calculating the standard deviation, as the formulas slightly differ.

For the following set of data, find the population standard deviation, to the nearest thousandth.
44, 46, 44, 46, 45, 36, 47

Learn how to calculate the population standard deviation for the data set 44, 46, 44, 46, 45, 36, 47. This problem involves key statistical concepts, including the mean, variance, and standard deviation formulas. Suitable for students in Grades 9-12.

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