12
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23
13
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17
15
18      find population variance

To calculate the **population variance**, follow these steps:

1. **Find the mean** (average) of the data set.
2. **Subtract the mean** from each number** (to find the deviation from the mean).
3. **Square each deviation**.
4. **Find the average** of these squared deviations.

### Given Data Set:
12, 17, 23, 13, 20, 16, 20, 17, 15, 18

---

#### Step 1: Find the Mean
\[
\text{Mean} = \frac{12 + 17 + 23 + 13 + 20 + 16 + 20 + 17 + 15 + 18}{10} = \frac{171}{10} = 17.1
\]

#### Step 2: Find the Deviation from the Mean
\[
\text{Deviations} = 12 - 17.1, 17 - 17.1, 23 - 17.1, \ldots
\]
\[
\text{Deviations} = -5.1, -0.1, 5.9, -4.1, 2.9, -1.1, 2.9, -0.1, -2.1, 0.9
\]

#### Step 3: Square Each Deviation
\[
\text{Squared Deviations} = (-5.1)^2, (-0.1)^2, (5.9)^2, \ldots
\]
\[
\text{Squared Deviations} = 26.01, 0.01, 34.81, 16.81, 8.41, 1.21, 8.41, 0.01, 4.41, 0.81
\]

#### Step 4: Find the Average of Squared Deviations (Population Variance)
\[
\text{Population Variance} = \frac{26.01 + 0.01 + 34.81 + 16.81 + 8.41 + 1.21 + 8.41 + 0.01 + 4.41 + 0.81}{10} = \frac{101.9}{10} = 10.19
\]

### Population Variance: \( \boxed{10.19} \)

Would you like more details on any step? Here are 5 related questions to explore further:

1. What is the difference between population and sample variance?
2. How do you calculate the standard deviation from the variance?
3. Can variance be negative?
4. How do outliers affect the variance of a data set?
5. What is the relationship between variance and standard deviation?

**Tip:** Variance gives insight into the spread of data, but standard deviation is often more useful as it's in the same units as the data.

12 17 23 13 20 16 20 17 15 18 find population variance

Learn how to calculate the population variance for the data set {12, 17, 23, 13, 20, 16, 20, 17, 15, 18} step by step. This solution involves finding the mean, calculating deviations, and determining the population variance using key statistical formulas. Ideal for students in Grades 10-12.

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