what is the sample variance of these numbers? 16,35,25,18,15

To calculate the sample variance of the numbers 16, 35, 25, 18, and 15, follow these steps:

1. **Calculate the mean** of the sample.
2. **Subtract the mean** from each number to find the deviation of each number from the mean.
3. **Square each deviation**.
4. **Sum the squared deviations**.
5. **Divide the sum** by the number of data points minus one (since it's a sample variance, not population variance).

Let's go through each step:

### Step 1: Calculate the mean
\[ \text{Mean} (\bar{x}) = \frac{\sum x_i}{n} = \frac{16 + 35 + 25 + 18 + 15}{5} = \frac{109}{5} = 21.8 \]

### Step 2: Calculate each deviation from the mean
\[
\begin{align*}
16 - 21.8 & = -5.8 \\
35 - 21.8 & = 13.2 \\
25 - 21.8 & = 3.2 \\
18 - 21.8 & = -3.8 \\
15 - 21.8 & = -6.8 \\
\end{align*}
\]

### Step 3: Square each deviation
\[
\begin{align*}
(-5.8)^2 & = 33.64 \\
(13.2)^2 & = 174.24 \\
(3.2)^2 & = 10.24 \\
(-3.8)^2 & = 14.44 \\
(-6.8)^2 & = 46.24 \\
\end{align*}
\]

### Step 4: Sum the squared deviations
\[ 33.64 + 174.24 + 10.24 + 14.44 + 46.24 = 278.8 \]

### Step 5: Divide by \( n - 1 \)
\[ \text{Sample Variance} (s^2) = \frac{\sum (x_i - \bar{x})^2}{n - 1} = \frac{278.8}{5 - 1} = \frac{278.8}{4} = 69.7 \]

So, the sample variance of the numbers 16, 35, 25, 18, and 15 is **69.7**.

Would you like further details on this calculation or have any questions?

Here are 5 related questions that might expand your understanding:
1. What is the difference between sample variance and population variance?
2. How would you calculate the standard deviation from the variance?
3. Can you explain the significance of using \( n - 1 \) instead of \( n \) for sample variance?
4. What are some common uses of variance in data analysis?
5. How does variance help in understanding the spread of data?

**Tip:** Always remember to check whether you need sample variance or population variance before calculating, as the formulas differ slightly by the denominator.

what is the sample variance of these numbers? 16, 35, 25, 18, 15

Learn how to calculate the sample variance of the numbers 16, 35, 25, 18, and 15 with a detailed step-by-step solution. This problem involves calculating the mean, deviations, and squared deviations, and is suitable for students in Grades 10-12 studying statistics.

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