Calculate the sample standard deviation of the data shown. Round to two decimal places.

x

25

24

20

29

15

18

To calculate the sample standard deviation, we follow these steps:

### Step 1: Calculate the mean (average)
\[
\text{Mean} = \frac{25 + 24 + 20 + 29 + 15 + 18}{6} = \frac{131}{6} = 21.83
\]

### Step 2: Calculate each data point's deviation from the mean
Now, subtract the mean from each data point:
- \( 25 - 21.83 = 3.17 \)
- \( 24 - 21.83 = 2.17 \)
- \( 20 - 21.83 = -1.83 \)
- \( 29 - 21.83 = 7.17 \)
- \( 15 - 21.83 = -6.83 \)
- \( 18 - 21.83 = -3.83 \)

### Step 3: Square each deviation
- \( 3.17^2 = 10.05 \)
- \( 2.17^2 = 4.71 \)
- \( (-1.83)^2 = 3.35 \)
- \( 7.17^2 = 51.41 \)
- \( (-6.83)^2 = 46.65 \)
- \( (-3.83)^2 = 14.67 \)

### Step 4: Calculate the average of these squared deviations (variance)
First, sum up the squared deviations:
\[
10.05 + 4.71 + 3.35 + 51.41 + 46.65 + 14.67 = 130.84
\]
Then, divide by \( n - 1 \) (since it's a sample, not the whole population, we divide by \( 6 - 1 = 5 \)):
\[
\text{Variance} = \frac{130.84}{5} = 26.17
\]

### Step 5: Take the square root of the variance to get the sample standard deviation
\[
\text{Sample Standard Deviation} = \sqrt{26.17} = 5.11
\]

Thus, the sample standard deviation is **5.11** (rounded to two decimal places).

Do you have any questions or want further details?

Here are some related questions:
1. What is the difference between sample standard deviation and population standard deviation?
2. How would the standard deviation change if one more data point was added?
3. Why is the square root used in the calculation of standard deviation?
4. How does standard deviation help in understanding data distribution?
5. Can standard deviation ever be negative?

**Tip:** Remember to always check if you're dealing with a sample or the entire population, as the formulas slightly differ!

Calculate the sample standard deviation of the data shown. Round to two decimal places.

Learn how to calculate the sample standard deviation for the dataset {25, 24, 20, 29, 15, 18}. This step-by-step guide walks through calculating the mean, variance, and standard deviation with clear explanations. Ideal for students in Grades 10-12 studying statistics.

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