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Let's solve this step by step.

### Step 1: Find the missing probability.
The sum of all probabilities in a probability distribution must equal 1. We are missing the probability for \( X = 26 \). So, we add up the probabilities we know and subtract from 1:

\[
P(23) + P(24) + P(25) + P(26) + P(27) = 1
\]

From the table:
\[
P(23) = 0.09, \quad P(24) = 0.13, \quad P(25) = 0.16, \quad P(27) = 0.11
\]

So:
\[
0.09 + 0.13 + 0.16 + P(26) + 0.11 = 1
\]

Add the known probabilities:
\[
0.09 + 0.13 + 0.16 + 0.11 = 0.49
\]

Now solve for \( P(26) \):
\[
P(26) = 1 - 0.49 = 0.51
\]

### Step 2: Calculate the mean (expected value).
The formula for the mean (expected value) of a probability distribution is:

\[
\mu = \sum X \cdot P(X)
\]

Using the probabilities and values of \( X \):

\[
\mu = (23 \cdot 0.09) + (24 \cdot 0.13) + (25 \cdot 0.16) + (26 \cdot 0.51) + (27 \cdot 0.11)
\]

Let's calculate each term:

\[
23 \cdot 0.09 = 2.07, \quad 24 \cdot 0.13 = 3.12, \quad 25 \cdot 0.16 = 4.00, \quad 26 \cdot 0.51 = 13.26, \quad 27 \cdot 0.11 = 2.97
\]

Now sum these up:
\[
\mu = 2.07 + 3.12 + 4.00 + 13.26 + 2.97 = 25.42
\]

So, the **mean number of students** is approximately **25.42**.

### Step 3: Calculate the standard deviation.
The formula for the standard deviation is:

\[
\sigma = \sqrt{\sum (X - \mu)^2 \cdot P(X)}
\]

We already know \( \mu = 25.42 \). Now calculate \( (X - \mu)^2 \cdot P(X) \) for each \( X \):

1. For \( X = 23 \):
   \[
   (23 - 25.42)^2 \cdot 0.09 = (-2.42)^2 \cdot 0.09 = 5.8564 \cdot 0.09 = 0.527076
   \]

2. For \( X = 24 \):
   \[
   (24 - 25.42)^2 \cdot 0.13 = (-1.42)^2 \cdot 0.13 = 2.0164 \cdot 0.13 = 0.262132
   \]

3. For \( X = 25 \):
   \[
   (25 - 25.42)^2 \cdot 0.16 = (-0.42)^2 \cdot 0.16 = 0.1764 \cdot 0.16 = 0.028224
   \]

4. For \( X = 26 \):
   \[
   (26 - 25.42)^2 \cdot 0.51 = (0.58)^2 \cdot 0.51 = 0.3364 \cdot 0.51 = 0.171564
   \]

5. For \( X = 27 \):
   \[
   (27 - 25.42)^2 \cdot 0.11 = (1.58)^2 \cdot 0.11 = 2.4964 \cdot 0.11 = 0.274604
   \]

Now sum all these values:
\[
0.527076 + 0.262132 + 0.028224 + 0.171564 + 0.274604 = 1.2636
\]

Finally, take the square root:
\[
\sigma = \sqrt{1.2636} \approx 1.12
\]

So, the **standard deviation** is approximately **1.12**.

### Final Answers:
- Missing probability \( P(26) = 0.51 \)
- Mean number of students \( \mu \approx 25.42 \)
- Standard deviation \( \sigma \approx 1.12 \)

Would you like further details or have any questions?

---

Here are 5 related questions to expand your understanding:
1. What is the significance of the mean in a probability distribution?
2. How would the mean change if the probability for \( X = 26 \) were smaller?
3. What does the standard deviation tell you about the distribution of students?
4. How do we interpret a probability distribution in terms of expected outcomes?
5. How can the variance of this distribution be used to understand data spread?

**Tip:** The variance is just the square of the standard deviation, and it's a key measure to understand how much values differ from the mean.

The probability distribution for the number of students in Statistics classes offered at a small college is given, but one value is missing. Fill in the missing value, then answer the questions that follow. Find the mean number of students in a Statistics class and the standard deviation.

Find the missing probability in a Statistics class probability distribution. Calculate the mean (expected value) and the standard deviation using the given probability distribution data. This step-by-step guide covers key concepts for high school students.

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