The histogram shows the reviewer ratings on a scale from 1​ (lowest) to 5​ (highest) of a recently published book.
​(a) Find the​ mean, variance, and standard deviation of the probability distribution.
​(b) Interpret the results.
Reviewer Ratings
1
2
3
4
5
0.0
0.1
0.2
0.3
0.4
0.5
Rating
Probability
0.020
0.069
0.219
0.248
0.444

To solve the problem, we need to follow these steps:

### (a) Find the mean, variance, and standard deviation of the probability distribution.

**Step 1: Find the Mean (μ)**
The mean (μ) of a discrete probability distribution is calculated using the formula:

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

Where:
- \(x\) is the rating (1, 2, 3, 4, 5)
- \(P(x)\) is the probability associated with each rating

Given:
\[
\begin{array}{c|c}
\text{Rating (x)} & \text{Probability } P(x) \\
\hline
1 & 0.020 \\
2 & 0.069 \\
3 & 0.219 \\
4 & 0.248 \\
5 & 0.444 \\
\end{array}
\]

Let's calculate the mean:

\[
\mu = (1 \cdot 0.020) + (2 \cdot 0.069) + (3 \cdot 0.219) + (4 \cdot 0.248) + (5 \cdot 0.444)
\]

**Step 2: Find the Variance (σ²)**
The variance (σ²) of a discrete probability distribution is calculated using the formula:

\[
\sigma^2 = \sum [(x - \mu)^2 \cdot P(x)]
\]

**Step 3: Find the Standard Deviation (σ)**
The standard deviation (σ) is the square root of the variance:

\[
\sigma = \sqrt{\sigma^2}
\]

### (b) Interpret the results.
The mean gives us the average rating, the variance tells us how much the ratings vary from the mean, and the standard deviation gives us the average distance of the ratings from the mean.

Let's calculate these values now.### (a) Results:
- **Mean (μ):** 4.027
- **Variance (σ²):** 1.118
- **Standard Deviation (σ):** 1.057

### (b) Interpretation:
- **Mean:** The average rating of the book is approximately 4.03, indicating that most reviewers rated the book positively.
- **Variance:** The variance of 1.118 suggests that the ratings are spread out to some degree around the mean, but not excessively so.
- **Standard Deviation:** The standard deviation of 1.057 implies that the ratings typically vary by about 1.06 points from the mean, showing a moderate spread in the reviewers' ratings.

Would you like further details or have any questions?

---

Here are 5 related questions to consider:
1. How does the skewness of the distribution affect the interpretation of the mean?
2. What would the implications be if the standard deviation were much higher?
3. How would adding more ratings at the extreme ends (1 and 5) affect the variance?
4. What other statistical measures could provide additional insights into this data?
5. How does the distribution of ratings compare to other books in the same genre?

**Tip:** The standard deviation is especially useful when comparing the spread of different datasets that have the same mean.

Learn how to calculate the mean, variance, and standard deviation of a probability distribution from reviewer ratings of a book. Understand how these statistical measures describe the spread and central tendency of the ratings.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation