Question content area top
Part 1
Use the display of data to find the​ mean, median,​ mode, and midrange.
8
9
10
11
12
13
14
15
16
0
1
2
3
4
5
6
7
Score
Frequency



A graph has a horizontal axis labeled "Score" from 8 to 16 in intervals of 1 and a vertical axis labeled "Frequency" from 0 to 7 in intervals of 1. A frequency polygon starts at (9, 0), ends at (16, 0), and goes through the following plotted points, (10, 2), (11, 1), (12, 5), (13, 6), (14, 1), (15, 1).
Question content area bottom
Part 1
The mean of the data is 
  
enter your response here. ​(Round to the nearest tenth as​ needed.)
Part 2
The median of the data is 
  
enter your response here. ​(Round to the nearest tenth as​ needed.)
Part 3
Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.
A.
The mode of the data is 
  
enter your response here. 
​(Use a comma to separate answers as​ needed.)
B.
There is no mode for the given data.
Part 4
The midrange of the data is 
  
enter your response here. ​(Round to the nearest tenth as​ needed.)

Let's go through the calculations step by step to find the mean, median, mode, and midrange for the given data.

### Part 1: Calculating the Mean
The mean is calculated using the formula:

\[
\text{Mean} = \frac{\sum (x \cdot f)}{\sum f}
\]

Where:
- \(x\) is the score.
- \(f\) is the frequency.

Given the data points:

\[
\begin{array}{c|c|c}
\text{Score } (x) & \text{Frequency } (f) & x \cdot f \\
\hline
9 & 0 & 9 \cdot 0 = 0 \\
10 & 2 & 10 \cdot 2 = 20 \\
11 & 1 & 11 \cdot 1 = 11 \\
12 & 5 & 12 \cdot 5 = 60 \\
13 & 6 & 13 \cdot 6 = 78 \\
14 & 1 & 14 \cdot 1 = 14 \\
15 & 1 & 15 \cdot 1 = 15 \\
16 & 0 & 16 \cdot 0 = 0 \\
\end{array}
\]

Sum of \(x \cdot f\):

\[
\sum (x \cdot f) = 0 + 20 + 11 + 60 + 78 + 14 + 15 + 0 = 198
\]

Sum of frequencies (\(\sum f\)):

\[
\sum f = 0 + 2 + 1 + 5 + 6 + 1 + 1 + 0 = 16
\]

Mean:

\[
\text{Mean} = \frac{198}{16} \approx 12.4
\]

So, the mean is **12.4**.

### Part 2: Calculating the Median
The median is the middle value in an ordered data set. Since we have an even number of data points (16), the median will be the average of the 8th and 9th data points.

The cumulative frequencies are:

\[
\begin{array}{c|c|c}
\text{Score } (x) & \text{Frequency } (f) & \text{Cumulative Frequency} \\
\hline
9 & 0 & 0 \\
10 & 2 & 2 \\
11 & 1 & 3 \\
12 & 5 & 8 \\
13 & 6 & 14 \\
14 & 1 & 15 \\
15 & 1 & 16 \\
16 & 0 & 16 \\
\end{array}
\]

The 8th value lies in the class interval for the score 12, and the 9th value lies in the class interval for the score 13. Therefore, the median is:

\[
\text{Median} = \frac{12 + 13}{2} = 12.5
\]

So, the median is **12.5**.

### Part 3: Determining the Mode
The mode is the value that appears most frequently in the data set. From the frequency table, the score 13 has the highest frequency (6), so:

The mode is **13**.

### Part 4: Calculating the Midrange
The midrange is the average of the maximum and minimum values in the data set:

\[
\text{Midrange} = \frac{\text{Max value} + \text{Min value}}{2}
\]

The minimum value is 9 and the maximum value is 15, so:

\[
\text{Midrange} = \frac{9 + 15}{2} = \frac{24}{2} = 12
\]

So, the midrange is **12**.

### Summary of Results
1. **Mean:** 12.4
2. **Median:** 12.5
3. **Mode:** 13
4. **Midrange:** 12

Would you like to explore any of these results further or have any other questions?

---

Here are some related questions you might find interesting:

1. What would the mean be if the highest score increased to 18?
2. How would the mode change if an additional score of 13 was added?
3. Can you calculate the range of this data set?
4. How do you find the variance and standard deviation for this data?
5. What happens to the median if all scores are doubled?
6. How does removing the outliers affect the mean?
7. What is the impact of changing the frequency of the mode on the mean?
8. Can you determine the skewness of the data distribution based on the given values?

**Tip:** When calculating the median for a frequency distribution, always use cumulative frequencies to determine the position of the median value.

Learn how to calculate the mean, median, mode, and midrange from a frequency distribution. Step-by-step solution for statistical analysis using data points and frequency distribution. Suitable for high school students studying statistics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation