https://file.aiquickdraw.com/mathbot/file/17294848864647rw9jcha.jpg

Let's solve the math problem shown in the image step by step. The problem appears to be focused on calculating the **mean (average)**, **variance (ragam)**, and **standard deviation (simpangan baku)** for the given data.

We are provided with a frequency distribution table of the following intervals (nilai) and their frequencies (F):

| Nilai (Range) | F (Frequency) |
|---------------|---------------|
| 50 - 59       | 8             |
| 60 - 69       | 10            |
| 70 - 79       | 16            |
| 80 - 89       | 11            |
| 90 - 99       | 5             |
| **Total (N)** | **50**        |

### Step 1: Calculate the Mean (Simpangan Rata-rata)
The mean for a frequency distribution is calculated by:

\[
\bar{x} = \frac{\sum{f \cdot x}}{N}
\]

Where:
- \( f \) is the frequency
- \( x \) is the midpoint of the class interval (Nilai range)
- \( N \) is the total frequency

**Midpoints (x)**:
- For 50-59: \( \frac{50 + 59}{2} = 54.5 \)
- For 60-69: \( \frac{60 + 69}{2} = 64.5 \)
- For 70-79: \( \frac{70 + 79}{2} = 74.5 \)
- For 80-89: \( \frac{80 + 89}{2} = 84.5 \)
- For 90-99: \( \frac{90 + 99}{2} = 94.5 \)

Now, compute the products of the frequencies and midpoints, then sum them up:

\[
\sum{f \cdot x} = (8 \cdot 54.5) + (10 \cdot 64.5) + (16 \cdot 74.5) + (11 \cdot 84.5) + (5 \cdot 94.5)
\]

\[
\sum{f \cdot x} = 436 + 645 + 1192 + 929.5 + 472.5 = 3675
\]

Now, calculate the mean:

\[
\bar{x} = \frac{3675}{50} = 73.5
\]

### Step 2: Calculate the Variance (Ragam)
Variance \( s^2 \) is calculated by:

\[
s^2 = \frac{\sum{f \cdot (x - \bar{x})^2}}{N}
\]

We first calculate \( (x - \bar{x})^2 \) for each midpoint:

\[
(54.5 - 73.5)^2 = 361, \quad (64.5 - 73.5)^2 = 81, \quad (74.5 - 73.5)^2 = 1, \quad (84.5 - 73.5)^2 = 121, \quad (94.5 - 73.5)^2 = 441
\]

Now, multiply these by their respective frequencies and sum them up:

\[
\sum{f \cdot (x - \bar{x})^2} = (8 \cdot 361) + (10 \cdot 81) + (16 \cdot 1) + (11 \cdot 121) + (5 \cdot 441)
\]

\[
\sum{f \cdot (x - \bar{x})^2} = 2888 + 810 + 16 + 1331 + 2205 = 7250
\]

Finally, calculate the variance:

\[
s^2 = \frac{7250}{50} = 145
\]

### Step 3: Calculate the Standard Deviation (Simpangan Baku)
The standard deviation is simply the square root of the variance:

\[
s = \sqrt{145} \approx 12.04
\]

### Summary:
1. **Mean (Simpangan Rata-rata)**: 73.5
2. **Variance (Ragam)**: 145
3. **Standard Deviation (Simpangan Baku)**: 12.04

Would you like more details or explanations on any step? 

### Related Questions:
1. How would the results change if the class intervals were different?
2. How do we interpret the standard deviation in this context?
3. What would the variance be if the frequency of the 90-99 interval doubled?
4. How do you calculate mean and variance for non-grouped data?
5. What are the implications of a higher standard deviation in a data set?

**Tip**: Standard deviation gives you an idea of how spread out the data is around the mean. The higher the standard deviation, the more spread out the data.

Tentukan Simpangan Rata-rata, Variasi atau Ragam (S²), dan Simpangan Baku (S) dari distribusi frekuensi nilai 50-99.

Learn how to calculate the mean (average), variance (S²), and standard deviation (S) for a frequency distribution of values ranging from 50-99. This problem covers important statistical concepts and provides a step-by-step solution using the relevant formulas. Suitable for high school students in grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation