Find Mean , Modal class , variance, and standard deviation for this grouped data of a sample: class limit f 22 - 27 28 - 33 16 2 34 - 39 40 - 45 46 - 51 1
Find Mean , Modal class , variance, and standard deviation for this grouped data of a sample: class limit f 22 - 27 28 - 33 16 2 34 - 39 40 - 45 46 - 51 1
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![To calculate the Mean, Modal class, Variance, and Standard Deviation for the given grouped data of a sample, follow the steps outlined below:
**Grouped Data Table:**
| Class Limit | Frequency (f) |
|-------------|---------------|
| 22 - 27 | 16 |
| 28 - 33 | 2 |
| 34 - 39 | 2 |
| 40 - 45 | 3 |
| 46 - 51 | 1 |
### 1. Mean
To calculate the mean, we first need to find the midpoints (x) of each class interval. The midpoint is calculated by taking the average of the upper and lower class limits.
\[
x = \frac{(Lower \, Class \, Limit + Upper \, Class \, Limit)}{2}
\]
Next, we multiply each midpoint by the frequency to get \( f \cdot x \), and then we sum these values.
\[
\text{Mean} = \frac{\sum (f \cdot x)}{\sum f}
\]
### 2. Modal Class
The modal class is the class interval with the highest frequency.
### 3. Variance
To calculate the variance, we'll use the formula:
\[
\text{Variance} = \frac{\sum (f \cdot (x - \bar{x})^2)}{\sum f}
\]
where \( \bar{x} \) is the mean of the data. First, compute \( x - \bar{x} \) for each class midpoint and then square it, multiply by the frequency, and sum these values.
### 4. Standard Deviation
The standard deviation is the square root of the variance:
\[
\text{Standard Deviation} = \sqrt{\text{Variance}}
\]
### Example Calculation (Illustrative):
Let's calculate the midpoint (x) for each class:
- For 22 - 27: \( x = \frac{22 + 27}{2} = 24.5 \)
- For 28 - 33: \( x = \frac{28 + 33}{2} = 30.5 \)
- For 34 - 39: \( x = \frac{34 + 39}{2} = 36.5 \)
- For 40 - 45: \( x = \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8d7dd878-68eb-4c6b-9d67-0903dd8963d8%2F2aca2107-db2a-4029-9fc2-14a8f59c49b0%2Feyey01p_processed.jpeg&w=3840&q=75)
Transcribed Image Text:To calculate the Mean, Modal class, Variance, and Standard Deviation for the given grouped data of a sample, follow the steps outlined below:
**Grouped Data Table:**
| Class Limit | Frequency (f) |
|-------------|---------------|
| 22 - 27 | 16 |
| 28 - 33 | 2 |
| 34 - 39 | 2 |
| 40 - 45 | 3 |
| 46 - 51 | 1 |
### 1. Mean
To calculate the mean, we first need to find the midpoints (x) of each class interval. The midpoint is calculated by taking the average of the upper and lower class limits.
\[
x = \frac{(Lower \, Class \, Limit + Upper \, Class \, Limit)}{2}
\]
Next, we multiply each midpoint by the frequency to get \( f \cdot x \), and then we sum these values.
\[
\text{Mean} = \frac{\sum (f \cdot x)}{\sum f}
\]
### 2. Modal Class
The modal class is the class interval with the highest frequency.
### 3. Variance
To calculate the variance, we'll use the formula:
\[
\text{Variance} = \frac{\sum (f \cdot (x - \bar{x})^2)}{\sum f}
\]
where \( \bar{x} \) is the mean of the data. First, compute \( x - \bar{x} \) for each class midpoint and then square it, multiply by the frequency, and sum these values.
### 4. Standard Deviation
The standard deviation is the square root of the variance:
\[
\text{Standard Deviation} = \sqrt{\text{Variance}}
\]
### Example Calculation (Illustrative):
Let's calculate the midpoint (x) for each class:
- For 22 - 27: \( x = \frac{22 + 27}{2} = 24.5 \)
- For 28 - 33: \( x = \frac{28 + 33}{2} = 30.5 \)
- For 34 - 39: \( x = \frac{34 + 39}{2} = 36.5 \)
- For 40 - 45: \( x = \
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