Find Mean , Modal class , variance, and standard deviation for this grouped data of a sample: class limit 22 - 27 28 - 33 34 - 39 40 - 45 46 - 51 o223r
Inverse Normal Distribution
The method used for finding the corresponding z-critical value in a normal distribution using the known probability is said to be an inverse normal distribution. The inverse normal distribution is a continuous probability distribution with a family of two parameters.
Mean, Median, Mode
It is a descriptive summary of a data set. It can be defined by using some of the measures. The central tendencies do not provide information regarding individual data from the dataset. However, they give a summary of the data set. The central tendency or measure of central tendency is a central or typical value for a probability distribution.
Z-Scores
A z-score is a unit of measurement used in statistics to describe the position of a raw score in terms of its distance from the mean, measured with reference to standard deviation from the mean. Z-scores are useful in statistics because they allow comparison between two scores that belong to different normal distributions.
![### Grouped Data Analysis: Mean, Modal Class, Variance, and Standard Deviation
We are given the following grouped data and are required to determine the mean, modal class, variance, and standard deviation.
#### Data Table:
| Class Limit | Frequency (f) |
|-------------|---------------|
| 22 - 27 | 16 |
| 28 - 33 | 2 |
| 34 - 39 | 2 |
| 40 - 45 | 3 |
| 46 - 51 | 1 |
### Steps for Calculating Descriptive Statistics:
#### 1. **Mean (Average):**
For grouped data, the mean can be estimated using the formula:
\[ \bar{X} = \frac{\sum (f \cdot x)}{\sum f} \]
Where \(x\) is the midpoint of each class interval.
#### 2. **Modal Class:**
The modal class is the class interval with the highest frequency.
#### 3. **Variance and Standard Deviation:**
Variance for grouped data is calculated by:
\[ \sigma^2 = \frac{\sum f \cdot (x - \bar{X})^2}{\sum f} \]
Standard deviation is the square root of variance:
\[ \sigma = \sqrt{\sigma^2} \]
### Detailed Breakdown:
#### Calculate Midpoints:
- \(22 - 27\): Midpoint \( x = \frac{22 + 27}{2} = 24.5\)
- \(28 - 33\): Midpoint \( x = \frac{28 + 33}{2} = 30.5\)
- \(34 - 39\): Midpoint \( x = \frac{34 + 39}{2} = 36.5\)
- \(40 - 45\): Midpoint \( x = \frac{40 + 45}{2} = 42.5\)
- \(46 - 51\): Midpoint \( x = \frac{46 + 51}{2} = 48.5\)
#### Calculate Mean:
\[ \bar{X} = \frac{(24.5 \cdot 16) + (30.5 \cdot 2) + (36.5 \cdot 2) + (42.5 \cdot 3)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8d7dd878-68eb-4c6b-9d67-0903dd8963d8%2F51fb3cbe-2fcc-41ad-9470-e6ebc5cdad98%2F60sb3gq_processed.jpeg&w=3840&q=75)
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