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.
![**Title: Calculating the Standard Deviation for Grouped Data**
**Objective:**
Learn how to find the standard deviation for grouped data using a frequency distribution table.
### Data Table
The table below provides the frequency distribution for a dataset:
| Interval | Frequency |
|----------|-----------|
| 0 - 3 | 8 |
| 4 - 7 | 15 |
| 8 - 11 | 10 |
| 12 - 15 | 16 |
| 16 - 19 | 0 |
| 20 - 23 | 1 |
### Calculation Steps
1. **Identify Midpoints:**
- Calculate the midpoint for each interval. The midpoint (\(x_i\)) is found by averaging the upper and lower boundaries of each interval.
2. **Calculate Mean:**
- Determine the mean using the formula:
\[
\bar{x} = \frac{\sum (f_i \cdot x_i)}{\sum f_i}
\]
where \(f_i\) is the frequency of the interval and \(x_i\) is the midpoint.
3. **Find Variance:**
- Calculate the variance using:
\[
\sigma^2 = \frac{\sum f_i \cdot (x_i - \bar{x})^2}{\sum f_i}
\]
4. **Standard Deviation:**
- Take the square root of the variance:
\[
\sigma = \sqrt{\sigma^2}
\]
### Multiple Choice Options
Choose the correct value for the standard deviation from the options below:
- ☐ 5.0
- ☐ 4.5
- ☐ 5.4
- ☐ 4.8
Use this framework to practice calculations and enhance your understanding of statistical analysis for grouped data.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7c5fa014-da02-4c57-a781-fd58c7d971af%2F8bfeca29-9d4c-4efe-9b97-5eed4fc52930%2Fp270d1_processed.jpeg&w=3840&q=75)
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