Given The Following Data, Find The Weight That Represents The $45^{\text {th }}$ Percentile.$\[ \begin{array}{|c|c|c|c|c|} \hline \multicolumn{5}{|c|}{\text{Weights Of Newborn Babies}} \\ \hline 7.8 & 5.5 & 7.6 & 8.3 & 7.1 \\ \hline 7.3
Percentiles are a measure of the distribution of a dataset, indicating the value below which a certain percentage of observations fall. In this article, we will explore how to find the weight that represents the 45th percentile in a given dataset of newborn babies' weights.
What is a Percentile?
A percentile is a value below which a certain percentage of observations in a dataset fall. For example, the 25th percentile is the value below which 25% of the observations fall, and the 50th percentile is the median, below which 50% of the observations fall.
Calculating Percentiles
To calculate a percentile, we need to first arrange the data in ascending order. Then, we need to find the value below which the desired percentage of observations fall. There are several methods to calculate percentiles, including:
- Direct Method: This method involves counting the number of observations below the desired percentile and finding the corresponding value.
- Inverse Method: This method involves finding the value below which the desired percentage of observations fall.
- Interpolation Method: This method involves interpolating between two values to find the desired percentile.
Given Data
The given data consists of the weights of newborn babies:
| Weight (lbs) |
|---|
| 7.8 |
| 5.5 |
| 7.6 |
| 8.3 |
| 7.1 |
| 7.3 |
Finding the 45th Percentile
To find the 45th percentile, we need to first arrange the data in ascending order:
| Weight (lbs) |
|---|
| 5.5 |
| 7.1 |
| 7.3 |
| 7.6 |
| 7.8 |
| 8.3 |
Next, we need to find the value below which 45% of the observations fall. Since there are 6 observations, we need to find the value below which 2.7 observations (45% of 6) fall.
Using the Direct Method
Using the direct method, we can count the number of observations below the desired percentile:
- 5.5 (1st observation)
- 7.1 (2nd observation)
- 7.3 (3rd observation)
Since 3 observations are below the 45th percentile, we need to find the value below which 3.7 observations (45% of 6) fall.
Using the Inverse Method
Using the inverse method, we can find the value below which the desired percentage of observations fall. Since 3 observations are below the 45th percentile, we need to find the value below which 3.7 observations fall.
To do this, we can use the following formula:
x = (n * p) / 100
where x is the value below which the desired percentage of observations fall, n is the number of observations, and p is the desired percentage.
Plugging in the values, we get:
x = (6 * 45) / 100 x = 2.7
Since x is not an integer, we need to interpolate between the two values below which 2.7 observations fall.
Interpolating Between Values
To interpolate between the two values below which 2.7 observations fall, we can use the following formula:
x = (y1 + (y2 - y1) * (x - n)) / (n + 1)
where x is the value below which the desired percentage of observations fall, y1 and y2 are the two values below which 2.7 observations fall, and n is the number of observations.
Plugging in the values, we get:
x = (7.1 + (7.3 - 7.1) * (2.7 - 3)) / (3 + 1) x = 7.2
Therefore, the weight that represents the 45th percentile is approximately 7.2 pounds.
Conclusion
In this article, we explored how to find the weight that represents the 45th percentile in a given dataset of newborn babies' weights. We used the direct method, inverse method, and interpolation method to calculate the percentile. The weight that represents the 45th percentile is approximately 7.2 pounds.
References
- National Institute of Child Health and Human Development. (2019). Birth Weight Percentiles.
- World Health Organization. (2019). Growth Charts for Children and Adolescents.
Future Work
In this article, we will answer some frequently asked questions about percentiles, including what percentiles are, how to calculate them, and how to use them in real-world scenarios.
Q: What is a percentile?
A: A percentile is a value below which a certain percentage of observations in a dataset fall. For example, the 25th percentile is the value below which 25% of the observations fall, and the 50th percentile is the median, below which 50% of the observations fall.
Q: How do I calculate a percentile?
A: There are several methods to calculate percentiles, including the direct method, inverse method, and interpolation method. The direct method involves counting the number of observations below the desired percentile and finding the corresponding value. The inverse method involves finding the value below which the desired percentage of observations fall. The interpolation method involves interpolating between two values to find the desired percentile.
Q: What is the difference between a percentile and a quartile?
A: A percentile is a value below which a certain percentage of observations in a dataset fall. A quartile is a value below which a certain percentage of observations in a dataset fall, but it is typically used to divide the data into four equal parts. The 25th percentile is the same as the first quartile, the 50th percentile is the same as the second quartile (median), and the 75th percentile is the same as the third quartile.
Q: How do I use percentiles in real-world scenarios?
A: Percentiles are used in a variety of real-world scenarios, including:
- Medical research: Percentiles are used to describe the distribution of patient outcomes, such as blood pressure or weight.
- Finance: Percentiles are used to describe the distribution of stock prices or returns.
- Education: Percentiles are used to describe the distribution of student test scores.
- Quality control: Percentiles are used to describe the distribution of product quality or performance.
Q: What are some common percentile values?
A: Some common percentile values include:
- 25th percentile: The value below which 25% of the observations fall.
- 50th percentile: The median, below which 50% of the observations fall.
- 75th percentile: The value below which 75% of the observations fall.
- 90th percentile: The value below which 90% of the observations fall.
Q: How do I interpret percentile values?
A: To interpret percentile values, you need to understand what the value represents. For example, if the 25th percentile is 10, it means that 25% of the observations are below 10. If the 50th percentile is 20, it means that 50% of the observations are below 20.
Q: What are some common mistakes to avoid when working with percentiles?
A: Some common mistakes to avoid when working with percentiles include:
- Misinterpreting the meaning of a percentile value: Make sure you understand what the value represents.
- Using the wrong method to calculate a percentile: Use the correct method to calculate the percentile.
- Not considering the distribution of the data: Consider the distribution of the data when interpreting percentile values.
Conclusion
In this article, we answered some frequently asked questions about percentiles, including what percentiles are, how to calculate them, and how to use them in real-world scenarios. We also discussed some common percentile values and how to interpret them. By understanding percentiles, you can better analyze and interpret data in a variety of fields.
References
- National Institute of Child Health and Human Development. (2019). Birth Weight Percentiles.
- World Health Organization. (2019). Growth Charts for Children and Adolescents.
- Wikipedia. (2022). Percentile.
Future Work
In future work, we can explore other methods for calculating percentiles, such as the linear interpolation method and the cubic spline interpolation method. We can also apply these methods to other datasets to see how they perform in different scenarios.