Find 10-th, 60-th and 75-th percentiles in each of your data set, then write down each value that corresponds to those percentiles. Now use those values to find sample percentiles empirically (Suppose value “a” corresponds to 10-th percentile in my data set. After finding the value “a” forget about the 10-th percentile, use only the data set to find value “a” corresponds to which percentile in my data set. Suppose I found the percentile value as “X”, is there any difference between “X” and 10? Should they be equal to each other? ). Are there any differences between the percentile values corresponding to the same observation in the data? If there exists any difference, what could be the reason. Also compare the findings in terms of 2 different data sets.
To find the 10th, 60th, and 75th percentiles in a dataset, follow these steps:
1. Arrange the data in ascending order.
2. Determine the position of the percentile values using the following formulas:
- For 10th percentile: (10/100) * (n + 1), where n is the number of observations.
- For 60th percentile: (60/100) * (n + 1).
- For 75th percentile: (75/100) * (n + 1).
3. Round the above computed positions to the nearest whole number.
- If you get a whole number for the position, take the value at that position in the sorted dataset as the percentile value.
- If you get a decimal number for the position, take the interpolated value between the two nearest positions. Here's how you can do that:
- Let's say the position is p, and the values at positions p and p+1 are v1 and v2 respectively.
- The percentile value is given by v1 + (p - floor(p)) * (v2 - v1).
4. Repeat these steps for each dataset, recording the values that correspond to the 10th, 60th, and 75th percentiles.
Now we can use the calculated percentile values to empirically find the corresponding percentiles.
1. Sort the dataset in ascending order, as before.
2. Locate the position of the value corresponding to the calculated percentile in the sorted dataset:
- For the value "a" that corresponds to the 10th percentile, find its position in the sorted dataset and denote it as p1.
- For the value "a" that corresponds to the 60th percentile, find its position in the dataset and denote it as p2.
- For the value "a" that corresponds to the 75th percentile, find its position in the dataset and denote it as p3.
3. Calculate the percentiles for each dataset using the positions p1, p2, and p3 as follows:
- For the 10th percentile: (p1 / n) * 100, where n is the number of observations.
- For the 60th percentile: (p2 / n) * 100.
- For the 75th percentile: (p3 / n) * 100.
Now let's address the questions you raised:
1. Is there any difference between the calculated percentile values and the corresponding percentiles found empirically?
- Ideally, the calculated percentile values and the corresponding percentiles found empirically should be equal.
- However, due to the interpolation process used when dealing with decimal positions, there might be slight differences between the two.
- The differences are generally small and may be due to rounding errors or the approximation caused by the interpolation formula.
2. Are there any differences between the percentile values corresponding to the same observation in the data?
- Yes, there can be differences between the percentile values corresponding to the same observation in different datasets.
- This can occur because the position of an observation relative to the rest of the dataset differs between datasets.
- For example, an observation that falls below the 10th percentile in one dataset may fall above the 10th percentile in another dataset if the datasets have different values and distributions.
In summary, while the calculated percentile values and the corresponding percentiles found empirically should ideally be equal, there may be small differences due to rounding or interpolation. Additionally, the percentile values for the same observation can differ between datasets if the datasets have different distributions.