Analyze Exam Results: Quartiles, Deciles, Percentiles
Hey guys! Let's dive into analyzing the results of a student admission exam using frequency tables and understanding how to calculate and interpret quartiles, deciles, and percentiles. This is super useful for understanding the distribution of scores and how individual students performed relative to the group. We'll break it down step-by-step, so it's easy to follow along. We are going to use a real data set to illustrate this.
The Frequency Table
First, let's take a look at the frequency table we'll be working with. This table summarizes the exam results, showing the score ranges and the number of people who scored within each range. It looks something like this:
| Result of Exam | No. of Persons |
|---|---|
| [54-60) | 5 |
| 60-66 | 9 |
| 66-72 | 3 |
| 72-78 | 11 |
| 78-84 | 7 |
| 84-90 | 5 |
| 90-96 | 3 |
| 96-102 | 4 |
This table tells us how many students scored within each score bracket. For example, 5 students scored between 54 and 60 (not including 60), 9 students scored between 60 and 66, and so on. Understanding this distribution is the first step in analyzing the exam results.
Breaking Down the Frequency Table
Before we jump into calculating quartiles, deciles, and percentiles, let's make sure we fully grasp what this frequency table is telling us. The first column, "Result of Exam," gives us the score intervals or classes. Notice that the intervals are continuous, meaning they flow from one to the next (e.g., 54-60, 60-66). The parentheses and brackets indicate whether the endpoint is included in the interval or not. In the interval [54-60), 54 is included, but 60 is not. This is important to keep in mind when we're doing our calculations.
The second column, "No. of Persons," shows the frequency of each interval. This is simply the number of students who scored within that range. The higher the frequency, the more students scored in that particular range. This gives us a sense of where the bulk of the scores lie.
To work with this data effectively, we'll need to calculate a few more things. The most important of these is the cumulative frequency. The cumulative frequency for an interval is the sum of the frequencies of that interval and all the intervals before it. This tells us how many students scored at or below a certain score. For example, to find the cumulative frequency for the 66-72 interval, we would add the frequencies of the 54-60, 60-66, and 66-72 intervals. The cumulative frequency will be essential when we calculate our quartiles, deciles, and percentiles.
Another useful value is the cumulative percentage. This is the cumulative frequency expressed as a percentage of the total number of students. To calculate the cumulative percentage for an interval, we divide the cumulative frequency of that interval by the total number of students and multiply by 100. The cumulative percentage tells us what percentage of students scored at or below a certain score. This can be helpful for interpreting our results in terms of student performance.
By understanding these basic concepts, we can start to delve deeper into the data and extract meaningful insights about student performance on the admission exam. Now, let's move on to the exciting part: calculating the quartiles, deciles, and percentiles!
Quartiles: Dividing the Data into Fourths
Now, let's talk about quartiles. Quartiles are values that divide a dataset into four equal parts. Imagine lining up all the exam scores from lowest to highest. The quartiles are the scores that mark the 25%, 50%, and 75% points. Specifically:
- Q1 (First Quartile): The value below which 25% of the data falls. Also known as the 25th percentile.
- Q2 (Second Quartile): The value below which 50% of the data falls. This is also the median.
- Q3 (Third Quartile): The value below which 75% of the data falls. Also known as the 75th percentile.
For this problem, we need to find Q1. To do this, we'll use a formula that takes into account the cumulative frequencies from our table.
Calculating Q1
The formula for calculating the first quartile (Q1) for grouped data is:
Q1 = L + [(N/4 - CF) / f] * w
Where:
- L is the lower limit of the quartile class (the class containing the 25th percentile).
- N is the total number of observations (students).
- CF is the cumulative frequency of the class preceding the quartile class.
- f is the frequency of the quartile class.
- w is the class width.
First, we need to find the quartile class. To do this, we calculate N/4. The total number of students (N) is the sum of the frequencies: 5 + 9 + 3 + 11 + 7 + 5 + 3 + 4 = 47. So, N/4 = 47/4 = 11.75. This means the first quartile lies in the class interval where the cumulative frequency is just greater than 11.75.
Let's add a cumulative frequency column to our table:
| Result of Exam | No. of Persons (f) | Cumulative Frequency (CF) |
|---|---|---|
| [54-60) | 5 | 5 |
| 60-66 | 9 | 14 |
| 66-72 | 3 | 17 |
| 72-78 | 11 | 28 |
| 78-84 | 7 | 35 |
| 84-90 | 5 | 40 |
| 90-96 | 3 | 43 |
| 96-102 | 4 | 47 |
Looking at the cumulative frequencies, we see that 11.75 falls within the 60-66 interval (since the cumulative frequency for 54-60 is 5, and for 60-66 it's 14). Therefore, the quartile class is 60-66.
Now we can identify the values for our formula:
- L (lower limit of the quartile class) = 60
- N (total number of students) = 47
- CF (cumulative frequency of the class preceding the quartile class) = 5
- f (frequency of the quartile class) = 9
- w (class width) = 6 (the difference between the upper and lower limits of the class interval)
Plugging these values into the formula, we get:
Q1 = 60 + [(47/4 - 5) / 9] * 6
Q1 = 60 + [(11.75 - 5) / 9] * 6
Q1 = 60 + [6.75 / 9] * 6
Q1 = 60 + 0.75 * 6
Q1 = 60 + 4.5
Q1 = 64.5
So, the first quartile (Q1) is 64.5. This means that approximately 25% of the students scored 64.5 or below on the exam.
Deciles: Dividing the Data into Tenths
Next up, we have deciles. Deciles are similar to quartiles, but they divide the data into ten equal parts. So, we have nine deciles (D1 to D9), representing the 10th, 20th, 30th, ..., 90th percentiles. We're asked to find the 3rd decile (D3), which represents the 30th percentile. This means we want to find the score below which 30% of the students fall.
Calculating D3
The formula for calculating the deciles is very similar to the quartile formula:
Dk = L + [(k * N/10 - CF) / f] * w
Where:
- Dk is the kth decile (in our case, D3).
- L is the lower limit of the decile class (the class containing the 30th percentile).
- N is the total number of observations (students).
- CF is the cumulative frequency of the class preceding the decile class.
- f is the frequency of the decile class.
- w is the class width.
- k is the decile we're calculating (in our case, 3).
To find the decile class, we calculate 3 * N/10 = 3 * 47/10 = 14.1. This means the 3rd decile lies in the class interval where the cumulative frequency is just greater than 14.1.
Looking at our cumulative frequency table (from the quartile calculation), we see that 14.1 falls within the 66-72 interval (since the cumulative frequency for 60-66 is 14, and for 66-72 it's 17). Therefore, the decile class is 66-72.
Now we can identify the values for our formula:
- L (lower limit of the decile class) = 66
- N (total number of students) = 47
- CF (cumulative frequency of the class preceding the decile class) = 14
- f (frequency of the decile class) = 3
- w (class width) = 6
- k = 3
Plugging these values into the formula, we get:
D3 = 66 + [(3 * 47/10 - 14) / 3] * 6
D3 = 66 + [(14.1 - 14) / 3] * 6
D3 = 66 + [0.1 / 3] * 6
D3 = 66 + 0.0333 * 6
D3 = 66 + 0.2
D3 = 66.2
So, the 3rd decile (D3) is 66.2. This means that approximately 30% of the students scored 66.2 or below on the exam.
Percentiles: Diving the Data into Hundredths
Finally, let's tackle percentiles. Percentiles divide the data into 100 equal parts. So, we have 99 percentiles (P1 to P99), representing the 1st, 2nd, 3rd, ..., 99th percentiles. We're asked to find the 70th percentile (P70), which means we want to find the score below which 70% of the students fall.
Calculating P70
You probably see the pattern by now! The formula for calculating percentiles is very similar to the quartile and decile formulas:
Pk = L + [(k * N/100 - CF) / f] * w
Where:
- Pk is the kth percentile (in our case, P70).
- L is the lower limit of the percentile class (the class containing the 70th percentile).
- N is the total number of observations (students).
- CF is the cumulative frequency of the class preceding the percentile class.
- f is the frequency of the percentile class.
- w is the class width.
- k is the percentile we're calculating (in our case, 70).
To find the percentile class, we calculate 70 * N/100 = 70 * 47/100 = 32.9. This means the 70th percentile lies in the class interval where the cumulative frequency is just greater than 32.9.
Looking at our cumulative frequency table, we see that 32.9 falls within the 78-84 interval (since the cumulative frequency for 72-78 is 28, and for 78-84 it's 35). Therefore, the percentile class is 78-84.
Now we can identify the values for our formula:
- L (lower limit of the percentile class) = 78
- N (total number of students) = 47
- CF (cumulative frequency of the class preceding the percentile class) = 28
- f (frequency of the percentile class) = 7
- w (class width) = 6
- k = 70
Plugging these values into the formula, we get:
P70 = 78 + [(70 * 47/100 - 28) / 7] * 6
P70 = 78 + [(32.9 - 28) / 7] * 6
P70 = 78 + [4.9 / 7] * 6
P70 = 78 + 0.7 * 6
P70 = 78 + 4.2
P70 = 82.2
So, the 70th percentile (P70) is 82.2. This means that approximately 70% of the students scored 82.2 or below on the exam.
Interpreting the Results
Alright, we've calculated Q1, D3, and P70. But what does it all mean? Let's break it down:
- Q1 = 64.5: This tells us that 25% of the students scored 64.5 or lower. This could be an indicator of the lower-performing students in the group. If the admission criteria are stringent, this might be a benchmark for students who may need additional support or may not meet the required standards.
- D3 = 66.2: This tells us that 30% of the students scored 66.2 or lower. This is a slightly higher threshold than Q1, but still represents the lower end of the score distribution. It can be useful for setting cutoff scores for scholarships or specific programs.
- P70 = 82.2: This tells us that 70% of the students scored 82.2 or lower. This is a good indication of the performance of a majority of the students. It also means that 30% of the students scored above 82.2, representing the higher-achieving group.
By analyzing these measures, we can get a good understanding of the distribution of scores and the relative performance of students. This information can be used to make decisions about admissions, scholarships, and academic support programs.
Conclusion
So, there you have it! We've walked through calculating quartiles, deciles, and percentiles from a frequency table, and we've discussed how to interpret these values. Understanding these concepts is super important for analyzing data and making informed decisions. Whether you're looking at exam scores, survey results, or any other kind of data, these tools will help you gain valuable insights. Keep practicing, and you'll become a data analysis pro in no time! Remember practice is the key.
