Finding The Third Quartile In A Stem And Leaf Plot
This article will guide you through the process of extracting information from a stem and leaf plot and calculating the third quartile. We'll use a specific example to illustrate the steps involved, making it easy to understand even if you're new to this concept. Stem and leaf plots are a fantastic way to visualize data, and understanding how to interpret them is crucial for statistical analysis. One of the most important things to grasp about stem and leaf plots is how they display data in an organized manner. Each data point is split into two parts: a "stem" (usually the leading digit or digits) and a "leaf" (the trailing digit). This allows us to see the distribution of the data at a glance. The third quartile, often denoted as Q3, is a crucial statistical measure that represents the value below which 75% of the data falls. It's a key component in understanding the spread and distribution of a dataset. To find the third quartile, we first need to understand how the data is arranged in the stem and leaf plot. The stem represents the tens digit, and the leaf represents the ones digit. For instance, a stem of 1 and a leaf of 3 represents the number 13. So, the stem and leaf plot provided is a visual representation of numerical data, allowing for a quick understanding of the data's distribution and central tendency. Unlike a simple list of numbers, a stem and leaf plot preserves the original data while grouping it in a way that reveals patterns.
Decoding the Stem and Leaf Plot
Let's take a closer look at the given stem and leaf plot:
| Stem | Leaf |
|---|---|
| 1 | 3 5 |
| 2 | 1 5 7 9 |
| 3 | 2 4 6 8 |
| 4 | 5 |
From this plot, we can reconstruct the original data set. Remember, each leaf represents a single data point. So, the first row (stem 1, leaves 3 and 5) represents the numbers 13 and 15. The second row (stem 2, leaves 1, 5, 7, and 9) represents 21, 25, 27, and 29, and so on. By carefully reading each row, we can list out all the data points. The data points are: 13, 15, 21, 25, 27, 29, 32, 34, 36, 38, 45. This gives us a total of 11 data points. Understanding the data points is the first step to finding the quartiles. Once we have the complete dataset, we can proceed with calculating the third quartile. The total number of data points, often denoted as 'n', is a critical factor in determining the position of the quartiles. With 'n' known, we can apply the appropriate formulas to pinpoint the exact values we need. We can see that the data is already arranged in ascending order, thanks to the nature of the stem and leaf plot. This is a huge advantage because finding quartiles requires the data to be sorted. The stem and leaf plot not only presents the data but also sorts it implicitly, saving us a step in the calculation process.
Calculating the Third Quartile (Q3)
Now that we have our data set: 13, 15, 21, 25, 27, 29, 32, 34, 36, 38, 45, we can move on to calculating the third quartile. The third quartile (Q3) represents the 75th percentile of the data. This means 75% of the data values fall below Q3. To find Q3, we use a specific formula based on the number of data points. The formula to find the position of Q3 in an ordered data set is: Q3 position = 0.75 * (n + 1), where 'n' is the number of data points. In our case, we have 11 data points (n = 11). Therefore, the position of Q3 = 0.75 * (11 + 1) = 0.75 * 12 = 9. This tells us that the third quartile is located at the 9th position in our ordered data set. Looking at our data set (13, 15, 21, 25, 27, 29, 32, 34, 36, 38, 45), the value at the 9th position is 36. Therefore, the third quartile (Q3) for this data set is 36. The calculation of Q3's position is crucial because it pinpoints the exact location of the quartile within the dataset. This formula ensures we are finding the value that correctly divides the data into the required percentile. Once we have the position, identifying the corresponding data point is a straightforward process of counting through the ordered data. This step-by-step approach makes quartile calculation manageable even for larger datasets.
Verifying the Result
We've determined that the third quartile is 36. To ensure our answer is correct, let's think about what this means in the context of our data. The third quartile, Q3, should be the value that separates the top 25% of the data from the bottom 75%. In our data set (13, 15, 21, 25, 27, 29, 32, 34, 36, 38, 45), Q3 is 36. This means that approximately 75% of the values should be less than 36, and 25% should be greater than 36. Let's count the values less than 36: 13, 15, 21, 25, 27, 29, 32, and 34. There are 8 values less than 36. Since we have a total of 11 values, 8 out of 11 is approximately 73%, which is close to 75%. The values greater than 36 are 38 and 45, which are 2 out of 11, or approximately 18%. This is also reasonably close to 25%, considering our data set is relatively small. Therefore, our result of Q3 = 36 seems correct. This verification step is crucial in statistical analysis. It helps to confirm that the calculated value aligns with the overall distribution of the data and the definition of the quartile. By cross-checking our result, we can increase our confidence in the accuracy of the analysis.
Conclusion
In conclusion, by carefully interpreting the stem and leaf plot and applying the formula for calculating the third quartile, we have successfully found that Q3 = 36. Understanding quartiles is essential for data analysis, as it helps us to divide the data into meaningful segments and understand its distribution. This process highlights the importance of organized data presentation and the power of statistical measures in interpreting that data. The third quartile is just one piece of the puzzle when it comes to understanding data distribution. Other measures, such as the first quartile (Q1), the median (Q2), and the interquartile range (IQR), provide additional insights into the central tendency and spread of the data. By combining these measures, we can gain a comprehensive understanding of the dataset and draw meaningful conclusions. The ability to extract and analyze information from stem and leaf plots is a valuable skill in statistics. It allows for a quick and visual assessment of data distribution, making it easier to identify patterns and outliers. This understanding is crucial for making informed decisions based on data. Therefore, the correct answer is B. 36.
Therefore, the correct answer is B. 36.