[নোটঃ এই আর্টিকেলটি (5) Statistics and D.I. বিভাগের অধীনে 〈2〉Percentile Problems চ্যাপ্টারের অন্তর্গত, যা 〈5.2.a〉চিহ্ন দিয়ে প্রকাশ করা হয়েছে]
Percentile
A percentile is a measure used in statistics indicating the value below which a given percentage of observations in a group of observations fall. For example, the [latex]20[latex]th percentile is the value (or score) below which [latex]20[latex] percent of the observations may be found.
Percentile এর একদম Fixed কোন সংজ্ঞা নেই। তাও এটাকে বেসিক ধরে নেওয়া যায়। এখন নিজের উদাহরণটি দেখুনঃ
Holly just took her first math test in her college algebra class. Her professor says she scored in the [latex]90[latex]th percentile for the class. Unfortunately, Holly isn't really sure what this means, and she's really curious about her performance in this class. Holly needs to learn more about finding percentiles in a data set to really understand how she performed on this math test.
A percentile is a measure that indicates what percent of the given population scored at or below the measure. Often, schools and colleges will use percentiles to rank students based on their academic performance. If Holly scored in the [latex]90[latex]th percentile, then that means she scored at or better than [latex]90[latex]% of her class. Since percentiles are based on a percentage, you will only see percentiles between the range of [latex]0-100[latex].
Holly’s professor posts a list of grades, without the names, on the blackboard. There are [latex]15[latex] students in the class and [latex]15[latex] grades on the board. The grades are:
[latex]85, 34, 42, 51, 84, 86, 78, 85, 87, 69, 94, 74, 65, 56, 97[latex].
To find Holly’s grade, we need to do the following steps:
1. Multiply the total number of values in the data set by the percentile, which will give you the index.
2. Order all of the values in the data set in ascending order (least to greatest).
3. If the index is a whole number, count the values in the data set from least to greatest until you reach the index, then take the index and the next greatest number and find the average.
4. If the index is not a whole number, round the number up, then count the values in the data set from least to greatest, until you reach the index.
Let’s start with step number one. The total number of values in the data set is [latex]15[latex]. I found this number by looking at how many students there are in the class. The percentile is 90 because that is the score Holly’s professor said she received. Therefore, [latex]15*.90=13.5[latex].
Okay, so I got [latex]13.5[latex]. Now, according to step two, I need to order the grades from least to greatest:
[latex]34, 42, 51, 56, 65, 69, 74, 78, 84, 85, 85, 86, 87, 94, 97[latex].
For step three I need to round my index up from [latex]13.5[latex] to [latex]14[latex]. Next, I need to count from the smallest number up to the 14th number in the list. The 14th number in this list is [latex]94[latex]. That tells us that Holly scored a [latex]94[latex] on her math test and only one person scored higher than she did.
Holly’s friend, Dave, scored in the [latex]80[latex]th percentile. We can use the same process to figure out Dave’s grade. First, multiply the percentile by the number of values in the data set: [latex]15*.80=12[latex].
From this list we can see that the 12th number in this list is [latex]86[latex]. However, because our index turned out to be a whole number, we need to take the 12th number and the [latex]13[latex]th number and find the average: [latex]86+87=173/2=86.5[latex].
From this information, we know that Dave scored at or better than [latex]86.5[latex]. We used the average in this step because the percentiles don’t always divide out perfectly. Since percentiles tell us that the value is at or better than the rest of the population, we have to use the average in this particular instance. In this case, we know that Dave actually scored better than [latex]86.5[latex] to be in the [latex]80[latex]th percentile.
Given Rank
Dave has a fantasy football league with his coworkers. Dave's team is ranked third out of 35 teams in his office. In what percentile is Dave's team?
We can find this information using the following formula: ([latex]k+.5r)/n=p[latex]
First we must consider Dave’s rank as x. In this formula k = the number of teams below [latex]x, r [latex]= the number of teams equal to [latex]x, n[latex] = the total number of teams and [latex]p[latex]=percentile.
