5.3.5. Percentiles

Percentiles divide the distribution into hundred equal parts, so you can get 99 dividing positions denoted by P1, P2, P3, ..., P99. P50 is the median value.

Percentiles are a way of dividing a dataset into 100 equal parts. This means that each percentile represents a position below which a certain percentage of data falls. For example, P1 is the 1st percentile and tells you that 1% of the data falls below this point. P50 represents the median, which means that half of the data points fall below this value and half above. Therefore, when we talk about percentiles, we help to understand the position of a particular score within a larger dataset.

Imagine a race with 100 runners. If you finish in the 50th percentile, it means that you are faster than 50 runners and slower than the rest. If you know your position in a race, it gives you a good understanding of how well you performed relative to others.

If you have secured 82 percentile in a management entrance examination, it means that your position is below 18 per cent of total candidates appeared in the examination. If a total of one lakh students appeared, where do you stand?

If you score in the 82nd percentile in an exam, it indicates that you performed better than 82% of the candidates who took the exam. To put it in perspective, if 100,000 students took the exam, this means you scored higher than 82,000 students, placing you below just 18,000 other students. This kind of information is very useful as it allows candidates to assess their performance relative to their peers.

Think about a competitive sports event like the Olympics. If you finish in the 82nd percentile for a specific swimming event, it means you did better than most swimmers and were ahead of a large number of competitors. It could motivate you to improve your skills even further, just like how knowing your rank in a competitive exam can guide your study efforts.

The method for locating the Quartile is the same as that of the median in case of individual and discrete series. The value of Q1 and Q3 of an ordered series can be obtained by the following formula where N is the number of observations. Q1 = size of (N + 1)/4 th item Q3 = size of 3(N+1)/4 th item.

To calculate quartiles, which divide the data into four equal parts, we use the same approach as finding the median. First, we need to sort the data in ascending order. Then we find Q1, the first quartile, which is the median of the first half of the data. Q3, the third quartile, is the median of the second half. The formulas involve finding specific positions in the sorted list based on the total number of observations, N.

Consider a classroom with a variety of exam scores. If you want to find out how students performed in the bottom 25% (Q1) and top 25% (Q3), calculating these quartiles helps you understand the performance range without focusing just on averages. It’s like observing not just how well a few top students did but also how many struggled.

Example 9 Calculate the value of lower quartile from the data of the marks obtained by ten students in an examination. 22, 26, 14, 30, 18, 11, 35, 41, 12, 32. Arranging the data in an ascending order, 11, 12, 14, 18, 22, 26, 30, 32, 35, 41. Q1 = size of (N +1)/4 th item = size of (10+1)/4 th item = size of 2.75th item = 2nd item + .75 (3rd item – 2nd item) = 12 + .75(14 –12) = 13.5 marks.

In this example, we have marks from ten students. First, we sort these scores, which gives us an ordered list. To find Q1, we use the formula to position it—here it's the 2.75th item. This means we take a fraction between the 2nd and 3rd item in the ordered list of data. By calculating this, we find the lower quartile (Q1) value to be 13.5 marks, indicating that 25% of students scored below this threshold.

Imagine you’re at a concert with over a hundred performers. Understanding who among them falls into the top 25% can help talent scouts focus on potential stars. The method of identifying these quartiles reflects insights that can guide future training and selections.

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