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Statistics 101: Principles of Statistics11 chapters | 144 lessons | 9 flashcard sets

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Lesson Transcript

Instructor:
*Cathryn Jackson*

Cat has taught a variety of subjects, including communications, mathematics, and technology. Cat has a master's degree in education and is currently working on her Ph.D.

Percentiles are often used in academics to compare student scores. Finding percentiles in a data set can be a useful way to organize and compare numbers in a data set.

Holly just took her first math test in her college algebra class. Her professor says she scored in the 90th 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 90th percentile, then that means she scored at or better than 90% of her class. Since percentiles are based on a percentage, you will only see percentiles between the range of 0-100.

Defining percentiles can get a little muddy, especially when it comes to rounding. For that reason, we will stick to the definition above when solving the following problems. Let's look at different ways you can use and find percentiles.

Holly's professor posts a list of grades, without the names, on the blackboard. There are 15 students in the class and 15 grades on the board. The grades are:

85, 34, 42, 51, 84, 86, 78, 85, 87, 69, 94, 74, 65, 56, 97.

To find Holly's grade, we need to do the following steps:

- Multiply the total number of values in the data set by the percentile, which will give you the index.
- Order all of the values in the data set in ascending order (least to greatest).
- 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.
- 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.

This probably sounds a little confusing. Feel free to pause the video and work the examples with me!

Let's start with step number one. The total number of values in the data set is 15. 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, 15 * .90 = 13.5.

Okay, so I got 13.5. Now, according to step two, I need to order the grades from least to greatest:

34, 42, 51, 56, 65, 69, 74, 78, 84, 85, 85, 86, 87, 94, 97.

For step three I need to round my index up from 13.5 to 14. Next, I need to count from the smallest number up to the 14th number in the list. The 14th number in this list is 94. That tells us that Holly scored a 94 on her math test and only one person scored higher than she did.

Holly's friend, Dave, scored in the 80th 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: 15 * .80 = 12.

From this list we can see that the 12th number in this list is 86. However, because our index turned out to be a whole number, we need to take the 12th number and the 13th number and find the average: 86 + 87= 173 / 2 = 86.5.

From this information, we know that Dave scored at or better than 86.5. 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 86.5 to be in the 80th percentile.

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: (*k* + .5*r*) / *n* = *p*

First, we must consider Dave's rank as *x*. In this formula *k* = the number of teams below *x*, *r* = the number of teams equal to *x*, *n* = the number of teams and *p* = percentile.

The first variable is *k*, which is the number of teams below Dave's rank at third place. Since there are 35 teams in Dave's fantasy football league that means there are 32 teams that rank under Dave. Therefore, *k* = 32.

The next variable is *r*, which is the number of teams equal to Dave's rank at third place. Since there are no ties for third place, we can assume that only one team is in third place. Therefore, *r* = 1.

Finally, we know that *n* = 35 because there are 35 teams in the league. Now our formula looks like this: (32 + .5(1)) / 35 = *p*

Solve the equation: 32.5 / 35 = 92.8%

Dave's team is in the 93rd percentile. That means that he is doing better than 93% of the teams in his league.

Percentiles can be a bit confusing to learn. For this lesson, we defined percentiles as a measure that indicates what percent of the given population scored at or below the measure. We used two different methods of analyzing percentiles today. The first method required that you knew the values in the data set and the percentile of one of those values. By using the following steps, you can determine the value you are looking for:

- Multiply the total number of values in the data set by the percentile; this will give you the index.
- Order all of the values in the data set in ascending order (least to greatest).
- 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.
- 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.

The second method required that you knew the total number of values in the data set and the rank, or index, of one of those values. Using the following formula you can find the percentile of that value: (*k* + .5*r*) / *n* = *p*

In this formula, *k* = the number of values in the data set below the rank, *r* = the number of values in the data set equal to the rank and *n* = the number of values in the data set.

Because there are different definitions and interpretations of percentiles, don't get confused if a calculator or another resource gives you a different answer when you are finding percentiles. Unfortunately, in mathematics there is no one strict definition. However, these two methods will help you out greatly when you need to find percentiles in the future!

Once you complete this lesson you should be able to:

- Define percentile and index
- Find a score based on the percentile rank and a score series
- Demonstrate how to find a percentile rank based on a score and the number of scores in the data set

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Statistics 101: Principles of Statistics11 chapters | 144 lessons | 9 flashcard sets

- What is the Center in a Data Set? - Definition & Options 5:08
- Mean, Median & Mode: Measures of Central Tendency 6:00
- How to Calculate Mean, Median, Mode & Range 8:30
- Calculating the Mean, Median, Mode & Range: Practice Problems 7:13
- Visual Representations of a Data Set: Shape, Symmetry & Skewness 5:22
- Unimodal & Bimodal Distributions: Definition & Examples 5:29
- The Mean vs the Median: Differences & Uses 6:30
- Spread in Data Sets: Definition & Example 7:51
- Maximums, Minimums & Outliers in a Data Set 4:40
- Quartiles & the Interquartile Range: Definition, Formulate & Examples 8:00
- Finding Percentiles in a Data Set: Formula & Examples 8:25
- The Effect of Linear Transformations on Measures of Center & Spread 6:16
- Population & Sample Variance: Definition, Formula & Examples 9:34
- Ordering & Ranking Data: Process & Example 6:54
- Go to Summarizing Data

- Go to Probability

- Go to Sampling

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