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Chapter 3: Problem 24

Find the percentile rank for each test score in the data set. 5,12,15,16,20,21 What test score corresponds to the 33 rd percentile?

Short Answer

Expert verified
The test score corresponding to the 33rd percentile is approximately 12.93.

Step by step solution

01

Order the Data Set

First, ensure the data set is ordered from smallest to largest. The given data set is already ordered: 5, 12, 15, 16, 20, 21.
02

Calculate Percentile Rank Formula

We use the formula for the percentile rank: \( R = \frac{P}{100} \times (N+1) \), where \( R \) is the rank, \( P \) is the percentile, and \( N \) is the number of scores in the data set.
03

Apply the Formula for the 33rd Percentile

Plug \( P = 33 \) and \( N = 6 \) (since there are 6 scores) into the formula: \( R = \frac{33}{100} \times (6+1) = 2.31 \). Since 2.31 is not a whole number, we'll need to approximate between the 2nd and 3rd scores.
04

Linear Interpolation to Find Exact Percentile Value

The 2.31st position is between the 2nd score (12) and 3rd score (15). Use linear interpolation: \( 12 + (0.31) imes (15-12) = 12 + 0.93 = 12.93 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Percentile Rank
The percentile rank is a measure that tells us where a particular score stands relative to other scores in a data set. It's expressed as a percentage.
For example, a test score at the 33rd percentile means that 33% of the scores in the data set are below that score.
To calculate the percentile rank using our set of scores, we rely on a specific formula:
  • First, identify the percentile you鈥檙e interested in, say 33rd percentile as in our example.
  • Use the rank formula: \( R = \frac{P}{100} \times (N+1) \).
  • Here, \( P \) is the desired percentile (33 in this case), and \( N \) is the total number of observations (6 for our example).
Once we substitute these values into the formula, we can find the rank position of the desired percentile.
Importance of Data Set Ordering
Data set ordering involves arranging data in sequence, from smallest to largest value. This ordering is crucial when calculating percentiles or performing any sort of rank-based analysis.
If your data isn鈥檛 ordered, the results of rankings and percentiles won鈥檛 be accurate, leading to incorrect conclusions.
Here鈥檚 what to keep in mind:
  • Always check if the data set is organized correctly before starting any calculations.
  • In our original task, the data set (5, 12, 15, 16, 20, 21) was already ordered.
  • Ordered data helps ensure accurate and reliable analytic outputs.
Remember, without proper ordering, all further steps could yield incorrect positions or interpolations.
Using Linear Interpolation
Linear interpolation is a simple mathematical method that helps us estimate the precise value between two known values.
We use it when the calculated rank is not an integer, meaning it falls between two scores in our data set.
Here鈥檚 how it works for our example:
  • Identify the data points your calculated rank falls between. For a rank of 2.31, you find it lies between the 2nd score (12) and the 3rd score (15).
  • Calculate the proportion the rank occupies between these two points. For 2.31, it's 0.31 into the interval from 2 to 3.
  • Use the calculation: starting value + (proportion \( \times \) difference in values).
This yields a more accurate percentile value within your data, ensuring precision in your statistical assessments.
Grasping Elementary Statistics
Elementary statistics form the basis for understanding and interpreting data with context and clarity.
Key concepts include measures of central tendency, variability, and position, like means, medians, and percentiles.
These foundational elements allow students to build strong analytical skills. Here鈥檚 why understanding these basics is important:
  • They provide a language for describing data.
  • They allow you to make informed decisions based on data.
  • They lay the groundwork for more complex statistical concepts later on.
Those applying these basic statistical concepts, like percentiles, can perform more detailed data analyses with confidence and precision.

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Most popular questions from this chapter

The mean of the waiting times in an emergency room is 80.2 minutes with a standard deviation of 10.5 minutes for people who are admitted for additional treatment. The mean waiting time for patients who are discharged after receiving treatment is 120.6 minutes with a standard deviation of 18.3 minutes. Which times are more variable?

The data for a recent year show the taxes (in millions of dollars) received from a random sample of 10 states. Find the first and third quartiles and the IQR. \(\begin{array}{llllllllll}13 & 15 & 32 & 36 & 11 & 24 & 6 & 25 & 11 & 71\end{array}\)

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The mean of a distribution is 20 and the standard deviation is 2 . Use Chebyshev's theorem. a. At least what percentage of the values will fall between 10 and \(30 ?\) b. At least what percentage of the values will fall between 12 and \(28 ?\)

For these situations, state which measure of central tendency - mean, median, or mode-should be used. a. The most typical case is desired. b. The distribution is open-ended. c. There is an extreme value in the data set. d. The data are categorical. e. Further statistical computations will be needed. f. The values are to be divided into two approximately equal groups, one group containing the larger values and one containing the smaller values.
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