Table of contents

1. Intro to Stats and Collecting Data55m
2. Describing Data with Tables and Graphs1h 55m
3. Describing Data Numerically1h 45m
4. Probability2h 16m
5. Binomial Distribution & Discrete Random Variables2h 33m
6. Normal Distribution and Continuous Random Variables1h 38m
7. Sampling Distributions & Confidence Intervals: Mean1h 53m
8. Sampling Distributions & Confidence Intervals: Proportion1h 12m
- Sampling Distribution of Sample Proportion
  29m
- Confidence Intervals for Population Proportion
  42m
9. Hypothesis Testing for One Sample2h 19m
10. Hypothesis Testing for Two Samples3h 22m
11. Correlation1h 6m
12. Regression1h 4m
13. Chi-Square Tests & Goodness of Fit1h 20m
14. ANOVA1h 0m

3. Describing Data Numerically

Percentiles & Quartiles

Statistics

3. Describing Data Numerically

Percentiles & Quartiles

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2:04 minutes

Problem 2.5.38

Textbook Question

Finding and Interpreting Percentiles In Exercises 37– 40, use the data set, which represents wait times (in minutes) for various services at a state’s Department of Motor Vehicles locations.
6 10 1 22 23 10 6 7 2 1 6 6 2 4 14 15 16 4
19 3 19 26 5 3 4 7 6 10 9 10 20 18 3 20 10 13
14 11 14 17 4 27 4 8 4 3 26 18 21 1 3 3 5 5
Which wait time represents the 50th percentile? How would you interpret this?

Verified step by step guidance

Step 1: Organize the data set in ascending order. This is necessary to determine percentiles, as percentiles are based on the position of values in an ordered data set.

Step 2: Identify the formula for finding the position of the 50th percentile (median). The formula is: P = (n + 1) * (percentile / 100), where n is the total number of data points.

Step 3: Calculate the position of the 50th percentile using the formula. Substitute the total number of data points (n) and the desired percentile (50) into the formula.

Step 4: Locate the value at the calculated position in the ordered data set. If the position is a whole number, the value at that position is the 50th percentile. If the position is not a whole number, interpolate between the two closest values.

Step 5: Interpret the 50th percentile. The 50th percentile represents the median wait time, meaning that 50% of the wait times are less than or equal to this value, and 50% are greater than or equal to this value.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Percentiles

A percentile is a measure used in statistics to indicate the value below which a given percentage of observations fall. For example, the 50th percentile, also known as the median, is the value that separates the higher half from the lower half of the data set. Understanding percentiles helps in interpreting the distribution of data and identifying relative standings within a dataset.

Median

The median is a specific type of percentile that represents the middle value of a dataset when it is ordered from least to greatest. If there is an even number of observations, the median is calculated as the average of the two middle numbers. It is a robust measure of central tendency, less affected by outliers than the mean, making it useful for skewed distributions.

Interpreting Percentiles

Interpreting percentiles involves understanding what the percentile value signifies in the context of the data. For instance, if a wait time is at the 50th percentile, it means that 50% of the wait times are less than or equal to this value. This interpretation provides insights into the typical experience of individuals in the dataset, allowing for better decision-making and understanding of service efficiency.

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Related Practice

Textbook Question

True or False? In Exercises 7–10, determine whether the statement is true or false. If it is false, rewrite it as a true statement.

The second quartile is the mean of an ordered data set.

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Textbook Question

Using and Interpreting Concepts

Using and Interpreting Concepts Finding Quartiles, Interquartile Range, and Outliers In Exercises 11 and 12,

(a) find the quartiles

56 63 51 60 57 60 60 54 63 59 80 63 60 62 65

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Textbook Question

Using and Interpreting Concepts

Using and Interpreting Concepts Finding Quartiles, Interquartile Range, and Outliers In Exercises 11 and 12,

56 63 51 60 57 60 60 54 63 59 80 63 60 62 65

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Textbook Question

Finding a Percentile In Exercises 33–36, use the data set, which represents the ages of 30 executives.

43 57 65 47 57 41 56 53 61 54

56 50 66 56 50 61 47 40 50 43

54 41 48 45 28 35 38 43 42 44

Which ages are above the 75th percentile?

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Textbook Question

Extending Concepts

Midquartile Another measure of position is called the midquartile. You can find the midquartile of a data set by using the formula below.

Midquartile = (Q₁ + Q₃) / 2

In Exercises 55 and 56, find the midquartile of the data set.

5 7 1 2 3 10 8 7 5 3

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Textbook Question

A student’s test grade of 75 represents the 65th percentile of the grades. What percent of students scored higher than 75?

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Textbook Question

Interpreting Percentiles In Exercises 29–32, use the ogive, which represents the cumulative frequency distribution for quantitative reasoning scores on the Graduate Record Examination in a recent range of years. (Adapted from Educational Testing Service)

What score represents the 65th percentile? How should you interpret this?

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Multiple Choice

Answer the questions below using the data in the table.

(A) Find $P sub 50$

(B) A playlist with 15 songs is in which percentile?

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rank

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