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Cauchon, Dennis, Paul Overberg. “Census data shows minorities now a majority of U.S. births.” USA Today, 2012. Available online at http://usatoday30.usatoday.com/news/nation/story/2012-05-17/minority-birthscensus/55029100/1 (accessed April 3, 2013).
Data from the United States Department of Commerce: United States Census Bureau. Available online at http://www.census.gov/ (accessed April 3, 2013).
“1990 Census.” United States Department of Commerce: United States Census Bureau. Available online at http://www.census.gov/main/www/cen1990.html (accessed April 3, 2013).
Data from San Jose Mercury News .
Data from Time Magazine ; survey by Yankelovich Partners, Inc.
The values that divide a rank-ordered set of data into 100 equal parts are called percentiles. Percentiles are used to compare and interpret data. For example, an observation at the 50 ^{th} percentile would be greater than 50 percent of the other obeservations in the set. Quartiles divide data into quarters. The first quartile ( Q _{1} ) is the 25 ^{th} percentile,the second quartile ( Q _{2} or median) is 50 ^{th} percentile, and the third quartile ( Q _{3} ) is the the 75 ^{th} percentile. The interquartile range, or IQR , is the range of the middle 50 percent of the data values. The IQR is found by subtracting Q _{1} from Q _{3} , and can help determine outliers by using the following two expressions.
$i=\left(\frac{k}{100}\right)\left(n+1\right)$
where i = the ranking or position of a data value,
k = the kth percentile,
n = total number of data.
Expression for finding the percentile of a data value: $\left(\frac{x\text{+}0.5y}{n}\right)$ (100)
where x = the number of values counting from the bottom of the data list up to but not including the data value for which you want to find the percentile,
y = the number of data values equal to the data value for which you want to find the percentile,
n = total number of data
Listed are 29 ages for Academy Award winning best actors in order from smallest to largest.
18; 21; 22; 25; 26; 27; 29; 30; 31; 33; 36; 37; 41; 42; 47; 52; 55; 57; 58; 62; 64; 67; 69; 71; 72; 73; 74; 76; 77
Listed are 32 ages for Academy Award winning best actors in order from smallest to largest.
18; 18; 21; 22; 25; 26; 27; 29; 30; 31; 31; 33; 36; 37; 37; 41; 42; 47; 52; 55; 57; 58; 62; 64; 67; 69; 71; 72; 73; 74; 76; 77
Jesse was ranked 37 ^{th} in his graduating class of 180 students. At what percentile is Jesse’s ranking?
Jesse graduated 37 ^{th} out of a class of 180 students. There are 180 – 37 = 143 students ranked below Jesse. There is one rank of 37.
x = 143 and y = 1. $\frac{x+0.5y}{n}$ (100) = $\frac{143+0.5(1)}{180}$ (100) = 79.72. Jesse’s rank of 37 puts him at the 80 ^{th} percentile.
On an exam, would it be more desirable to earn a grade with a high or low percentile? Explain.
Mina is waiting in line at the Department of Motor Vehicles (DMV). Her wait time of 32 minutes is the 85 ^{th} percentile of wait times. Is that good or bad? Write a sentence interpreting the 85 ^{th} percentile in the context of this situation.
When waiting in line at the DMV, the 85 ^{th} percentile would be a long wait time compared to the other people waiting. 85% of people had shorter wait times than Mina. In this context, Mina would prefer a wait time corresponding to a lower percentile. 85% of people at the DMV waited 32 minutes or less. 15% of people at the DMV waited 32 minutes or longer.
In a survey collecting data about the salaries earned by recent college graduates, Li found that her salary was in the 78 ^{th} percentile. Should Li be pleased or upset by this result? Explain.
In a study collecting data about the repair costs of damage to automobiles in a certain type of crash tests, a certain model of car had $1,700 in damage and was in the 90 ^{th} percentile. Should the manufacturer and the consumer be pleased or upset by this result? Explain and write a sentence that interprets the 90 ^{th} percentile in the context of this problem.
The manufacturer and the consumer would be upset. This is a large repair cost for the damages, compared to the other cars in the sample. INTERPRETATION: 90% of the crash tested cars had damage repair costs of $1700 or less; only 10% had damage repair costs of $1700 or more.
The University of California has two criteria used to set admission standards for freshman to be admitted to a college in the UC system:
Suppose that you are buying a house. You and your realtor have determined that the most expensive house you can afford is the 34 ^{th} percentile. The 34 ^{th} percentile of housing prices is $240,000 in the town you want to move to. In this town, can you afford 34% of the houses or 66% of the houses?
You can afford 34% of houses. 66% of the houses are too expensive for your budget. INTERPRETATION: 34% of houses cost $240,000 or less. 66% of houses cost $240,000 or more.
Use [link] to calculate the following values:
First quartile = _______
Second quartile = median = 50 ^{th} percentile = _______
4
Third quartile = _______
Interquartile range ( IQR ) = _____ – _____ = _____
6 – 4 = 2
10 ^{th} percentile = _______
70 ^{th} percentile = _______
6
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