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NAME:
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INTEGRATED ALGEBRA 1
MR. THOMPSON
Li LESSON 16.
QUARTILES, PERCENTILES, AND CUMULATIVE FREQUENCY
HOMEWORK ASSIGNMENT # 127: PAGES 707-710: # 2 - 18 EVENS
EXAMPLE I
Find the five statistical summary for the following set of data:
8, 5, 12, 9, 6, 2, 14, 7, 10, 17, 11, 8, 14, 5
Solution (1) Arrange the data in numerical order: 2, 5, 5, 6, 7, 8, 8, 9,10, 11, 12, 14, 14, 17 We can see that 2 is the minimum value and 17 is the maximum value.
(2) Find the median. Since there are 14 data values in the set, the median is the average of the 7th and 8th values.
Median = 8 1- 9 = 8. Therefore, 8.5 is the second quartile.
(3) Find the first quartile. There are seven values less than 8 5 The middle value is the 4th value from the lower end of the set of data, 6. Therefore, 6 is the first, or lower, quartile
(4) Find the third quartile. There are seven values greater than 8.5. The middle value is the 4th value from the upper end of the set of data, 12. Therefore, 12 is the third, or upper, quartile.
Answer The minimum is 2, first quartile is 6, the second quartile is 8.5, the third quartile is 12, and the maximum is 17.
Note: The quartiles 6, 8.5, and 12 separate the data values into four equal parts even though the original number of data values, 14, is not divisible by 4:
The first and third quartile values, 6 and 12, are data values. If we think of each of these as a half data value in the groups that they separate, each group con- tains 31 data values, which is 25% of the total
Stem (^) Leaf 3.0 0 2.9 8 2.8 (^02) 2.7 (^59) 2.6 5 2.5 (^0357) (^14 ) 2.3 0
Key: 2.91 8 = $2.
EXAMPLE 2 gr706E'VMAIA4ggOigK*Agtcrar.
Find the percentile rank of 87 in the following set of 30 marks: 56, 65, 65, 67, 72, 73, 75, 77, 77, 78, 78, 78, 80, 80, 80, 82, 83, 85, 85, 85, 86, 87, 87, 87, 88, 90, 92, 93, 95, 98
Solution (1) Find the sum of the number of marks less than 87 and half of the number of 87's: Number of marks less than 87 = 21 Half of the number of 87's (0.5 X 3) =
(2) Divide the sum by the total number of marks: 965 0.
(3) Change the decimal value to a percent: 0.75 = 75%.
Answer: A mark of 87 is at the 75th percentile.
Note: 87 is also the upper quartile mark.
EXAMPLE 3
A reporter for the local newspaper is preparing an article on the ice cream stores in the area. She listed the following prices for a two-scoop cone at 15 stores.
$2.48, $2.57, $2.30, $2.79, $2.25, $3.00, $2.82, $2.75, $2.55, $2.98, $2.53, $2.40, $2.80, $2.50, $2.
a.List the data in a stem-and-leaf diagram. b.Find the median. c.Find the first and third quartiles. d.Construct a box-and-whisker plot. e.Draw a cumulative frequency histogram.
E Find the percentile rank of a price of $2.75.
Solution (^) a. (^) The first two digits in each price will be the stem. The lowest price is $2 25 and the highest price is $3.00.
- (^) Since there are 15 prices, the median is the 8th from the top or from the bottom. The median is $2.57. c. The middle value of the set of numbers below the median is the first quartile. That price is $2.48.
The middle value of the set of numbers above the median is the third quartile. That price is $2.80.
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b. Is it possible to determine the percentile rank of a given score if the set of scores is shown on a cumulative frequency histogram? Explain why or why not.
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Developing Skills In 3-6, for each set of data: a. Find the five numbers of the statistical summary b. Draw a box-and- whisker plot.
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In 7-9, data are grouped into tables. For each set of data: a. Construct a cumulative frequency histogram. b. Find the interval in which the lower quartile lies.
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s.) Find the interval in which the median lies.
d. Find the interval in which the upper quartile lies.
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d. What percent of scores are 17 or less?^ e. In what interval is the 25th percentile?
c. Find the first and third quartiles.
Applying Skills
13. (^) A journalism student was doing a study of the readability of the daily newspaper. She chose several paragraphs at random and listed the number of letters in each of 88 words. She pre- pared the following chart. a. (^) Copy the chart, adding a column that lists the cumulative frequency b.^ Find the median.
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In 17 and 18, select, in each case, the numeral preceding the correct answer.
- On a standardized test, Sally scored at the 80th percentile. This means that (1)Sally answered 80 questions correctly. (2)Sally answered 80% of the questions correctly. (3)Of the students who took the test, about 80% had the same score as Sally. [(4)Of the students who took the test, at least 80% had scores that were less than or equal to Sally's score.
