percentiles and normal distribution. percentiles there are 2 definitions for a percentile. a value x...
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Ch 11 – Probability & Statistics
Percentiles and Normal Distribution
PercentilesThere are 2 definitions for a percentile.
A value x is in the nth percentile if x is ABOVE OR EQUAL to n% of the values.A value x is in the nth percentile if x is ABOVE n% of the values.
Neither the state (CRT tests) nor the AP Statistics people can decide on which one is best to use. So, be prepared to calculate both!
Use this one for homework
Look at the following data set, representing test scores.23, 24, 26, 27, 28, 28, 29, 31, 37, 39, 40, 48, 50
Suppose you scored the 27 and wanted to know what your percentile was.
Definition #1: Definition #2:
The dilemma of two definitions is apparent, but as the number of data values increases, the difference diminishes.
percentileth8.30308.013
4
percentilest1.23231.013
3
Where is the 30th percentile?
012345678
9.9 -19.9
19.9 -29.9
29.9 -39.9
39.9 -49.9
49.9 -59.9
0.000.050.100.150.200.250.300.350.40
9.9 -19.9
19.9 -29.9
29.9 -39.9
39.9 -49.9
49.9 -59.9
• This histogram shows the frequency on the left column.
• Since there are 19 total items and we are looking for the 30th percentile 19 x 0.30 = 5.7
• The 5.7 item occurs somewhere in the 2nd interval (19.9-29.9).
• This histogram shows the percentages on the left column.
• Here we simply need to count up the percents. We’re looking for the 30th, so there are 20% in the first interval, and another in the second, so 30% must be below (or equal to) some number within the 2nd interval (19.9- 29.9).
A frequency distribution shows how data are spread out over the range of values.
Median: 28Mean: 30.3Standard Deviation: 19
9th Grade ScoresScores Frequency
-19 – -10 1
-9 – 0 14
1 – 10 47
11 – 20 71
21 – 30 106
31 – 40 80
41 – 50 57
51 – 60 27
61 – 70 11
71 – 80 4
81 – 90 1
91 – 100 3
101 – 110 4
2008 Utah State Math Contest Results - 9th Grade
0
20
40
60
80
100
120
Scores
Fre
qu
en
cy
A bell curve is a symmetric curve that is the general shape of the graph of a normal distribution, which indicates that the frequencies are concentrated around the center portion of the distribution. Normal distribution is a data distribution that gives a bell-shaped, symmetric graph.
mean
68%
99.8%
95%
The SAT is designed so that scores are normally distributed with a mean of 500 and a standard deviation of 100.
A. What percent of SAT scores are between 400 and 600?
B. What is the probability that an SAT score is above 600?
C. What is the probability that an SAT score is less than 300 or greater than 700?
34% + 34% = 68%
50% - 34% = 16%
50% - 47.5% = 2.5% 2(2.5%) = 5%
mean
68%
99.8%
95%