Find the following for the Weight of Football Players x Sample standard deviation n=
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3-3 Measures of Position Z-score Percentile Quartile Outlier Bluman, Chapter 3 3
Measures of Position: Z-score A z-score or standard score for a value is obtained by following formulas: z X s X A z-score represents the number of standard deviations a value is above or below the mean. z X Bluman, Chapter 3 4
Rounding Rule for Z score Round the Z score to 2 decimal places.
Chapter 3 Data Description Section 3-3 Example 3-29 Page #142 Bluman, Chapter 3 6
Example 3-29: Test Scores A student scored 65 on a calculus test that had a mean of 50 and a standard deviation of 10; she scored 30 on a history test with a mean of 25 and a standard deviation of 5. Compare her relative positions on the two tests. X X 65 50 z 1.5 Calculus s 10 X X 30 25 z 1.0 History s 5 She has a higher relative position in the Calculus class. Bluman, Chapter 3 7
Percentiles Percentiles divides the data into 100 equal groups. Percentiles are symbolized by P 1, P 2, P 3,,P 99 Please take a look at data on page 135 of your text.
Formula Percentile formula: Number of values below X Total # of values 0. 5 100%
Check this website out! There are variation on the formula for Percentiles. The website below does a fantastic job of displaying and explaining the variations. http://www.regentsprep.org/regents/mat h/algebra/ad6/quartiles.htm
Definition 1: A percentile is a measure that tells us what percent of the total frequency scored at or below that measure. A percentile rank is the percentage of scores that fall at or below a given score. Formula: To find the percentile rank of a score, x, out of a set of n scores, where x is included: Definition 2: A percentile is a measure that tells us what percent of the total frequency scored below that measure. A percentile rank is the percentage of scores that fall below a given score. Where B = number of scores below x E = number of scores equal to x n = number of scores
About Percentile Ranks: Percentile rank is a number between 0 and 100 indicating the percent of cases falling at or below that score. Scores are divided into 100 equally sized groups. Percentile ranks are usually written to the nearest whole percent: 74.5% = 75% = 75 th percentile.
About Percentile Ranks: Scores are arranged in rank order from lowest to highest There is no 0 percentile rank-the lowest score is the 1 st percentile. There is no 100 th percentile- the highest score is the 99 th percentile. You can t perform the same mathematical operations on percentile that you can on raw scores. You can t, for example, compute the mean of percentile scores.
Example 3-31 The frequency distribution for the systolic blood pressure readings (mm of mercury) of 200 randomly selected college students is shown here. Construct a percentile graph
A B C D Class Boundaries Frequency Cumulative Freq Cumulative Percent 89.5-104.5 24 104.5-119.5 62 119.5-134.5 72 134.5-149.5 26 149.5-164.5 12 164.5-179.5 4 200
Measures of Position: Example of a Percentile Graph Bluman, Chapter 3 16
Chapter 3 Data Description Section 3-3 Example 3-32 Page #147 Bluman, Chapter 3 17
Example 3-32 A teacher gives a 20-point test to 10 students. The scores are shown here. 18,15,12,6,8,2,3,5,20,10 Find the percentile rank of a score of 12? 6?
Example 3-32: Test Scores A teacher gives a 20-point test to 10 students. Find the percentile rank of a score of 12. 18, 15, 12, 6, 8, 2, 3, 5, 20, 10 Sort in ascending order. 2, 3, 5, 6, 8, 10, 12, 15, 18, 20 Percentile 6 values # of values below X 0.5 100% total # of values 6 0.5 100% 10 65% A student whose score was 12 did better than 65% of the class. Bluman, Chapter 3 19
Refer to the same data Find the value of the 25 th percentile? 60 th? If the percentile is known, then the formula is c n p 100 See page 149 for full explanation
Example 3-34: Test Scores A teacher gives a 20-point test to 10 students. Find the value corresponding to the 25 th percentile. 18, 15, 12, 6, 8, 2, 3, 5, 20, 10 Sort in ascending order. 2, 3, 5, 6, 8, 10, 12, 15, 18, 20 c n p 100 10 25 2.5 3 100 This is the location of data. i.e. 3 rd value. The value 5 corresponds to the 25 th percentile. Bluman, Chapter 3 21
Example 3-35: Test Scores A teacher gives a 20-point test to 10 students. Find the value corresponding to the 60 th percentile. 18, 15, 12, 6, 8, 2, 3, 5, 20, 10 Sort in ascending order. 2, 3, 5, 6, 8, 10, 12, 15, 18, 20 c = np 100 = 10 60 100 = 6 The answer is between the 6 th and 7 th data point. The value 11 corresponds to the 60 th percentile. Bluman, Chapter 3 22
Measures of Position: Deciles Deciles separate the data set into 10 equal groups. D 1 =P 10, D 4 =P 40 Please Pages 150-151 for further instructions. Bluman, Chapter 3 23
Measures of Position: Quartiles Quartiles separate the data set into 4 equal groups. Q 1 =P 25, Q 2 =MD, Q 3 =P 75 Q 2 = median(low,high) Q 1 = median(low,q 2 ) Q 3 = median(q 2,High) The Interquartile Range, IQR = Q 3 Q 1. Bluman, Chapter 3 24
Chapter 3 Data Description Section 3-3 Example 3-36 Page #150 Bluman, Chapter 3 25
Example 3-36: Quartiles Find Q 1, Q 2, and Q 3 for the data set. 15, 13, 6, 5, 12, 50, 22, 18 Sort in ascending order. 5, 6, 12, 13, 15, 18, 22, 50 6 12 Q 1 median Low, MD 9 2 1315 Q 2 median Low, High 14 2 18 22 Q 3 median MD, High 20 2 Bluman, Chapter 3 26
Measures of Position: Outliers An outlier is an extremely high or low data value when compared with the rest of the data values. A data value less than Q 1 1.5(IQR) or greater than Q 1 + 1.5(IQR) can be considered an outlier. Please refer to page 152 of your test for the procedure of how to identify the outliers. Bluman, Chapter 3 27
Homework Read section 3-3 and study the relevant examples. Page 153 #1-8 all, 11-31 odds