Chapter 3.3 – 3.4 Applications of the Standard Deviation and Measures of Relative Standing

Review of the Mean and Standard Deviation  The MEAN of a distribution is a measure (a number) that is used to describe the center of the distribution.  The STANDARD DEVIATION is a measure that helps describe the spread of the data values within the distribution.  In this section, we’ll use techniques to help INTERPRET the spread of data values within a distribution.

Empirical Rule  The empirical rule will only apply for a distribution that is approximately bell shaped (not skewed). There are 3 parts to the rule.  1. Approximately 68% of the data values will lie within 1 standard deviation of the mean. ( ) 68%

Empirical Rule  2. Approximately 95% of the data values will lie within 2 standard deviations of the mean. ( ) 68% 95%

Empirical Rule  3. Approximately all (99% to100%) of the data values will lie within 3 standard deviations of the mean. ( ) 68% 95% 99% - 100%

Example 3.22 Page 118  The results of a standardized achievement test are approximately bell-shaped with. According to the Empirical Rule, approximately what percentage of test scores would you expect to fall between:  265 to 435  180 to 520  95 to 605  If there were a test score equal to 625, could you consider such a score an outlier?

Draw a picture  In this problem, the mean is 350, so below the center of the bell curve, label 350.  Calculate.  Label the curve at with 265 and with

Continue to label the horizontal (x) axis  Calculate : 350

 According to the Empirical Rule, approximately what percentage of test scores would you expect to fall between:  265 to 435?  68%, since 265 to 435 marks 1 standard deviation below the mean to 1 standard deviation above the mean. 350

 What percentage would fall between 180 to 520?  95%, since 180 to 520 marks 2 standard deviations below the mean to 2 standard deviations above the mean. 350

 What percentage would fall between 95 to 605?  99% to 100%, since 95 to 605 marks 3 standard deviations below the mean to 3 standard deviations above the mean.  Would you consider a data value of 625 an outlier?  Yes, 625 is an outlier since it lies far from the other test scores. According to the empirical rule, data values that lie more than 3 standard deviations from the mean occur less than 1% of the time. 625 is a very rare score. 350

Use the Empirical Rule to Divide the Sections of the Bell Curve  1. Approximately 68% of the data values will lie within 1 standard deviation of the mean. ( ) 34% 68%

Use the Empirical Rule to Divide the Sections of the Bell Curve  2. Approximately 95% of the data values will lie within 2 standard deviations of the mean. ( ) 34% 68% 13.5% 95%

Use the Empirical Rule to Divide the Sections of the Bell Curve  3. Approximately all (99% to100%) of the data values will lie within 3 standard deviations of the mean. ( ) 34% 68% 13.5% 95% 2.5%

Exampleusing the Empirical Rule  For a data distributions with a mean of 25 and a sample standard deviation of 3, what percentage of the data values:  Lie between 22 and 28?  Lie between 22 and 31?  Lie between 19 and 25?

2.5% 13.5% 34%

3.4 - Measures of Relative Standing  A statistical measure that describes the position of a particular data value relative to the other value of the data set is called a Measure of Relative Standard.  In in a distribution of test scores out of 100, the position of a score of 99 would be far right of average.  However, if the distribution were describing IQ scores, a data value of 99 would be average.  A data value can have different interpretations based on its standing within a distribution.

Z-Score  The z-score of a data value indicates the number of standard deviations that the data value deviates from the mean.  Each data value of a distribution has its own z-score that indicates how far from the mean that data value lies.  For a data distributions with a mean of 25 and a sample standard deviation of 3:  The data value 28 has a z-score of 1 because it is 1 standard deviation away from the mean.  The data value 31 has a z-score of 2 because it is 2 standard deviations away from the mean.  The data value 22 has a z-score of -1 because it is 1 standard deviation below the mean.

Z-Score 2.5% 13.5% 34% Raw scores Z scores

Z Score  The formula for the z-score of a data value is:  For a sample  For a population

Example 3.25 page 123  Debbie earned a grade of 87 on her History exam, and a grade of 39 on her English exam. She would like to determine on which exam she did better, relative to the students in each class. How can this be accomplished using the information in Table 3.16?

Example 3.25 page 123  If we can find how many standard deviations each grade lies from the mean of the class, we can determine which grade is “better” relative to the rest of the class.  Better grades will be far above the mean; the number of standard deviations will be greater than that of lower grades.

Example 3.25 page 123  A z-score will give the number of standard deviations a score lies from the mean.  For the History grade,  For the English grade,

Debbie did better on the English exam because the z-score of 3 for the English exam is Greater than the z-score of 2 for the History exam.

Interpreting the Sign of a Z-Score

Detecting Outliers Using z Scores  A z-score can be used to identify an outlier since the farther the data value is from a z-score of z = 0, the more likely the data value is an outlier.  According to the Empirical Rule, any data value within an approximately bell-shaped distribution having a z-score less than − 3 or greater than +3 can be considered as an outlier since such a data value is considered an extreme data value. Z=0 Positive z scores Negative z scores

Baseball Z-Score Example  Determine which league leader had the best average relative to the league he plays in. Baseball Statistics League Batting Average League Standard Deviation Highest Individual Average American National

Baseball Z-Score Example  Z-score for the American league player: League Batting Average League Standard Deviation Highest Individual Average American National

Baseball Z-Score Example  Z-score for the National league player: League Batting Average League Standard Deviation Highest Individual Average American National

Baseball Z-Score Example  The Z-score for the American League player was higher than that of the National League player, therefore the American League player had the BEST average relative to the league he plays in.  Additionally, the American player’s z-score is greater than 3 standard deviations away from the mean…  This implies that score is an outlier.

Review: What is a z-score and how to find it  The z-score of a data value indicates the number of standard deviations that the data value deviates from the mean. The formula for the z-score of a data value is:  Each raw data value of a distribution has its own z-score that indicates how far from the mean that data value lies.

Converting Z-Scores to Raw Scores  Raw Score Formula:  For a sample, the raw score formula:  For a population, the raw score formula:

Example 3.29 page 127  On each of her student’s test papers, a statistics instructor writes the z score rather than the actual test score. The statistics for the test are:  Determine the student’s test score (to the nearest whole number) if the test grade: a) Has a z score of b) Is 2 standard deviations below the mean test grade. c) Has a z score of 0.

Example 3.29  Z Score of 1.65  A student with a z score of 1.65 has a test score (raw score) approximately equal to 88.  2 Standard deviations below the mean test grade.  A test grade 2 standard deviations below the mean has a z score of -2  A student with a test score of approximately 53 is 2 standard deviations below the mean.

Example 3.29  Z Score of 0  A student with a z score of 0 has a test score (raw score) equal to 72, the mean.

Percentile Rank  The percentile rank of a data value, say data value X, indicates the percent of data values within the distribution less than the data value X.  For example, consider an SAT verbal score of 687 with a percentile rank of 90.  This means that 90% of all the students who took the test scored below 687 and 10% of the students scored higher than 687.

Percentile vs Percentile Rank  Verbally, they are used in different ways.  Example:  “Exactly 62 percent of the students in a given school system received IQ scores less than 112.”  112 has a percentile rank of 62, and 112 is the 62 nd percentile.  “In a third grade class, 70% of the students have heights less than 54 inches.”  54 has a percentile rank of 70, and 54 is the 70 th percentile.

Deciles  The nine deciles which divide the data set into tenths or ten equal parts are:  The 10th percentile is called the 1st decile  The 20th percentile is called the 2nd decile  The 30th percentile is called the 3rd decile  The 40th percentile is called the 4th decile  The 50th percentile is called the 5th decile  The 60th percentile is called the 6th decile  The 70th percentile is called the 7th decile  The 80th percentile is called the 8th decile  The 90th percentile is called the 9th decile

Quartiles  The three quartiles which divide the data set into quarters or four equal parts are:  The 25th percentile is called the 1st quartile, Q1  The 50th percentile is called the 2nd quartile, Q2  The 75 percentile is called the 3 rd quartile, Q3

Deciles and Quartiles  The relationship between Deciles and Percentiles is shown here in figure 3.22 :  The relationship between Quartiles and Percentiles is shown in figure 3.23 :

Example 3.32  Some results from the mathematics reasoning part of the Scholastic Aptitude Test (SAT) are given in Table Using Table 3.20 determine:  a) The score which is the 1st decile.  320  b) The median.  474  c) The score which has a percentile rank of 90.  650  d) The percentile rank of the score 380, and state what quartile it represents.  PR of 380 is 25, it is the first quartile  e) The 3rd quartile.  570  f) The score which is the 6th decile.  510

5 Number Summary  A 5-number summary uses the following five numbers to describe a data set:  1. The smallest data value  2. The first quartile, Q1  3. The median, Q2  4. The third quartile, Q3  5. The largest data value

Get the 5 number summary from the calculator  Put the data in a list: 56, 23, 130, 12, 67, 34, 29, 78, 52, 46  Press STAT -> CALC -> 1 VAR STATS L1 -> Enter  Scroll down to get to the five number summary