Perhaps the most important measure of location is the mean (average). Sample mean: where n = sample size. Arrange the values from smallest to largest:

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1 1 Chapter 3 - Descriptive stats: Numerical measures 3.1 Measures of Location Mean Perhaps the most important measure of location is the mean (average). Sample mean: where n = sample size Example: The number of students per class is as follows: The mean is: Median The median is another measure of location for a variable. The median is the value in the middle when the data are arranged in ascending order (smallest to largest value). Computation: o Arrange the data in ascending order (smallest to largest value) o For an odd number of observations, the median is the middle value o For an even number of observations, the median is the average of the middle 2 values Example: The number of students per class is as follows: The median is: Arrange the values from smallest to largest: Middle value = Median = 46 Copyright Reserved 1

2 2 Example The yearly income (R1000 s) of 8 workers is as follows: Calculate the mean and the median. Answers: Mean/average: Median: For the median, we arrange the values from smallest to largest: Mode Median = Although the mean is the more commonly used measure of central location, in some situations the median is preferred. The mean is influenced by extremely small and large data values, while the median is not influenced by extreme values. Definition: The mode is the value that occurs with greatest frequency. Example: The number of students per class is as follows: The mode is: 46 Note: Bi-modal: If the data have exactly 2 modes. Example of a bi-modal data set: Multimodal: If data have more than 2 modes. Copyright Reserved 2

3 3 Example: Give the appropriate measure of location for the following data: Soft drink Frequency Coke Classic 19 Diet Coke 8 Dr. Pepper 5 Pepsi-Cola 13 Sprite 5 The mode is: Coke Classic For this type of data it obviously makes no sense to speak of the mean or median. Using Microsoft Excel 2007 to compute the mean, median and mode Formula worksheet Value worksheet Copyright Reserved 3

4 4 Percentiles Definition: The p th percentile is a value such that at least p percent of the observations are less than or equal to this value and at least (100 p) percent of the observations are greater than or equal to this value. Calculating the p th percentile: Arrange the data in ascending order (smallest to largest value) Compute an index i Example: ( ) where p = percentile of interest n = sample size (a) If i is not an integer, round up (b) If i is an integer, the p th percentile is the average of the values in positions i and (i +1) Determine the 85 th percentile ( ) for the starting salary data: Step 1: Arrange the data in ascending order Step 2: ( ) ( ) ( ) Step 3: In the 11 th position (after being arranged in ascending order):. Interpretation: 85% of the graduates have a starting salary of R3 730 or less. Copyright Reserved 4

5 5 Determine the 33 rd percentile ( starting salary: ) for the Determine the median ( salary: ) for the starting Step 1: Arrange the data in ascending order Step 1: Arrange the data in ascending order Step 2: ( ) ( ) ( ) Step 2: ( ) ( ) ( ) i + 1 = 7 Step 3: In the 4 th position (after being arranged in ascending order):. Step 3: The median is the average of the values in the 6 th and 7 th positions: Interpretation: 33% of the graduates have a starting salary of R3 480 or less. Interpretation: 50% of the graduates have a starting salary of R3 505 or less. Copyright Reserved 5

6 6 Determine the 25 th percentile ( starting salary: ) for the Determine the 75 th percentile ( starting salary: ) for the Step 1: Arrange the data in ascending order Step 1: Arrange the data in ascending order Step 2: ( ) ( ) ( ) i + 1 = 4 Step 2: ( ) ( ) ( ) i + 1 = 10 Step 3: is the average of the values in the 3 rd and 4 th positions: Step 3: is the average of the values in the 9 th and 10 th positions: Interpretation: 25% of the graduates have a starting salary of R3 465 or less. Interpretation: 75% of the graduates have a starting salary of R3 600 or less. Copyright Reserved 6

7 7 Quartiles First quartile, 25 th percentile Second quartile, 50 th percentile, median Third quartile, 75 th percentile 3.2 Measures of variability Range Range = Largest Value Smallest Value Range Example of the salary data. The range is: = = 615 Advantages: o Easy to calculate Disadvantages: o It s sensitive to just 2 data values: the Largest Value and the Smallest Value. o Unstable, it is influenced by extreme values. Suppose one of the graduates received a starting salary of per month. Then the range is equal to: The range is: = = Copyright Reserved 7

8 8 Interquartile Range - IQR It s the range for the middle 50% of the data Example of the salary data. The interquartile range for the salary data is: Advantages: o Easy to interpret o Is not influenced by extreme values Disadvantages: o It s only based on the middle 50% of the data. Variance The variance is a measure of variability that utilizes all the data Example Given: The Sample Variance ( ) Standard Deviation Sample Standard Deviation and therefore ( ) Copyright Reserved 8

9 9 Example Calculate the standard deviation of the class sizes. Number of students in class ( ) Mean class size ( ) Deviation about the mean ( ) Squared deviation about the mean ( ) ( ) ( ) ( ) and OR ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) and Interpretation: The average deviation of the class sizes from the average class size (44) is 8 students. Coefficient of Variation It s a relative measure of variability It measures the standard deviation relative to the mean Coefficient of Variation: The coefficient of variation tells us that the sample standard deviation is a % of the value of the sample mean. Copyright Reserved 9

10 10 Example: The class test mark (out of 10) and the semester test mark (out of 50) of 5 students are investigated. Class test (out of 10) Semester test (out of 50) Average of class test marks = 6 Average of semester test marks = 26 Variance of class test marks = 2.5 Variance of semester test marks = Which test has the biggest relative variation? Calculate the relevant numerical measures. Coefficient of variation for the class test marks: Coefficient of variation for the semester test marks: Therefore, the semester test has the biggest relative variation. Using Microsoft Excel s 2007 Descriptive Statistics Tool Self-study (see page 115) 3.3 Measures of Distribution Shape, Relative Location and Detecting Outliers Distribution Shapes Read through by yourself. z- Scores z - Scores: The z -score is called the standardized value. It can be interpreted as the number of standard deviations x is from the mean. Copyright Reserved 10

11 11 Example: z -scores of the class sizes dataset. (We calculated the mean and standard deviation previously: and s = 8). Number of students in class ( ) Deviation about the mean ( ) z-score ( ) Interpretation: 54 is 1.25 standard deviations above the mean. 32 is 1.5 standard deviation below the mean. Example: The Mathematics marks of 2 students are compared. Student 1 75% (in School A) Student 2 80% (in School B) Which one has done the best, relatively to his school? Student 1: School s A B Student 1 s mark is 2.5 standard deviations above the mean. Student 2: Student 2 s mark is exactly the same value as the mean. Conclusion: Student 1 has done relatively better in his school than Student 2. Copyright Reserved 11

12 12 Chebyshev s Theorem Not for examination Empirical Rule Empirical Rule: 68% of the data values will be within 1 std dev of. 95% of the data values will be within 2 std dev of. 100% of the data values will be within 3 std dev of. Copyright Reserved 12

13 13 Example of the application of the empirical rule: Suppose IQ scores have a bell-shaped distribution with a mean of 100 and a standard deviation of 15. a) What percentage of people should have an IQ score between 85 and 115? Answer = 68% b) What percentage of people should have an IQ score between 70 and 130? Answer = 95% c) What percentage of people should have an IQ score of more than 130? Answer = 2.5% 100% - 95% = 5% and = 2.5% Copyright Reserved 13

14 14 d) The 16 th percentile ( ) is equal to: 100% - 68% = 32% and = 16%. Therefore, P 16 = 85. e) The 84 th ( ) percentile is equal to: 16% + 68% = 84%. P 84 = 115 f) Is a person with an IQ score of 160 seen as an outlier? Yes, since approximately 100% of the values are between 55 and 145, an IQ score of 160 is seen as an outlier. OR > 3 (see the next Section on outliers). Copyright Reserved 14

15 15 Detecting Outliers Sometimes a data set will have one or more observations with unusually large or unusually small values. Extreme values are called outliers. Standardized values (z-scores) can be used to identify outliers. In the case of a bell-shaped distribution, the following rule can be applied: Since 100% of the data will be within 3 std dev of the mean, we recommend treating any data value with a (z-score <-3) OR a (z score >3) as an outlier. 3.4 Exploratory Data Analysis Five-Number Summary The following 5 numbers are used to summarize the data: 1. Smallest Value 2. First Quartile ( ) 3. Second Quartile ( ) 4. Third Quartile ( ) 5. Largest Value The five-number summary of the salary data is: Smallest value = 3310 Largest value = 3925 (Median) (These values have been calculated previously). Copyright Reserved 15

16 16 Box Plot A box plot is a graphical summary of data that is based on a five-number summary. A box plot provides another way to identify outliers. Upper limit = Q 3 + (1.5)(IQR) = (1.5)(135) = Lower limit = Q 1 - (1.5)(IQR) = (1.5)(135) = If a point falls above the upper limit or below the lower limit, the point is seen as an outlier. Copyright Reserved 16

17 17 Box-plots and skewness: The median is in the middle of the box, indicating symmetry. The median is not centered in the middle of the box. The median is closer to, indicating that the shape of the distribution is skewed to the right. The median is not centered in the middle of the box. The median is closer to, indicating that the shape of the distribution is skewed to the left. Skewness: Skewed to the left (negative skew): The left tail is longer; the mass of the distribution is concentrated on the right of the figure. It has relatively few low values. Skewed to the right (positive skew): The right tail is longer; the mass of the distribution is concentrated on the left of the figure. It has relatively few high values. Symmetric Note: A normal distributions is symmetric Copyright Reserved 17

18 Measures of association between two variables Covariance Sample Covariance: Measure of the linear relationship between x and y. ( )( ) Note: Positive linear relationship Negative linear relationship No linear relationship Note: (Not in the textbook) ( )( ) ( ) where denotes the sample variance of the x observations. Similarly: ( )( ) ( ) where denotes the sample variance of the y observations. Calculations for the variance and standard deviation of x, the variance and standard deviation of y and the covariance between x and y: x y ( ) ( ) ( ) ( ) ( )( ) and Copyright Reserved 18

19 19 1. Calculate the variance and the standard deviation of x: ( ) and 2. Calculate the variance and the standard deviation of y: ( ) and 3. Calculate and interpret the covariance between x and y: ( )( ). There is a positive linear relationship between x and y. Copyright Reserved 19

20 y y 20 Interpretation of sample covariance 25 A positive linear relationship x 25 A negative linear relationship x Correlation Coefficient To measure the strength of the linear relationship between x and y. ( )( ) Strong positive linear relationship between x and y. where Sample covariance between x and y. Sample standard deviation of x. Sample standard deviation of y. Copyright Reserved 20

21 21 Interpretation of the Correlation Coefficient Measures the linear relationship between x and y i. Positive linear relationship Perfect positive linear relationship ii. Negative linear relationship Perfect negative linear relationship iii. Non-linear relationship Strong negative linear relationship between x and y Weak negative linear relationship between x and y Weak positive linear relationship between x and y Strong positive linear relationship between x and y No linear relationship between x and y Copyright Reserved 21

22 22 Using Microsoft Excel 2007 to compute the covariance and correlation coefficient Formula worksheet: Value worksheet: Note: We have to adjust the Excel result of 9.9 for the covariance, since the COVAR function in Excel calculates the population covariance. = sample covariance = population covariance ( ) ( ) Copyright Reserved 22

23 23 Homework (work through the following example on your own): The class test mark (out of 10) (x) and the semester test mark (out of 50) (y) of 5 students are investigated. Class test (out of 10) (x) Semester test (out of 50) (y) (a) Calculate the mean mark and the variance for the class test: and ( ) ( ) ( ) ( ) ( ) ( ). (b) Calculate the mean mark and the variance for the semester test: and ( ) ( ) ( ) ( ) ( ) ( ). (c) Calculate and interpret the standard deviation for the semester test:. The average deviation of the semester test marks from the average ( ) is (d) Calculate and interpret the covariance: Answer: ( )( ) x y ( ) ( ) ( )( ) There is a positive linear relationship between x and y. (e) Calculate and interpret the correlation coefficient:. There is a strong positive linear relationship between x and y. (f) Suppose a student obtained 6/10 for the class test and 30/50 for the semester test. In which test did the student perform the best, relative to the other students? and test, relative to the other students.. The student performed the best in the semester Copyright Reserved 23

24 The weighted mean and working with grouped data Weighted Mean Example Consider the following sample of 5 purchases of raw material Purchase Cost per pound ($) Number of pounds Question: The mean cost per pound for the raw material? The weighted mean: ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) Example: The net full supply capacity (FSC) (in millions of cubic metres) in the various regions and catchment areas in South Africa, and also the percentage content as on 31 August 1992 are given in the table below. Region/catchment area FSC % content Vaaldam Bloemhofdam Sterkfonteindam Question: Calculate the weighted mean for the % content in the catchment area: ( )( ) ( )( ) ( )( ) Copyright Reserved 24

25 25 Grouped data The audit times for 20 clients were as follows: Audit times (in days) Frequency Class Midpoint Sample mean for grouped data: The midpoint for class i The frequency for class i ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) Sample variance for grouped data: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = 30 The standard deviation: Copyright Reserved 25

26 26 Homework (go through this example on your own) Automobiles traveling on a road that has a posted speed limit of 55 miles per hour are checked for speed by a state police radar system. Following is a frequency distribution of speeds. Speed (miles per hour) (a) Calculate the average speed of the automobiles. (b) Calculate the variance and the standard deviation ( ) Copyright Reserved 26

27 27 Typical exam questions: The annual amounts (in $ millions) spent on research and development for a random sample of 30 electronic component manufacturers are given in the following Excel spreadsheets. By using the Sort-option in Excel the data set is sorted according to the amount spent. Unsorted Sorted Annual amounts (in $ millions) for electronic component manufacturers has a bell-shaped distribution with a mean of 20 and a standard deviation of 7. Question 1 The range is: Answer 1 Range = x max x min = 38 6 = 32. Question 2 The median is: Answer 2 ( ) ( ). We need to take the average of the values in the 15 th and 16 th positions. In position 15 we have 20 and in position 16 we have 20, therefore. Question 3: The data type of annual amounts is: Answer 3: Continuous Question 4 According to the coefficient of variation: Answer 4. The standard deviation is 35% of the average. Copyright Reserved 27

28 28 Questions 5 to 8 are based on the following information: The relationship between the age (in years) of a motorist and the speed (in km/h) of the car on the highway is summarised in the following Excel spreadsheet: Formula sheet: Value sheet: Question 5 The variance of the age of the motorists is: Answer 5 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Question 6 The coefficient of variation of the age of the motorists is: Answer 6 Copyright Reserved 28

29 29 Question 7 The sample covariance is: Answer 7 Sample covariance = Population covariance Question 8 The relationship between the age of a motorist and the speed of the car on the highway can be described as: (A) (B) (C) (D) (E) no linear relationship a strong negative linear relationship a weak negative linear relationship a strong positive linear relationship a weak positive linear relationship Answer 8 r = which is close to -1. Consequently, we have a strong negative linear relationship. Questions 9 to 11 are based on the following information: Consider the following set of Descriptive Statistics on time per week (in hours) spent on campaigning for the upcoming general election for a specific political party: Descriptive statistic Value Smallest value 8 Largest value 36 Question 9 The distribution of time per week (in hours) is: (A) Bimodal (C) Symmetrical (E) Skewed to the left (B) Multimodal (D) Skewed to the right Answer 9 Q 1 and Q 3 are equally far away from the median, therefore, the distribution is symmetrical. The boxplot, for example, will look something like this: The median is in the middle of the box, indicating symmetry. Copyright Reserved 29

30 30 Question 10 Using the box and whisker plot approach, an outlier is a value greater than: Answer 10 ( ) ( ). Question 11 The z-score (standardised value) for the largest value in the data set is: Answer 11 Copyright Reserved 30