MATH 117 Statistical Methods for Management I Chapter Three

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1 Jubail University College MATH 117 Statistical Methods for Management I Chapter Three This chapter covers the following topics: I. Measures of Center Tendency. 1. Mean for Ungrouped Data (Raw Data) 2. Mean for a Frequency Distribution Table (Weighted Mean) 3. Mean for a Grouped Frequency Table (Optional) 4. Median 5. Mode II. III. IV. Measures or Location. 1. Percentiles 2. Quartiles Measures of Variations (Dispersion). 1. Range 2. Variance for Ungrouped Data (Raw data) 3. Variance for Frequency Distribution Table 4. Standard Deviation 5. Inter quartile Range Measures of Relative Variations. 1. Coefficient of Variations 2. Z-Values V. Measures of shape 1. Skewness 2. Kurtosis VI. Central Limit Theorem 1. Empirical Rule Ms. Ghaida Barghouthi, JUC Page 1

2 I. Measures of Center Tendency Often it is desirable to have a certain number to describe a set of data. In other words, this one number would be representative of the data. Since a representative number should be close to the "middle" of the data, we call these measures of central tendency. 1. The mean is the most powerful, and usually the most accurate and reliable, measure of central tendency. When we usually hear the word "average", what we are really thinking about is the mean. To find the mean for a set of data, we take the sum of all of the values, and divide the sum by how many values there are. If we are looking for the mean of a sample, we denote that mean by. This is read "x-bar": The formula for the sample mean, where is the sum of all the values, and n is the number of values in the set. The formula for the population mean, where is the sum of all the values and N is the number of values in the set. If we are looking for the mean of a population, we denote that mean by the Greek letter µ, mu. The way to calculate this mean is the same. The difference in notation is to tell a sample statistic, from a population parameter µ. Example: Calculate the mean for the given sample data 10, 12, 13, 10 and 15 Properties of the mean: a. Numerical center of the data b. The mean can be affected be extreme values. c. The data level interval or ratio. d. The sum of deviation from the mean is zero. That is (10-12)+ (12-12) + (13-12) + (10-12) + (15-12) = =0 2. Mean for a Frequency Distribution Table (Weighted Mean): A measure of central tendency that weights each data value according to its relative frequency. One example of the weighted mean is the GPA. The course credit hours are the weights. Also to calculate a student total mark in English I or II at JUC the weight is the contact hours per week for the 4 skills (Writing, Grammar, Reading and Listening). Example 1: Given the following data set {5, 6, 6, 7, 7, 8, 8, 8, 9, 16} The sample mean To find the mean if the data is given in a frequency table below: Frequency ( ) * - - )* Total Ms. Ghaida Barghouthi, JUC Page 2

3 a. Multiply each value by its frequency. b. Sum the result. c. Divide the total by the sum of frequencies. The weighted mean = Example 2: Calculate the semester average for student taking the following scores: Courses Scores Credit Hours ( W ) X*W Accounting Statistics Java I Health II Arabic Grammar Ethics Total The Average (the weighted mean) =.0 Example 3: Calculate the semester average for student taking the following grades if the university is using the 4 point system. Courses Grade Credit Hours ( W ) Points Total points(w*f) Accounting B Statistics D Java I B Health II A Arabic Grammar C Ethics A Total GPA= = =2.91 Example 4: Calculate a student average in English I. English Skill Scores Contact Hours ( W ) X*W Grammar Writing Reading Listening Total The average (weighted mean) = =77.0 Ms. Ghaida Barghouthi, JUC Page 3

4 Mean for a Grouped Frequency Table: When data are grouped in classes, we no longer know the exact values of each observation, we assume estimate the midpoint. Classes F Mid (M) M*F Total The mean =275/15=18.33 Median: The value in the middle of the data set when the values are ordered numerical. Example 1: Find the median for the following {14, 11, 12, 13, 10, 11, 10, 25, 14, 19, 10, 28, 22, 10, 15} Sort the data: 10, 10, 10, 10, 11, 11, 12, 13, 14, 14, 15, 19, 22, 25, and 28 When n =15 is odd number: The location of the median = =7.5, rounded up to 8, the value in the 8 th position is the median. The median =13 When n =16 is even number: 10, 10, 10, 10, 11, 11, 12, 13, 14, 14, 15, 19, 22, 25, 28, 30 The location which is an integer, the median is the average of the two values in the 8 th and 9 th position. The median =13.5 Properties of the median: a. Not sensitive to extreme values. b. The data level is interval, ratio or ordinal. c. Computed only from the center of the data, does not use information from all the data. 5. Mode: The most frequent observation in the set. For the data set 10, 10, 10, 10, 11, 11, 12, 13, 14, 14, 15, 19, 22, 25, 28, 30, the mode is 10 For the data set 10, 10, 10, 10, 11, 11, 12, 13, 14, 14, 14, 14, 22, 25, 28, 30, two modes 10, 14 For the data set 10, 10, 11, 11, 12, 12, 14, 14, no mode Properties of the mode: a. Not sensitive to extreme values. b. The data level interval, ratio, ordinal or nominal. c. It may not exist, or may have multiple values; if it does exist it is one of the data. II. Measures or Location. 1. Quartiles: A measure of location that divides a sorted set of data into four equal parts, each containing approximately 25% of the values. The first quartile Q1, the second quartile Q2 and the third quartile Q3. The median divides the observation into two halves. 50% below the median and 50% above the median which is another name for the second quartile. Example: Find for the set of data {14, 11, 12, 13, 10, 11, 10, 15, 14, 13, 10, 28, 22, 10, and 25} Sort the data: 10, 10, 10, 10, 11, 11, 12, 13, 14, 14, 15, 19, 22, 25, 28 When n=15 (n is an odd number) Ms. Ghaida Barghouthi, JUC Page 4

5 III. Find the first quartile : The location of the first quartile i = 0.25(15) = 3.75, rounded up to 4, the value in the 4 th position is, so =10 Find the second quartile : The location of the second quartile i = 0.50(15) = 7.5, rounded up to 8, the value in the 8 th position is, so =13 Find the third quartile : The location of the third quartile i = 0.75(15) = 11.25, rounded up to 12, the value in the 12 th position is, so When n =16 (n is an even number and multiple of 4): 10, 10, 10, 10, 11, 11, 12, 13, 14, 14, 15, 19, 22, 25, 28, 30 Find the first quartile : The location of the first quartile, which is an integer, the first quartile Q1 is the average of the two values in the 4 th and 5 th position. = (10+11)/2=10.5 =10.5 Find the second quartile : The location of the second quartile, which is an integer, the second quartile is the average of the two values in the 8 th and 9 th position. The second quartile = (13+14)/2=13.5 =13.5. The median is another name for the second quartile. Find the third quartile : The location of the third quartile, which is an integer, third quartile is the average of the two values in the 12 th and 13 th position. = (19+22)/2=20.5 = Percentiles: Are like quartiles but divide the sorted set of data into 100 equal parts. Each group represents 1% of the data set. The percentiles are denoted by the median is another name for Also. Example: Find for the set of data {14, 11, 12, 13, 10, 11, 10, 25, 14, 13, 10, 28, 22, 10, 15} Sort the data: 10, 10, 10, 10, 11, 11, 12, 13, 14, 14, 15, 19, 22, 25, 28 Find : The location of i = 0.35(15) =5.25, rounded up to 6, the value in the 6 th position is, so =11 Find The location of i = 0.60(15) = 9, which is an integer, the average of the two values in the 9 th and 10 th position is so = Measures of Variations. 1. Range: The range is the simplest measure of dispersion Range=Maximum-Minimum Example: Find the range for the set of data {10, 10, 11, 11, 12, 13, 14, 14, 15, 19, 22, 25, 28, 30} The max=30, min=10 Range=30-10=20 It uses only two values of the data and is mostly affected by the extreme values in the data. 2. Variance for ungrouped data: The variance is the average of the squared deviation from the mean of a set of data. The unit of the variance will be the unit of the raw data square. Ms. Ghaida Barghouthi, JUC Page 5

6 Population variance is denoted by σ The sample variance is denoted by S ²=, Example 1: Find the variance for the data set 92, 75, 95, 90 and ² = = = = = =, µ = If the data represent a sample the variance S²= = If the data represent a population σ = = Variance for a frequency table: - - ² - ² Total The weighted mean S² = If the data represent a population σ = = = Standard Deviation: The standard deviation is the square root of the variance. Is used more than the variance because the unit of the standard deviation is the same unit as the data. For the data in example1: 92, 75, 95, 90 and 98 If the data is a sample, the sample standard deviation S= If the data is a population the population standard deviation σ= = Inter quartile Range: Is the range of the values between the third quartile and the first quartile. Interquartile Range IQR= In other words it is the range of the middle 50% of the observations. It is used to overcome the problem of the range being affected by extreme observations in a data set. Ms. Ghaida Barghouthi, JUC Page 6

7 Box and Whisker Plot: The variance measures the dispersion associated with the mean. The quartiles are represented graphically to measure the dispersion that is associated with the median In Box and Whisker plot five numbers are listed: Minimum,, Median, and Maximum. Outlier: Is a value that is inconsistent with the rest of the observation. Outliers < or Outliers > Example: Construct a Box and Whisker plot for the sample data set, 2, 0, 2, 11, 3, 4, 3, 5, 6, 2 and 27 The sorted data is {0, 2, 2, 2, 3, 3, 4, 5, 6, 11, 27} By looking at the dot plot below we can see the 27 is inconsistent with the data. Let us calculate the quartiles and check whether 27 is an outlier or not? DotPlot # 1 To construct a Box and Whisker plot we need to find the, Min=0, Max=27, and the quartiles. Find the first quartile : The location of the first quartile = 0.25(11) = 2.75, rounded up to 3, the value in the 3 rd position is, so =2 Find the second quartile : The location of the second quartile = 0.50(11) = 5.5, rounded up to 6, the value in the 6 th position is, so =3 Find the third quartile : The location of the third quartile = 0.75(11) = 8.25, rounded up to 9, the value in the 9 th position is, so IQR= =6-2=4 Lowe limit = Upper limit = Since 27 > upper limit=12, this means that 27 is an outlier of the data. BoxPlot Ms. Ghaida Barghouthi, JUC Page 7

8 IV. Measures of Relative Variations. 1. Coefficient of Variations: Is used to compare the spread of two sets of data with different means or data have different scales. The coefficient of variation for a sample is CV= The coefficient of variation for a population is Example 1: Consider a midterm exam out of 20 with mean =16.5 and standard deviation 4.5 and a quiz out of 5 with mean=3.5 and standard deviation =1.4. Which test has a larger relative spread? The midterm = The quiz = The quiz has greater relative dispersion, because Example 2: A sample 20 saving accounts in Bank A with average balance of $450 and a standard deviation of 150 another sample of 25 customers in Bank B with average balance of $1,500 and standard deviation of $380. Which sample has greater relative dispersion? Why?,, Sample in Bank A has greater relative dispersion, because Example 2: If the coefficient of variation for a sample is 12.5% and the sample mean is =20 Find the sample standard deviation? CV= (100%) CV=12.5% =(S/20). (100%), 12.5%=S/20 S = (12.5%)(20)=2.5 Example 3: If the coefficient of variation for a sample is 12.5% and the sample standard deviation 1.5 find the sample mean? CV= (100%) 12.5% = (1.5/ ). (100%) ) =1.5. Therefore =1.5/12.5%=1.5/0.125=12 2. Z-Values The number of standard deviation a value X is above or below the mean of a set of observation. The z-value for a sample, the z-value for a population z = The Z- scores redefine the raw values in terms of its distance in standard deviation from the mean. Positive z-value mean the raw data is above the mean. Negative means it is below the mean. The mean itself has a z-value of zero. Example: Student Amal scored 75 in MATH-1 final exam and 80 in the HEALTH-1 final exam. If the MATH exam had a mean of 65 and a standard deviation of 8 and the HEALTH-1 exam has a mean of 85 and a standard deviation of 5. Compare Amals result in both exams. The Z- Value for the MATH exam Z1= (75-65)/8= 1.25 The Z- Value for the HEALTH exam Z2= (80-85)/5= -1.0 Amal scored relatively better in MATH than in the HEALTH, because she Z1=1.25 in MATH compared to Z2=-1.0 in the HEALTH. Ms. Ghaida Barghouthi, JUC Page 8

9 V. Empirical Rule ( 3 Sigma Rule) Bell shaped distributions are a family of distributions as shown below that have the following properties: a. Mean=Mode=Median b. Symmetric around the mean c. For a bell -shaped distribution nearly all the values fall within 3 standard deviations of the mean. Empirical rule: 68 % of the data is within 1 standard deviation of the mean, µ ± 95% of the data is within 2 standard deviation of the mean, µ ± 99.7% of the data is within 3 standard deviation of the mean, µ ± Bell-shaped distribution Example: Suppose a manufacturer claims that the mean lifespan of a battery is 60 months, with a standard deviation of 10 months, and suppose also that the distribution is bell-shaped. a. What percentage of batteries will last between 50 months and 70 months? Z1 = (50-60)/10=-1, Z2 = (90-60)/10=1 By the empirical rule, 68% of the batteries will last between 50 and 70 months b. What percentage of batteries will last between 40 months and 80 months? Z1 = (40-60)/10=-2 Z2 = (80-60)/10=2 By the empirical rule, 95% of the batteries will last between 40 and 80 months c. What percentage of batteries will last between 50 months and 80 months? Z1 = (50-60)/10=-1, Z2 = (80-60)/10=2 By the empirical rule, 95% of the batteries will last between Z=-2 and Z=2 and 68% of the batteries will last between Z=-1 and Z=1 Therefore 95%-68%=27% will fall between (Z=-2 and Z=-1) plus between (Z=1 and Z=2) The curve is symmetric, implies 27%/2=13.5% is between Z-2 and Z=-1 Therefore 68%+13.5=81.5% will fall between Z=-1 and Z=2 d. What percentage of batteries will last less than 50 months? Z1 = (50-60)/10=-1 By the empirical rule, 68% of the batteries will last between Z=-1 and Z=1 Therefore 100%-68%=32% will fall less Z=-1 plus greater than Z=1 The curve is symmetric, implies 32%/2=16% is less than Z=-1 e. What percentage of batteries will last more than 80 months? Z1 = (80-60)/10=2 By the empirical rule, 95% of the batteries will last between Z=-2 and Z=2 Therefore 100%-95%=5% will be less Z=-2 plus greater than Z=2 The curve is symmetric, implies 5%/2=2.5% is more than Z=2 Ms. Ghaida Barghouthi, JUC Page 9

10 VI. Measures of Shape 1. Skewness: A measure of symmetry of the distribution. Data in a distribution can be: a. Symmetric if the values are evenly spread around the center. For symmetric data mean=mode=median b. Skewed distributions if it is not symmetric. The mean is greater than median or larger than median. Symmetric Skewed Left (Negatively skewed) Skewed Right (Positively skewed) Tails are balanced Long tail points to the left Long tail points to the right 1. Kurtosis: Kurtosis in statistics (from the Greek word κσρτός, kyrtos or kurtos, meaning bulging) is a measure of the peakedness of a distribution. The kurtosis shows the extent to which values are concentrated around the mode. It gives indication whether the curve is peaked or flat. The larger the kurtosis, the more peaked will be the distribution. A negative kurtosis implies a flatter distribution and is called platykrutic. A positive kurtosis implies a more peaked distribution and is called leptokrutic. When the kurtosis is equal zero the distribution is called mesokrutic. Ms. Ghaida Barghouthi, JUC Page 10

11 The sample mean is Formulas/ Chapter 3 The population mean is µ = The weighted mean is, The weighted mean for a frequency distribution is The sample variance is S S=, Sample standard deviation The sample variance for a frequency table is S The population variance is σ, Population standard deviation σ= The population variance for a frequency distribution is σ= The short cut formula for the sample variance is The short cut formula for the population variance is The coefficient of variation for a sample is The coefficient of variation for a population is The z-value for The z-value for a sample is is taken from a population is Ms. Ghaida Barghouthi, JUC Page 11