Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 1 – Slide 1 of 27 Chapter 3 Section 1 Measures of Central Tendency

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 1 – Slide 2 of 27 Chapter 3 – Section 1 ●Learning objectives  The arithmetic mean of a variable  The median of a variable  The mode of a variable  Identifying the shape of a distribution

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 1 – Slide 3 of 27 Chapter 3 – Section 1 ●Analyzing populations versus analyzing samples ●For populations  We know all of the data  Descriptive measures of populations are called parameters  Parameters are often written using Greek letters ( μ ) ●Analyzing populations versus analyzing samples ●For populations  We know all of the data  Descriptive measures of populations are called parameters  Parameters are often written using Greek letters ( μ ) ●For samples  We know only part of the entire data  Descriptive measures of samples are called statistics  Statistics are often written using Roman letters ( )

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 1 – Slide 4 of 27 Chapter 3 – Section 1 ●Learning objectives  The arithmetic mean of a variable  The median of a variable  The mode of a variable  Identifying the shape of a distribution

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 1 – Slide 5 of 27 Chapter 3 – Section 1 ●The arithmetic mean of a variable is often what people mean by the “average” … add up all the values and divide by how many there are ●Compute the arithmetic mean of 6, 1, 5 ●The arithmetic mean of a variable is often what people mean by the “average” … add up all the values and divide by how many there are ●Compute the arithmetic mean of 6, 1, 5 ●Add up the three numbers and divide by 3 ( ) / 3 = 4.0 ●The arithmetic mean is 4.0

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 1 – Slide 6 of 27 Chapter 3 – Section 1 ●The arithmetic mean is usually called the mean ●For a population … the population mean  Is computed using all the observations in a population  Is denoted μ  Is a parameter ●The arithmetic mean is usually called the mean ●For a population … the population mean  Is computed using all the observations in a population  Is denoted μ  Is a parameter ●For a sample … the sample mean  Is computed using only the observations in a sample  Is denoted  Is a statistic

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 1 – Slide 7 of 27 Chapter 3 – Section 1 ●One interpretation ●The arithmetic mean can be thought of as the center of gravity … where the yardstick balances

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 1 – Slide 8 of 27 Chapter 3 – Section 1 ●Learning objectives  The arithmetic mean of a variable  The median of a variable  The mode of a variable  Identifying the shape of a distribution

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 1 – Slide 9 of 27 Chapter 3 – Section 1 ●The median of a variable is the “center” ●When the data is sorted in order, the median is the middle value ●The median of a variable is the “center” ●When the data is sorted in order, the median is the middle value ●The calculation of the median of a variable is slightly different depending on  If there are an odd number of points, or  If there are an even number of points

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 1 – Slide 10 of 27 Chapter 3 – Section 1 ●To calculate the median (M) of a data set  Arrange the data in order  Count the number of observations, n ●To calculate the median (M) of a data set  Arrange the data in order  Count the number of observations, n ●If n is odd  There is a value that’s exactly in the middle  That value is the median M ●To calculate the median (M) of a data set  Arrange the data in order  Count the number of observations, n ●If n is odd  There is a value that’s exactly in the middle  That value is the median M ●If n is even  There are two values on either side of the exact middle  Take their mean to be the median M

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 1 – Slide 11 of 27 Chapter 3 – Section 1 ●An example with an odd number of observations (5 observations) ●Compute the median of 6, 1, 11, 2, 11 ●An example with an odd number of observations (5 observations) ●Compute the median of 6, 1, 11, 2, 11 ●Sort them in order 1, 2, 6, 11, 11 ●An example with an odd number of observations (5 observations) ●Compute the median of 6, 1, 11, 2, 11 ●Sort them in order 1, 2, 6, 11, 11 ●The middle number is 6, so the median is 6

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 1 – Slide 12 of 27 Chapter 3 – Section 1 ●An example with an even number of observations (4 observations) ●Compute the median of 6, 1, 11, 2 ●An example with an even number of observations (4 observations) ●Compute the median of 6, 1, 11, 2 ●Sort them in order 1, 2, 6, 11 ●An example with an even number of observations (4 observations) ●Compute the median of 6, 1, 11, 2 ●Sort them in order 1, 2, 6, 11 ●Take the mean of the two middle values (2 + 6) / 2 = 4 ●The median is 4

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 1 – Slide 13 of 27 Chapter 3 – Section 1 ●One interpretation ●The median splits the data into halves 62, 68, 71, 74, 77, 82, 84, 88, 90, 94 M = , 68, 71, 74, 77 5 on the left 82, 84, 88, 90, 94 5 on the right

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 1 – Slide 14 of 27 Chapter 3 – Section 1 ●Learning objectives  The arithmetic mean of a variable  The median of a variable  The mode of a variable  Identifying the shape of a distribution

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 1 – Slide 15 of 27 Chapter 3 – Section 1 ●The mode of a variable is the most frequently occurring value ●Find the mode of 6, 1, 2, 6, 11, 7, 3 ●The mode of a variable is the most frequently occurring value ●Find the mode of 6, 1, 2, 6, 11, 7, 3 ●The values are 1, 2, 3, 6, 6, 7, 11 ●The mode of a variable is the most frequently occurring value ●Find the mode of 6, 1, 2, 6, 11, 7, 3 ●The values are 1, 2, 3, 6, 6, 7, 11 ●The value 6 occurs twice, all the other values occur only once ●The mode is 6

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 1 – Slide 16 of 27 Chapter 3 – Section 1 ●Qualitative data  Values are one of a set of categories  Cannot add or order them … the mean and median do not exist  The mode is the only one of these three measurements that exists ●Qualitative data  Values are one of a set of categories  Cannot add or order them … the mean and median do not exist  The mode is the only one of these three measurements that exists ●Qualitative data  Values are one of a set of categories  Cannot add or order them … the mean and median do not exist  The mode is the only one of these three measurements that exists ●Find the mode of blue, blue, blue, red, green ●The mode is “blue” because it is the value that occurs the most often

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 1 – Slide 17 of 27 Chapter 3 – Section 1 ●Quantitative data  The mode can be computed but sometimes it is not meaningful  Sometimes each value will only occur once (which can often happen with precise measurements) ●Quantitative data  The mode can be computed but sometimes it is not meaningful  Sometimes each value will only occur once (which can often happen with precise measurements) ●Quantitative data  The mode can be computed but sometimes it is not meaningful  Sometimes each value will only occur once (which can often happen with precise measurements) ●Find the mode of 5.1, 6.6, 6.8, 9.3, 1.9 ●Quantitative data  The mode can be computed but sometimes it is not meaningful  Sometimes each value will only occur once (which can often happen with precise measurements) ●Find the mode of 5.1, 6.6, 6.8, 9.3, 1.9 ●Each value occurs only once ●The mode is not a meaningful measurement

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 1 – Slide 18 of 27 Chapter 3 – Section 1 ●One interpretation ●In primary elections, the candidate who receives the most votes is often called “the winner” ●One interpretation ●In primary elections, the candidate who receives the most votes is often called “the winner” ●Votes (data values) are CandidateNumber of votes Henry194 Kayla215 Jason172 ●One interpretation ●In primary elections, the candidate who receives the most votes is often called “the winner” ●Votes (data values) are ●The mode is “Kayla” … Kayla is the winner

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 1 – Slide 19 of 27 Chapter 3 – Section 1 ●Learning objectives  The arithmetic mean of a variable  The median of a variable  The mode of a variable  Identifying the shape of a distribution

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 1 – Slide 20 of 27 Chapter 3 – Section 1 ●The mean and the median are often different ●This difference gives us clues about the shape of the distribution  Is it symmetric?  Is it skewed left?  Is it skewed right?  Are there any extreme values?

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 1 – Slide 21 of 27 Chapter 3 – Section 1 ●Symmetric – the mean will usually be close to the median ●Skewed left – the mean will usually be smaller than the median ●Skewed right – the mean will usually be larger than the median

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 1 – Slide 22 of 27 Chapter 3 – Section 1 ●If a distribution is symmetric, the data values above and below the mean will balance  The mean will be in the “middle”  The median will be in the “middle” ●If a distribution is symmetric, the data values above and below the mean will balance  The mean will be in the “middle”  The median will be in the “middle” ●Thus the mean will be close to the median, in general, for a distribution that is symmetric

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 1 – Slide 23 of 27 Chapter 3 – Section 1 ●If a distribution is skewed left, there will be some data values that are larger than the others  The mean will decrease  The median will not decrease as much ●If a distribution is skewed left, there will be some data values that are larger than the others  The mean will decrease  The median will not decrease as much ●Thus the mean will be smaller than the median, in general, for a distribution that is skewed left

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 1 – Slide 24 of 27 Chapter 3 – Section 1 ●If a distribution is skewed right, there will be some data values that are larger than the others  The mean will increase  The median will not increase as much ●If a distribution is skewed right, there will be some data values that are larger than the others  The mean will increase  The median will not increase as much ●Thus the mean will be larger than the median, in general, for a distribution that is skewed right

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 1 – Slide 25 of 27 Chapter 3 – Section 1 ●For a mostly symmetric distribution, the mean and the median will be roughly equal ●Many variables, such as birth weights below, are approximately symmetric

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 1 – Slide 26 of 27 Chapter 3 – Section 1 ●What if one value is extremely different from the others? ●What if we made a mistake and 6, 1, 2 was recorded as 6000, 1, 2 ●What if one value is extremely different from the others? ●What if we made a mistake and 6, 1, 2 was recorded as 6000, 1, 2 ●The mean is now ( ) / 3 = 2001 ●The median is still 2 ●The median is “resistant to extreme values”

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 3 Section 1 – Slide 27 of 27 Summary: Chapter 3 – Section 1 ●Mean  The center of gravity  Useful for roughly symmetric quantitative data ●Median  Splits the data into halves  Useful for highly skewed quantitative data ●Mode  The most frequent value  Useful for qualitative data