Measures of Central Tendency 3.1

● 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 ) ● For samples  We know only part of the entire data  Descriptive measures of samples are called statistics  Statistics are often written using Roman letters ( ) PARAMETER VS. STATISTIC

● 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 ARITHMETIC MEAN

● 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 MEAN

 Whole Class (Population)  Height in Inches  Find the arithmetic mean  5 Students (Sample…statistic)  Height in Inches  Find the arithmetic mean LET’S TRY IT

● 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  The middle number  If there are an even number of points  Take the 2 middle numbers and find the mean MEDIAN

● 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 MEDIAN

● 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 MEDIAN

● 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 MODE

● 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 WEIRD MODE

● 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 ● Mode is what is used in elections! MODE (NO REPEATS)

 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? SHAPE

 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 SHAPE

● 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 SYMMETRIC

● 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 SKEWED LEFT

● 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 SKEWED RIGHT

Birth Weights FINDING MEAN AND MEDIAN ON CALCULATOR Find the mean and median Make a Histogram to discuss the shape of the data How to sort data in lists

 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 SUMMARY (IMPORTANT STUFF!)