March 19, 2012 Clemons Chapter 8 (pp )

Measures of Central Tendency Used to describe and summarize a distribution Three measures of central tendency Mean Median Mode Source: Clemens and McBeth (2000)

Measures of Central Tendency Mean arithmetic average of a distribution of numbers (add all the values and divide by the number of cases) Calculated with interval-ratio data Median the middle point of a distribution of numbers (if even number of values: the median is the mean of the two middle numbers) calculated with ordinal OR interval-ratio data Mode the value of a frequency distribution that occurs the most Calculated with ordinal, interval-ratio, AND nominal (categorical) data Source: Clemens and McBeth (2000)

Measures of Central Tendency In other words: With interval-ratio data, you can calculate all three measures of central tendency With nominal (categorical) data, you can only calculate the mode With ordinal data, you can calculate both the median and the mode (but not the mean) Source: Clemens and McBeth (2000)

Measures of Central Tendency Boondocks Example Let’s assume that we are asked to provide data on the average number of crimes based on previous statistics. We can calculate the mean, median, and mode in the following way. Source: Clemens and McBeth (2000)

Measures of Central Tendency Mean = Sum of crimes / number of years (arithmetic average) = 5,105 / 10 = crimes per year Year Reported Crimes in Boondocks TOTAL5,105 Source: Clemens and McBeth (2000)

Measures of Central Tendency For median and mode, we first need to reorder the data into a ranked distribution Year Reported Crimes in Boondocks Source: Clemens and McBeth (2000)

Measures of Central Tendency Median = Mid-point of distribution (between 550 in 1994 and 560 in 1997, since we have an even number of reported crimes) = Need to find the average of 550 and 560 = ( ) / 2 = 555 crimes per year Year Reported Crimes Source: Clemens and McBeth (2000)

Measures of Central Tendency Mode = the value of a frequency distribution that occurs the most = 600 crimes per year (occurs 3 times: 1993, 1996, 1998) Year Reported Crimes Source: Clemens and McBeth (2000)

Measures of Central Tendency Q: Given that we have 3 different measures of central tendency (for crimes per year), which one should we report to the mayor? (Mean) 555 (Median) 600 (Mode) Source: Clemens and McBeth (2000)

Measures of Central Tendency Best to report all the three figures and explain the differences between the three If there are outliers (unlike this case) choose the median means are unreliable (because they are vulnerable to the effects of outliers) Source: Clemens and McBeth (2000)

Measures of Central Tendency Q: Which measure of central tendency would the mayor be inclined to use: If s/he wants to attract businesses? If s/he wants to receive more funding for police services? Mean: 510.5; Median: 555; Mode: 600 Source: Clemens and McBeth (2000)

Measures of Central Tendency How would mean, median, and mode change if we were to look at the last 5 years instead of 10? Source: Clemens and McBeth (2000)

Measures of Central Tendency Year Reported Crimes in Boondocks TOTAL2,960

Measures of Central Tendency Mean 2,960 / 5 = 592 Median = 600 Mode = 600

Measures of Central Tendency Pay attention to the data each time. In this case, Data represents only “crimes reported” (The numbers in the last few years may be up due to a new “Neighborhood Watch” program which was introduced—more reporting of crime) Do not know if crimes fell per capita No distinguishing between serious crimes and lesser offenses (maybe murder down, petty theft up) Maybe the city has more males years old Perhaps the police started to include juvenile activity in their crime reports recently