Measures of Central Tendency Chapter 4 Homework: 1, 2, 3, 5, 6, 13 Ignore parts with eye-ball estimation

3 essential characteristics of distributions Conveys most info for most distributions 1. Where is middle of distribution? 2. How wide is distribution? 3. What is shape of the distribution? ~

Central Tendency Middle of distribution measures: mode, median, mean Portable & compact communication further simplification of data lose more detail Which most appropriate? Depends on level of measurement intent of your communication ~

Mode Most frequently occurring value appropriate for any measurement level nominal, ordinal, interval/ratio ~

Computing the Mode Frequency distribution most frequently occurring value Grouped frequency distribution find interval with highest frequency report midpoint e.g., interval: 150 to 160 report: (160 + 150)/2 = 155 Methods may produce different results ~

Computing the Mode mode = 11 mode = X f Grouped Frequency Distribution 19 1 18 2 16 3 15 3 14 5 13 2 12 6 11 7 10 3 9 6 8 5 7 3 6 2 5 2 50 Grouped Frequency Distribution X f 19-20 1 17-18 2 15-16 6 13-14 7 11-12 13 9-10 9 7- 8 8 5- 6 4 50 mode =

Grouped Frequency Distribution X f 81-100 1 61-80 3 41-60 4 21-40 9 81-100 1 61-80 3 41-60 4 21-40 9 1-20 2 mode =

Median Midpoint of a data set values ½ smaller, ½ larger appropriate for ordinal & interval/ratio NOT nominal ~

10 20 30 40 50 60 70 80 90

Average Daily Temperature (oF) 10 20 30 40 50 60 70 80 90 Average Daily Temperature (oF)

Finding the Median 1. List all values from largest---> smallest if f=3, then list 3 times 2. Odd # entries median = middle value middle = (n + 1)/2 3. Even # entries = half way b/n middle 2 values ~

Finding the Median: odd # f 9 7 5 3 1 X 9 7 5 3 1 f 2 1 3 11 (n + 1)/2 =

Finding the Median: even # f 9 7 5 3 1 X 9 7 5 3 1 f 2 1 3 12 Average middle 2 values median = n /2 = (n /2) + 1 =

Mean Average value on X-axis may not be actual value in data set Computing the mean Sample mean Population mean

Reporting Central Tendency Depends on level of measurement Nominal: mode only appropriate Ordinal: mode & median not mean ---> uneven intervals Interval/ratio: all 3 appropriate ~

Comparing the Measures Normal distribution all 3 coincide Skewed will not be same values greatest effect of mean less on median, least on mode positive: mode -->median-->mean negative: mean <--median<--mode