1 a value at the center or middle of a data set Measures of Center

2 Mean (Arithmetic Mean) AVERAGE the number obtained by adding the values and dividing the total by the number of values Definitions

3 Notation  denotes the addition of a set of values x is the variable usually used to represent the individual data values n represents the number of data values in a sample N represents the number of data values in a population

4 Notation is pronounced ‘x-bar’ and denotes the mean of a set of sample values x = n  x x x

5 Notation µ is pronounced ‘mu’ and denotes the mean of all values in a population is pronounced ‘x-bar’ and denotes the mean of a set of sample values Calculators can calculate the mean of data x = n  x x x N µ =  x x

6 Definitions  Median the middle value when the original data values are arranged in order of increasing (or decreasing) magnitude

7 Definitions  Median the middle value when the original data values are arranged in order of increasing (or decreasing) magnitude  often denoted by x (pronounced ‘x-tilde’) ~

8 Definitions  Median the middle value when the original data values are arranged in order of increasing (or decreasing) magnitude  often denoted by x (pronounced ‘x-tilde’)  is not affected by an extreme value ~

no exact middle -- shared by two numbers (even number of values) MEDIAN is 5.02

(in order - odd number of values) exact middle MEDIAN is no exact middle -- shared by two numbers (even number of values) MEDIAN is 5.02

11 Definitions  Mode the score that occurs most frequently Bimodal Multimodal No Mode denoted by M the only measure of central tendency that can be used with nominal data

12 Measures of Variation Range value highest lowest value

13  Midrange the value midway between the highest and lowest values in the original data set Definitions

14  Midrange the value midway between the highest and lowest values in the original data set Definitions Midrange = highest score + lowest score 2

15  5% trimmed mean the mean of the middle 90% of the scores Definitions

16 Carry one more decimal place than is present in the original set of values Round-off Rule for Measures of Center

17 a measure of variation of the scores about the mean (average deviation from the mean) Measures of Variation Standard Deviation

18 Sample Standard Deviation Formula calculators can compute the sample standard deviation of data  ( x - x ) 2 n - 1 S =S =

19 Sample Standard Deviation Shortcut Formula n ( n - 1) s = n (  x 2 ) - (  x ) 2 calculators can compute the sample standard deviation of data

20 Population Standard Deviation calculators can compute the population standard deviation of data 2  ( x - µ ) N  =

21 Symbols for Standard Deviation Sample Population   x x  n s S x x  n-1 Book Some graphics calculators Some non-graphics calculators Textbook Some graphics calculators Some non-graphics calculators

22 Measures of Variation Variance standard deviation squared

23 Measures of Variation Variance standard deviation squared s  2 2 } use square key on calculator Notation

24 Sample Variance Population Variance  ( x - x ) 2 n - 1 s 2 =  (x - µ)2 (x - µ)2 N  2 =

25 Find the range, midrange, mean, median, and 5% trimmed mean Data Set A Range = 98 Mean = 50 Midrange = 50 Median = 50 5% trimmed = 50 Data Set B Range = 98 Mean = 50 Midrange = 50 Median = 50 5% trimmed= 50 AB

26 Ax- µ(x - µ)  ( x - µ ) 2 N  =  10  =  = Find the Standard Deviation

27 Ax- µ(x - µ)  ( x - µ ) 2 N  =  10  =  = Find the Standard Deviation

28 Coefficient of Variance Used to compare the variability of two data sets that do not measure the same thing.

29 Coefficient of Variance AgeShoe Size / Use the coefficient of variance to determine which varies more. Age µ = 16.6  = Shoe Size µ = 8.7  = Age varies more than shoe size

30 Histogram a bar graph in which the horizontal scale represents classes and the vertical scale represents frequencies Sturgess’ Rule The number of intervals in a histogram be approximately log(N) Less-Than cummulative frequency curve ( ] Greater-Than cummulative frequency curve [ ) Frequency Polygon A straight line graph that is formed by connecting the midpoints of the top of the bars of the histogram Frequency Curve A smoothed out frequency polygon Ogive A frequency curve formed from a cummulative frequency histogram by connecting the corners with a smooth curve

31 Qwertry Keyboard Word Ratings Table

32 Frequency Table of Qwerty Word Ratings Table RatingFrequency

33 Histogram of Qwerty Word Ratings Rating Frequency

34 Relative Frequency Histogram of Qwerty Word Ratings Figure % % % % % Rating Relative Frequency

35 Histogram and Relative Frequency Histogram Figure 2-2 Figure 2-3

36 Figure 2-4 Frequency Polygon

37 Figure 2-5 Ogive

38 Bachelor High School Degree Diploma Figure 1-1 Salaries of People with Bachelor’s Degrees and with High School Diplomas $40,000 30,000 25,000 20,000 $40,500 $24,400 35,000 $40,000 20,000 10,000 0 $40,500 $24,400 30,000 Bachelor High School Degree Diploma (a)(b)

39  Symmetric Data is symmetric if the left half of its histogram is roughly a mirror of its right half.  Skewed Data is skewed if it is not symmetric and if it extends more to one side than the other. Definitions

40 Skewness Mode = Mean = Median SYMMETRIC Figure 2-13 (b)

41 Skewness Mode = Mean = Median SKEWED LEFT (negatively ) SYMMETRIC Mean Mode Median Figure 2-13 (b) Figure 2-13 (a)

42 Skewness Mode = Mean = Median SKEWED LEFT (negatively ) SYMMETRIC Mean Mode Median SKEWED RIGHT (positively) Mean Mode Median