Measures of Center Section 3-2 Definitions Mean (Arithmetic Mean)

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1 Measures of Center Section 3-1 Mean (Arithmetic Mean) AVERAGE the number obtained by adding the values and dividing the total by the number of values 1 Mean as a Balance Point 3

2 Mean as a Balance Point Mean 4 S x n N Notation denotes the addition of a set of values is the variable usually used to represent the individual data values represents the number of data values in a sample represents the number of data values in a population 5 x Notation is pronounced x-bar and denotes the mean of a set of sample values x = S x n µ is pronounced mu and denotes the mean of all values in a population µ = S x N Calculators can calculate the mean of data 6

3 Median the middle value when the original data values are arranged in order of increasing (or decreasing) magnitude 7 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 ) 3 8 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 9

4 (even number of values) no exact middle -- shared by two numbers MEDIAN is (even number of values) no exact middle -- shared by two numbers MEDIAN is (in order - odd number of values) exact middle MEDIAN is 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 1

5 Examples a b c Mode is 5 Bimodal - and 6 No Mode 13 Midrange the value midway between the highest and lowest values in the original data set 5 14 Midrange the value midway between the highest and lowest values in the original data set Midrange = highest score + lowest score 15

6 Round-off off Rule for Measures of Center Carry one more decimal place than is present in the original set of values x = = x = 5.05 = Measures of Center Mean Median Mode Midrange 6 17 Mean from a Frequency Distribution use class midpoint of classes for variable x 18

7 Mean from a Frequency Distribution use class midpoint of classes for variable x S (f x) x = Formula 3- S f 19 Mean from a Frequency Distribution use class midpoint of classes for variable x S (f x) x = Formula 3- S f 7 x = class midpoint f = frequency S f = n 0 Weighted Mean x = S (w x) S w 1

8 Mean for a Frequency Distribution Quiz Scores Frequency Mean for a Frequency Distribution Quiz Scores Midpoints Frequency Mean for a Frequency Distribution Quiz Scores Midpoints Frequency x = S (f x) S f 4

9 Calculator Basics for Statistical Data 1. Put calculator into statistical mode. Clear previous data 3. Enter data (and frequency) 4. Select key(s) that calculate x 5 Mean for a Frequency Table Quiz Scores Midpoints Frequency Mean for a Frequency Table x = 14.4 ( rounded to one more decimal place than data ) Quiz Scores Midpoints Frequency

10 Quiz Scores 1 10 Frequency Quiz Scores ( Midpoints) 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 Skewness Figure 3- (b) Mode = Mean = Median SYMMETRIC 30

11 Skewness Figure 3- (b) Mode = Mean = Median SYMMETRIC Figure 3- (a) Mean Mode Median SKEWED LEFT (negatively) 31 Skewness Mode = Mean = Median SYMMETRIC Figure 3- (b) 11 Figure 3- (a) Mean Mode Median SKEWED LEFT (negatively) Mode Median Mean SKEWED RIGHT (positively) Figure 3- (c) 3 Important Distributions Normal 33

12 Important Distributions Normal Uniform 34 Important Distributions Normal 1 Uniform 35 Important Distributions Normal Uniform Skewed Right 36

13 Important Distributions Normal Uniform Skewed Right Skewed Left