Exploring Different Types of Means in Statistics
Discover the concepts of arithmetic mean, weighted mean, geometric mean, harmonic mean, and more through examples and explanations from a presentation by Professor Takao Ito at the Graduate School of Advanced Science and Engineering, Hiroshima University.
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Presentation Transcript
Mean and Variance Graduate School of Advanced Science and Engineering, Hiroshima University Professor Takao Ito 28 September 2021
Mean Arithmetic mean Weighted mean Geometric mean Harmonic mean Moving Mean Generalized Mean Arithmetic Geometric Mean Root Mea Square
Average and Mean Average: measure of central tendency Average includes Mean, median, Mode, etc. Average > Mean
Arithmetic Mean Mark s Grades Literature Mathematics Sociology Music 75 80 90 95 x Mean (Average Score) of Mark s 1 n 1 n n = ? =75 + 80 + 90 + 95 = + + + ( ) x x x x x = 85( ) 1 2 i n 4 = 1 i
Weighted Mean Data of Bananas and Apples Bananas 5 150 Apples 10 180 Pieces price Average price of bananas and apples = i n 1 180 10 150 5 + + + + 1 n f x f x f x x = = 1 1 2 2 n n f x i i + + + f f 2550 f 1 2 n + x = = = 170 ( / ) 5 10 15
Geometric Mean Record of David s Body Height Age Body Height 16 100 17 156 18 166 Growth Rate of David s Body Height Age 16 100 17 156 1.56 1.064 18 166 Body Height Growth Rate
. 1 + . 1 56 064 x = . 1 = 312 2 n = = x x x x x n n 1 2 i n = 1 i x = = . 1 . 1 = 2 . 1 56 064 288 x x 2 1 2
Average Growth Rate of Davids Confirmation Age 16 100 17 156 1.56 1.064 18 166 Body Height Growth Rate (Estimate) Growth rate 129 166
Harmonic Mean From Fukuoka to Tokyo On the way On the way back Speed 40 km/hour 120km/hour Average Speed + 40 120 = = 80 ( / ) x km hour 2
n n = = x 1 1 1 1 + + + x x x x i 1 2 i n 2 = = 60 ( /hour ) x km 1 1 + 40 120
Confirmatio Actual distance (if the distance from Fukuka to Tokyo is x.) Actual hours / / tan Dis ce = Speed Hours tan Hours 2 Dis ce x = = = 60 ( /hour ) Speed km x x + 40 120
Preliminaries Descriptive statistics Mean Variance x 2s Probability Statistics Mean Variance 2
Standard Deviation Standard Deviation: difference between the observation and Mean DataDifference between data and Mean 5 Square value of the difference between data and Mean 2 4 4 1 1 3 0 0 2 -1 1 1 -2 4 Mean 3 Summation 0 10 1 n x 1 n x = = = 2 2 = ( ) 2 x ( ) 0 x i i i i
Standard Deviation The standard deviation is a statistic that measures the dispersion of a dataset relative to its mean and is calculated as the square root of the variance. The standard deviation is calculated as the square root of variance by determining each data point's deviation relative to the mean. 1 n x i = 2= ( ) 2 x i
