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# Data analysis: measures of central tendency MAT117

Pace

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This 3 page Class Notes was uploaded by WB on Wednesday September 30, 2015. The Class Notes belongs to MAT117 at Pace University taught by Dr. Glenn Miller in Fall 2015. Since its upload, it has received 33 views. For similar materials see Elementary statistics in Mathematics (M) at Pace University.

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Date Created: 09/30/15

Data Analysis Measures of Central Tendency One number or observation that represents the middle of the data set 1 Mean add up all numbers and divide by the number of numbers Sample mean i x barquot 39 u n 2X Populatlon mean u mu 7 Ex weights of 10yearold girls 75 79 69 82 56 70 find the mean gt 2 757969282567 718333333 g 718 I Round Off Rule round your final answer to one more degree of accuracy than is present in the data Here the data was whole numbers so the answer should be rounded to the nearest tenth If the data had been given with values accurate to the tenths place then we would round our statistic to the hundredths place Why we still need other measures of central tendency median mode 2 Two problems with using the mean 1 According to the definition of the mean it cannot be used for qualitative data 2 For data sets that contain outliers an extremely high or low value relative to the rest of the data set ie a weirdo more than two standard deviations away from the mean 2 Mode the observation with the highest frequency So for qualitative data we can use the mode I There can be more than one mode if two observations have the highest frequency OR no mode if no observation is repeated Ex data set 1 3345612 gt mode 3 2 33451212 gt mode 312 3 13579 gt mode no mode 3 Median f the number in the middle of the ordered data list So if there s a weirdo existing the median is the preferable measure of center Ex Suppose you own a store and pay your employees the following hourly wages 6 6 7 7 54 The last one 54 is the hourly wage You pay Ioe because he is your cousin Obviously 54 is a weirdo The employees are so mad and say they want a raise But you calculate the mean of the wages 667754 5 per hour so be quiet and get back to work What s wrong here 16 and show them and say Look at the mean salary I pay Iquot 2 Because of the outlier Ioe s wages the mean is much higher than the 4 out of the 5 values and not a good measure of center here The median is especially useful when outliers are present Ex 566970757980 I Before calculating the median you MUST put the numbers in order from least to greatest n1 2 the position of the median T When n is even the median of the data set is the mean of two middle numbers Here n is even 62 35 so the middle two numbers are 3rd 70 4th 75 The median m 725 Why we can t use the median all the time I It is robust against wildly different numbers present in the set Ex 56 69 70 83 79 75 8000 Obviously the median cannot measure the center of the data Exercises Find mean median and mode for a frequency distribution The following table shows the results of a survey of a class of college students age Age x Frequency f xf 19 7 19x7133 20 1 20 21 4 84 22 2 44 Total 14 students 2 f 281 Z xf 2 Mode 19 Mean 2 2 201 14 Here the mean of a frequency distribution is given by fo fo x 2f n 14 1 Median n is even the position ofthe median 75 2 gt 7th number 19 and the 8th 20 number I Count down the list of values taking frequencies into account 7th 8th 19 20 195 2 2 f

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