# Chapter 3 (numerical measures part a) student

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Chapter Statistical measures Measure center and location Measure variation/dispersion Summary Statistical measures Center and location - Mean (arithmetic, weighted, geometric) - Mode, Median Variation/Disper sion - Range - Variance - Standard deviation Part A Measures of center and location Arithmetic mean Weighted mean Geometric mean Harmonic mean Median Mode Percentile, Quartile Arithmetic mean   The mean of a data set is the average of all the data values Arithmetic mean of a data set is defined as ‘ the sum of the values’ divided by the ‘number of values’ Arithmetic mean = The sum of all values The number of values Formula x1  x2   xn x  n xi � x  n wher e - are the 1st x-value, 2nd x-value, x2 , , xn ….x nth ,x-value - n is the number of data values in the set Example  If a firm received orders worth: £151, £155, £160, £90, £270 for five consecutive months, their average value of orders per month would be calculated as: Limits of arithmetic mean Weighted mean  Simple frequency distribution  Grouped frequency distribution Weighted mean of a simple frequency distribution xi f 10 12 13 14 16 17    Is the arithmetic mean appropriate to a simple frequency distribution? Why? n Formula:�xi f i x  i 1 n �f i 1 i Example x f 12 18 30 20 15 Tota 100 l xf (x): Number of newspapers/maga zines/journals a student read a week (f): Number of students Estimating the median graphically  Read at home 50% poin t Characteristics of the median Mode   The mode of a data set is the value which occurs most often or equivalently, has the largest frequency Example: The mode of the set 2, 3, 2, 4, 5, 2, is: The mode of a simple frequency distribution  Mode is the value which has the largest frequency Mod e? xi f 10 15 17 20 18 The mode of a grouped frequency distribution   Step 1: Find the modal class Step 2: Estimate the mode by the formula M  LM  cM LM cM f M  f M 1 ( f M  f M 1 )  ( f M  f M 1 ) Lower limit of modal class Modal class width fM0 f M 1 f M 1 Frequency of modal class Frequency of the class immediately prior to the modal classof the class Frequency immediately following to the modal class Example  Amount of food per person in province A Amount of food (kg/person) 400-500 Number of people 10 500-600 30 600-700 45 700-800 80 800-900 30 900-1000 Mode? Modal class? Amount Number of food of people (kg/perso n) 400-500 10 500-600 30 600-700 45 700-800 80 800-900 30 900-1000 Estimate the Mode by the formula Graphical estimation of the mode Characteristics of the mode Graphical comparison of mean, median and mode  Symmetric Relative Frequency 35 30 25 20 15 10 05 Mean Media n Mode Graphical comparison of mean, median and mode  Moderately Skewed Left Relative Frequency 35 30 25 20 15 10 05 Mea n Media n Mode Graphical comparison of mean, median and mode  Moderately Right Skewed Relative Frequency 35 30 25 20 15 10 05 Mod Media Mean Graphical comparison of mean, median and mode  Highly Skewed Right Relative Frequency 35 30 25 20 15 10 05 Percentile and quartile  Read at home ... productivity of workers in a factory: Productivi ty (items/h) Number of workers 0-9 1019 2029 30 39 4049 5059 15 25 30 35 28 17  Weighted mean of a grouped frequency distribution Formula: n x  �x i 1... frequency distribution Productiv Numbe ity r of (items/h) worker s 0-9 10-19 20-29 30 -39 40-49 50-59 15 25 30 35 28 17 xi xif  Weighted mean of a grouped frequency distribution The average productivity... banke rs 200 200 200 200 200 200 200 200 220 250 262 284 30 0 31 2 Example Year 200 200 200 200 200 200 200 No of 200 220 250 262 284 30 0 31 2 banke rs Propo rtiona l multi pliers Example  The average
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