# Bài giảng nguyên lý thông kê chương 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 Variation/Dispersion - Mean (arithmetic, weighted, geometric) - Range - Mode, Median - Percentile, Quartile - Variance - Standard deviation - Coefficient of variation 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’ The sum of all values Arithmetic mean = The number of values Formula x1 + x2 + + xn x = n xi ∑ x = n where - x1 , x2 , , xn are the 1st x-value, 2nd xvalue, … 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 fi 10 12 13 14 16 17    Is the arithmetic mean appropriate to a simple frequency distribution? Why? Formula: n x f ∑ x = i =1n i f ∑ i= i i Example x f 12 18 30 20 15 Total 100 xf (x): Number of newspapers/magazines /journals a student read a week (f): Number of students Estimating the median graphically  Read at home 50% point 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 Mode? xi fi 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 Frequency of modal class f M −1 Frequency of the class immediately prior to the modal class fM0 +1 Frequency of the class 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 500-600 30 600-700 45 700-800 80 800-900 30 900-1000 Mode? 10 Modal class? Amount of food (kg/person) Number of people 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 Median Mode Graphical comparison of mean, median and mode Moderately Skewed Left 35 Relative Frequency  30 25 20 15 10 05 Mean Median Mode Graphical comparison of mean, median and mode Moderately Right Skewed 35 Relative Frequency  30 25 20 15 10 05 Mode Median Mean Graphical comparison of mean, median and mode Highly Skewed Right 35 Relative Frequency  30 25 20 15 10 05 Percentile and quartile  Read at home [...]... of bankers over the period is:  Example Year 1994 1995 1996 1997 1998 1999 2000 2001 2002 20 03 pm (%) Year pm (%) Example gmm = ∑fi n fi (1 + p ) ∏ i i =1 4 Harmonic mean  Read at home Characteristics of the mean 5 Median Median of a data set is the value of the item in the middle when the data items are arranged in ascending order  The median is considered as an alternative average to the mean ... Multiply by (proportional increases) (proportional multipliers) 3 Geometric mean   - A specialized measure, used to average proportional increases Formula: Step 1: Express the proportional increases (p) as proportional multipliers (1+p) 3 Geometric mean - Step 2: Calculate the geometric mean multiplier (i) Simple geometric mean multiplier: applied when each proportional increase appears once only gmm... of data values are meaningful Proportional increases and multipliers: Example: The number of students attending the music class last Tuesday was 160 This Tuesday, the number is expected to increase by 15% How many of them are likely to attend this Tuesday? 3 Geometric mean  The number of students likely to attend this Tuesday  Proportional increase?  Proportional multiplier? Example To add Multiply... mid-point as representative value of each class - f: frequency of each class  Weighted mean of a grouped frequency distribution Productivity Number (items/h) of workers 0-9 10-19 15 25 20-29 30 -39 30 35 40-49 50-59 28 17 Total xi xifi Weighted mean of a grouped frequency distribution  The average productivity (mean) of workers in the factory is: 3 Geometric mean    Applicable when the products of data... + pn ) 3 Geometric mean - Step 2: Calculate the geometric mean multiplier (ii) Weighted geometric mean multiplier: applied when each proportional increase repeatedly appears n gmm = ∑ fi i =1 (1 + p1 ) (1 + p2 ) (1 + pn ) f1 n fi fi ∑ gmm = ∏ (1 + pi ) i =1 f2 fn 3 Geometric mean - Step 3: Subtract 1 from the gm multiplier to obtain the average proportional increase average proportional increase = gm... multiplier - 1 Example  The number of bankers of a small bank over the period 2000-2006 is presented in the table below: Year No of bankers 2000 2001 2002 20 03 2004 2005 2006 200 220 250 262 284 30 0 31 2 Example Year 2000 2001 2002 20 03 2004 2005 2006 No of 200 bankers Proport ional multipli ers 220 250 262 284 30 0 31 2 Example  The average proportional multiplier: The average proportional increase in the... mean of a simple frequency distribution  The mean number of newspapers/magazines/journals a student read a week is: Weighted mean of a grouped frequency distribution  Example: The following data relates to the productivity of workers in a factory: Productivity (items/h) Number of workers 0-9 10-19 20-29 30 -39 40-49 50-59 15 25 30 35 28 17 Weighted mean of a grouped frequency distribution  Formula:... the value(s) that correspond to the middle item(s) - For an even-number data set: 2m, the median is the average value of mth item and (m+1)th item xm + xm +1 Me = 2 Median for a simple frequency distribution  Step 2: Find the value(s) that correspond to the middle item(s) - For an odd-number data set: 2m+1, the median is the value of (m+1)th item M e = xm +1 Example xi fi 1 8 2 15 3 20 4 13 5 9 Total... Median? Middle item? Median for a grouped frequency distribution    Step 1: Find the middle item(s) Step 2: Find the class(es) containing the middle item(s) Step 3: Estimating the median by formula n ∑f i =1 M e = LM e +cM e 2 i −FM e −1 fMe Where: LM e Lower limit of the median class FM e −1 fMe Cumulative frequency of class immediately prior to the median class Actual frequency of the median class... alternative average to the mean  Example: The productivity of 5 workers (items/h): 20, 22, 24, 100, 22 Calculate the mean? What is the problem?  Notes  When a data set contains an even number of items: 2m there are two middle items: the mth one and the (m+1)th one When a data set contains an odd number of items: (2m+1)  The middle item is the (m+1)th one Median for a simple frequency distribution 
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