Chapter 3 (numerical measures part b) 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 B Measures of variation/dispersion Range Mean deviation Variance Standard deviation Coefficient of variation The range   The range is defined as the numerical difference between the smallest and largest values of the items in a set or distribution Formula: R = largest value – smallest value Example  Ages of two groups of people on survey: Group A 20 30 40 50 60 Group B 38 39 40 41 42 Advantages and disadvantages of the range  Advantages:  Disadvantages: Implication The mean deviation   The mean deviation is a measure of dispersion that gives the average difference (i.e ignoring ‘-’ signs) between each item and mean Formula: - For a data set n d = ∑x i =1 i n −x Formulae - For a frequency distribution k d = ∑f i =1 i xi − x k ∑f i =1 i Example Group A Group B 20 30 40 50 60 38 39 40 41 42 n dA = ∑ i =1 xi − x n For a set of values n σ = ∑ (x i =1 i − x) n n or σ2 = x ∑i i =1 n − ( x )2 = x − ( x )2 The mean of the squares less the square of the mean For a frequency distribution k ( x − x ) fi ∑ i i =1 σ = k ∑f i =1 i k or σ2 = x ∑ i fi i =1 k ∑f i =1 − ( x )2 = x − ( x )2 i The mean of the squares less the square of the mean Example Group A Group B 20 30 40 50 60 38 39 40 41 42 n σ2 = ( x − x ) ∑ i i =1 n Example  The data in table below relates to the productivity (kg/person) of 100 workers in a small factory Variance? Productivi ty (kg/perso n) >10 10 – 20 20 – 30 30 – 35 35 – 40 ≥ 40 Number of workers 18 25 20 18 12 Characteristics of the variance    A better measure of dispersion than the range Complicated since it multiply the discrepancies The unit of the variance is not meaningful Standard deviation   Standard deviation is defined as the square root of the variance Formula For a set of values n ∑( x σ = n or σ= i =1 n − x) n ∑x i =1 i 2 i − (x ) = x − (x ) 2 For a frequency distribution k ∑( x σ = i =1 i − x) fi k ∑f i =1 i k or σ = x ∑ i fi i =1 k ∑f i =1 i − ( x )2 = x2 − ( x )2 Example Group A Group B 20 30 40 50 60 38 39 40 41 42 σ= σ Example  The data in table below relates to the productivity (kg/person) of 100 workers in a small factory Standard deviation? Productivi ty (kg/perso n) >10 10 – 20 20 – 30 30 – 35 35 – 40 ≥ 40 Number of workers 18 25 20 18 12 Characteristics of Standard Deviation   Can be regarded as one of the most useful and appropriate measure of dispersion For distribution that are not too skewed: - 99.7% of the data items should lie within three standard deviation of the mean - 95% of the data items should lie within two standard deviation - 68% of the data items should lie within one standard deviation of the mean 68 – 95 – 99.7 rule −3σ −2σ −1σ µ 1σ 2σ 3σ Coefficient of Variation   A standard measure used to compare the relative variation Formula: σ cv = × 100% x Example  Over a period of three months, the daily number of components produced by two comparable machines was measured, giving the following statistics: - Machine A: xA = 242.8 σ = 20.5 - Machine xB = 281.3 B: σ = 23.0 Example - The coefficient of variation for machine A: - The coefficient of variation for machine B: ... =1 i xi − x k ∑f i =1 i Example Group A Group B 20 30 40 50 60 38 39 40 41 42 n dA = ∑ i =1 xi − x n Example Group A Group B 20 30 40 50 60 38 39 40 41 42 n dB = ∑ i =1 xi − x n Example  The data... 100 workers in a small factory Mean deviation? Productivi ty (kg/perso n) 10 10 – 20 20 – 30 30 – 35 35 – 40 ≥ 40 Number of workers 18 25 20 18 12 Characteristics of the variance    A better
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