Describing Data Using Numerical Measures Mean The mean
![Describing Data Using Numerical Measures Describing Data Using Numerical Measures](https://slidetodoc.com/presentation_image/c7563dcf2c77f9ec598b8bd6e9273f40/image-1.jpg)
Describing Data Using Numerical Measures
![Mean The mean is a numerical measure of the center of a set of Mean The mean is a numerical measure of the center of a set of](http://slidetodoc.com/presentation_image/c7563dcf2c77f9ec598b8bd6e9273f40/image-2.jpg)
Mean The mean is a numerical measure of the center of a set of quantitative measures computed by dividing the sum of the values by the number of values in the data set.
![Population Mean where: = population mean (mu) N = number of data values xi Population Mean where: = population mean (mu) N = number of data values xi](http://slidetodoc.com/presentation_image/c7563dcf2c77f9ec598b8bd6e9273f40/image-3.jpg)
Population Mean where: = population mean (mu) N = number of data values xi = ith individual value of variable x
![Population Mean Example 3 -1 Table 3 -1: Foster City Hotel Data Population Mean Example 3 -1 Table 3 -1: Foster City Hotel Data](http://slidetodoc.com/presentation_image/c7563dcf2c77f9ec598b8bd6e9273f40/image-4.jpg)
Population Mean Example 3 -1 Table 3 -1: Foster City Hotel Data
![Population Mean Example 3 -1 The population mean for the number of rooms rented Population Mean Example 3 -1 The population mean for the number of rooms rented](http://slidetodoc.com/presentation_image/c7563dcf2c77f9ec598b8bd6e9273f40/image-5.jpg)
Population Mean Example 3 -1 The population mean for the number of rooms rented is computed as follows:
![Sample Mean where: = sample mean (pronounced “x-bar”) = sample size xi = ith Sample Mean where: = sample mean (pronounced “x-bar”) = sample size xi = ith](http://slidetodoc.com/presentation_image/c7563dcf2c77f9ec598b8bd6e9273f40/image-6.jpg)
Sample Mean where: = sample mean (pronounced “x-bar”) = sample size xi = ith individual value of variable x n
![Sample Mean Housing Prices Example {xi} = {house prices} = {$144, 000; 98, 000; Sample Mean Housing Prices Example {xi} = {house prices} = {$144, 000; 98, 000;](http://slidetodoc.com/presentation_image/c7563dcf2c77f9ec598b8bd6e9273f40/image-7.jpg)
Sample Mean Housing Prices Example {xi} = {house prices} = {$144, 000; 98, 000; 204, 000; 177, 000; 155, 000; 316, 000; 100, 000}
![Median The median is the center value that divides data that have been arranged Median The median is the center value that divides data that have been arranged](http://slidetodoc.com/presentation_image/c7563dcf2c77f9ec598b8bd6e9273f40/image-8.jpg)
Median The median is the center value that divides data that have been arranged in numerical order (i. e. an ordered array) array into two halves.
![Median Housing Prices Example {xi} = {house prices} = {$144, 000; 98, 000; 204, Median Housing Prices Example {xi} = {house prices} = {$144, 000; 98, 000; 204,](http://slidetodoc.com/presentation_image/c7563dcf2c77f9ec598b8bd6e9273f40/image-9.jpg)
Median Housing Prices Example {xi} = {house prices} = {$144, 000; 98, 000; 204, 000; 177, 000; 155, 000; 316, 000; 100, 000} Ordered array: $98, 000; 100, 000; 144, 000; 155, 000; 177, 000; 204, 000; 316, 000 Middle Value Median = 155, 000
![Median Another Housing Prices Example {xi} = {house prices} = {$144, 000; 98, 000; Median Another Housing Prices Example {xi} = {house prices} = {$144, 000; 98, 000;](http://slidetodoc.com/presentation_image/c7563dcf2c77f9ec598b8bd6e9273f40/image-10.jpg)
Median Another Housing Prices Example {xi} = {house prices} = {$144, 000; 98, 000; 204, 000; 177, 000; 155, 000; 316, 000; 100, 000; 177, 000; 170, 000} Ordered array: $98, 000; 100, 000; 144, 000; 155, 000; 170, 000; 177, 000; 204, 000; 316, 000 Middle Values Median = (170, 000 + 177, 000)/2 = 173, 500
![Skewed Data 4 Right-skewed data: Data are right skewed if the mean for the Skewed Data 4 Right-skewed data: Data are right skewed if the mean for the](http://slidetodoc.com/presentation_image/c7563dcf2c77f9ec598b8bd6e9273f40/image-11.jpg)
Skewed Data 4 Right-skewed data: Data are right skewed if the mean for the data is larger than the median. 4 Left-skewed data: Data are left skewed if the mean for the data is smaller than the median.
![Skewed Data (Figure 3 -3) Median Mean a) Right-Skewed Mean Median b) Left-Skewed Mean Skewed Data (Figure 3 -3) Median Mean a) Right-Skewed Mean Median b) Left-Skewed Mean](http://slidetodoc.com/presentation_image/c7563dcf2c77f9ec598b8bd6e9273f40/image-12.jpg)
Skewed Data (Figure 3 -3) Median Mean a) Right-Skewed Mean Median b) Left-Skewed Mean = Median c) Symmetric
![Percentiles The pth percentile in a data array is a value that divides the Percentiles The pth percentile in a data array is a value that divides the](http://slidetodoc.com/presentation_image/c7563dcf2c77f9ec598b8bd6e9273f40/image-13.jpg)
Percentiles The pth percentile in a data array is a value that divides the data into two parts. The lower segment contains at least p% and the upper segment contains at least (100 - p)% of the data. The median is the 50 th percentile.
![Quartiles in a data array are those values that divide the data set into Quartiles in a data array are those values that divide the data set into](http://slidetodoc.com/presentation_image/c7563dcf2c77f9ec598b8bd6e9273f40/image-14.jpg)
Quartiles in a data array are those values that divide the data set into four equal-sized groups. The median corresponds to the second quartile.
![Measures of Variation A set of data exhibits variation if all of the data Measures of Variation A set of data exhibits variation if all of the data](http://slidetodoc.com/presentation_image/c7563dcf2c77f9ec598b8bd6e9273f40/image-15.jpg)
Measures of Variation A set of data exhibits variation if all of the data are not the same value.
![Range The range is a measure of variation that is computed by finding the Range The range is a measure of variation that is computed by finding the](http://slidetodoc.com/presentation_image/c7563dcf2c77f9ec598b8bd6e9273f40/image-16.jpg)
Range The range is a measure of variation that is computed by finding the difference between the maximum and minimum values in the data set. R = Maximum Value - Minimum Value
![Interquartile Range The interquartile range is a measure of variation that is determined by Interquartile Range The interquartile range is a measure of variation that is determined by](http://slidetodoc.com/presentation_image/c7563dcf2c77f9ec598b8bd6e9273f40/image-17.jpg)
Interquartile Range The interquartile range is a measure of variation that is determined by computing the difference between the first and third quartiles. Interquartile Range = Third Quartile - First Quartile
![Variance & Standard Deviation The population variance is the average of the squared distances Variance & Standard Deviation The population variance is the average of the squared distances](http://slidetodoc.com/presentation_image/c7563dcf2c77f9ec598b8bd6e9273f40/image-18.jpg)
Variance & Standard Deviation The population variance is the average of the squared distances of the data values from the mean. The standard deviation is the positive square root of the variance.
![Population Variance where: = population mean N = population size 2 = population variance Population Variance where: = population mean N = population size 2 = population variance](http://slidetodoc.com/presentation_image/c7563dcf2c77f9ec598b8bd6e9273f40/image-19.jpg)
Population Variance where: = population mean N = population size 2 = population variance (sigma squared)
![Population Variance (Bryce Lumber Example) Population Variance (Bryce Lumber Example)](http://slidetodoc.com/presentation_image/c7563dcf2c77f9ec598b8bd6e9273f40/image-20.jpg)
Population Variance (Bryce Lumber Example)
![Population Standard Deviation (Bryce Lumber Example) Population Standard Deviation (Bryce Lumber Example)](http://slidetodoc.com/presentation_image/c7563dcf2c77f9ec598b8bd6e9273f40/image-21.jpg)
Population Standard Deviation (Bryce Lumber Example)
![Sample Variance where: = sample mean n = sample size s 2 = sample Sample Variance where: = sample mean n = sample size s 2 = sample](http://slidetodoc.com/presentation_image/c7563dcf2c77f9ec598b8bd6e9273f40/image-22.jpg)
Sample Variance where: = sample mean n = sample size s 2 = sample variance
![Sample Standard Deviation where: = sample mean n = sample size s = sample Sample Standard Deviation where: = sample mean n = sample size s = sample](http://slidetodoc.com/presentation_image/c7563dcf2c77f9ec598b8bd6e9273f40/image-23.jpg)
Sample Standard Deviation where: = sample mean n = sample size s = sample standard deviation
![The Empirical Rule If the data distribution is bell-shaped, then the interval: contains approximately The Empirical Rule If the data distribution is bell-shaped, then the interval: contains approximately](http://slidetodoc.com/presentation_image/c7563dcf2c77f9ec598b8bd6e9273f40/image-24.jpg)
The Empirical Rule If the data distribution is bell-shaped, then the interval: contains approximately 68% of the values in the population or the sample contains approximately 95% of the values in the population or the sample contains virtually all of the data values in the population or the sample
![The Empirical Rule (Figure 3 -11) 95% 68% X The Empirical Rule (Figure 3 -11) 95% 68% X](http://slidetodoc.com/presentation_image/c7563dcf2c77f9ec598b8bd6e9273f40/image-25.jpg)
The Empirical Rule (Figure 3 -11) 95% 68% X
![Tchebysheff’s Theorem Regardless of how the data are distributed, at least (1 - 1/k Tchebysheff’s Theorem Regardless of how the data are distributed, at least (1 - 1/k](http://slidetodoc.com/presentation_image/c7563dcf2c77f9ec598b8bd6e9273f40/image-26.jpg)
Tchebysheff’s Theorem Regardless of how the data are distributed, at least (1 - 1/k 2) of the values will fall within k = 1 standard deviations of the mean. For example: At least (1 - 1/12) = 0% of the values will fall within k=1 standard deviation of the mean ä At least (1 - 1/22) = 3/4 = 75% of the values will within k=1 standard deviation of the mean ä ä At least (1 - 1/32) = 8/9 = 89% of the values will within k=1 standard deviation of the mean fall
![6 Sigma Quality 4 Specification for a quality characteristic is six standard deviation away 6 Sigma Quality 4 Specification for a quality characteristic is six standard deviation away](http://slidetodoc.com/presentation_image/c7563dcf2c77f9ec598b8bd6e9273f40/image-27.jpg)
6 Sigma Quality 4 Specification for a quality characteristic is six standard deviation away from the mean of the process distribution. 4 Translates into process output that does not meet specifications two out of one billion times.
![Sigma Quality Levels Sigma ( ) Quality Level 1 s 2 s 3 s Sigma Quality Levels Sigma ( ) Quality Level 1 s 2 s 3 s](http://slidetodoc.com/presentation_image/c7563dcf2c77f9ec598b8bd6e9273f40/image-28.jpg)
Sigma Quality Levels Sigma ( ) Quality Level 1 s 2 s 3 s 4 s 5 s 6 s Defects per Million Opportunities for Defects 317, 400 45, 400 2700 63 0. 57 0. 002
![Sigma Quality Level Concepts Sigma ( ) Quality Level 1 s 2 s 3 Sigma Quality Level Concepts Sigma ( ) Quality Level 1 s 2 s 3](http://slidetodoc.com/presentation_image/c7563dcf2c77f9ec598b8bd6e9273f40/image-29.jpg)
Sigma Quality Level Concepts Sigma ( ) Quality Level 1 s 2 s 3 s 4 s 5 s 6 s 7 s Equated to Relative Area Floor space of a typical factory Floor space of a typical supermarket Floor space of a small hardware store Floor space of a typical living room Area under a typical desk telephone Top surface of a typical diamond Point of a sewing needle
![Standardized Data Values A standardized data value refers to the number of standard deviations Standardized Data Values A standardized data value refers to the number of standard deviations](http://slidetodoc.com/presentation_image/c7563dcf2c77f9ec598b8bd6e9273f40/image-30.jpg)
Standardized Data Values A standardized data value refers to the number of standard deviations a value is from the mean. The standardized data values are sometimes referred to as zscores.
![Standardized Data Values STANDARDIZED SAMPLE DATA where: x = original data value = sample Standardized Data Values STANDARDIZED SAMPLE DATA where: x = original data value = sample](http://slidetodoc.com/presentation_image/c7563dcf2c77f9ec598b8bd6e9273f40/image-31.jpg)
Standardized Data Values STANDARDIZED SAMPLE DATA where: x = original data value = sample mean s = sample standard deviation z = standard score
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