STATISTICS FOR BUSINESS

STATISTICS FOR BUSINESS Chapter 5 : Continuous distributions and probability Your can of beer, bar of chocolate, or length of fabric

STATISTICS FOR BUSINESS (Continuous distributions and probability) Your can of beer, bar of chocolate, or length of fabric Normal distribution is a continuous distribution • The Normal distribution is most important tenet in statistics. • It is a probability distribution, or frequency of occurrence • It is a polygon and describes a continuous random variable. • Developed by the German, Karl Friedrich Gauss (1777-1855) • Also known as the Gaussian distribution. • Can give information about probability outcomes in business

STATISTICS FOR BUSINESS (Continuous distributions and probability) Your can of beer, bar of chocolate, or length of fabric NORMAL DISTRIBUTION When we have set target values, bottling or filling machine (wine, water, beer, yogurt ) then for a large quantity of data, then distribution will be almost normal. Mean value is target value • Bell shaped and symmetrical • Left side mirror image of ride side • Tails can extend far at left and right • Mean, median, mode all at hump -3s +3s

STATISTICS FOR BUSINESS (Continuous distributions and probability) Your can of beer, bar of chocolate, or length of fabric Normal distributions All Normal distributions even though mean and standard deviations may be different

STATISTICS FOR BUSINESS (Continuous distributions and probability) Your can of beer, bar of chocolate, or length of fabric • Kurtosis value is 5.66 • A small standard deviation • Curve is leptokurtic (slender) • Peak is sharp Target distribution for six-sigma Kurtosis value indicates shape of Normal distribution • Kurtosis value is 0.60 • An intermediate standard deviation • Curve is mesokurtic (intermediate) • Peak is flat Inappropriate for six-sigma but better Leptokurtic has less dispersion thus more reliable for analysis Six-sigma basis • Kurtosis value is -1.37 • A large standard deviation • Curve is platykurtic (broad or flat) • Peak is flat Inappropriate for six-sigma

STATISTICS FOR BUSINESS (Continuous distributions and probability) Your can of beer, bar of chocolate, or length of fabric Areas under a Normal distribution Total area under curve is 100% Standard Normal distribution (x-axis is value z)

STATISTICS FOR BUSINESS (Continuous distributions and probability) Your can of beer, bar of chocolate, or length of fabric Transformation relationship for Normal distribution Transformation formula of continuous data; volume, weight, length, speed, noise (decibels), area, density, viscosity, ….. into a standard normal distribution x is value of random variable; µx is mean of all random variables sxis standard deviation of distribution; z is number of standard deviations from mean • Since units in numerator and denominator of the equation are the same, z has no units • When x is > than µ, z is positive; when x is < than µ, z is negative; Thus we write  z • Consider can of beer • Nominal (average) volume of 33 cl; Standard deviation of the filling operation is 1.25 cl • Sample taken from filling line has a volume of 34.25 cl. z = (34.25 – 33.00)/1.25 = 1.00 (no units) • Consider slab of chocolate • Nominal (average) weight of 100 gm; Standard deviation of molding operation is 2.05 g • Sample taken from production line has weight of 102.05 g; z = (102.05 – 100.00)/2.05 = 1.00 (no units) • Consider piece of fabric • Fabric has a nominal (average) length of 2 m; Standard deviation of cutting operation is 0.25 m • Sample taken from operation has a length of 2.25 m; z = (2.25 – 2.00)/0.25 = 1.00 (no units)

STATISTICS FOR BUSINESS (Continuous distributions and probability) Your can of beer, bar of chocolate, or length of fabric With transformation relationship • there is a standardized normal distribution • these are functions in Excel • A standardized normal distribution • Is one whose : • Random variable is z (not x) • Mean µz = 0 • Standard deviation , s = 1. -3s +3s

STATISTICS FOR BUSINESS (Continuous distributions and probability) Your can of beer, bar of chocolate, or length of fabric By extension for right side of distribution P(x) 1 - P(x) 0 x

STATISTICS FOR BUSINESS (Continuous distributions and probability) Your can of beer, bar of chocolate, or length of fabric Normal distribution and binomial distribution Binomial distribution related to Normal distribution if following applies Product of sample size and probability of "success “ is greater than five n*p  5 Product of sample size and probability of "failure“ is greater than five n*q  5 or n*(1 –p)  5 In binomial distribution , µ = n*p and s = √ (n*p*q) From normal transformation formula Substituting Relationship used in statistical process control (SPC)

STATISTICS FOR BUSINESS (Continuous distributions and probability) Your can of beer, bar of chocolate, or length of fabric Normal distribution and box and whisker plot Normal distribution Mean, median, and mode are equal Area to left of median is mirror image of area to right of median Box and whisker plot Interquartilerange (Q3– Q1) = 1.33 standard deviations

STATISTICS FOR BUSINESS (Continuous distributions and probability) Your can of beer, bar of chocolate, or length of fabric Right skewed distribution and box and whisker plot Right skewed data • Right or positively skewed • Mean is greater than the median • Mode is less than median • In sequential order from left: mode, median, mean • More lower values to left of mean–more cheaper hotel rooms Box and whisker plot

STATISTICS FOR BUSINESS (Continuous distributions and probability) Your can of beer, bar of chocolate, or length of fabric Left skewed distribution and box and whisker plot Left skewed data • Left or negatively skewed • Mean is less than the median • Mode is higher than median • In sequential order from left: mean, median, mode • More higher values to right of mean Box and whisker plot

STATISTICS FOR BUSINESS