Random Variable Probability Distribution X=Amt in next bottle X=total of 2 tossed dice X=#G in 4 N(μ=10.2,σ=0.16) B(n=4,p=.5) Summary Characteristics Mean Median Mode Std dev Variance skew We must admit that we cannot know exactly what value X will take…. …so that we can do the intelligent thing and talk about something we CAN know, the probability distribution of X. There are summary characteristics of any probability distribution… But knowing these summary measures do not replace our need to know the probability distribution.

Class 06: Descriptive Statistics EMBS: 3.1, 3.2, first part of 3.3

Characteristics of probability distributions Measures of Location – Mean – Median – Mode Measures of Variability – Standard Deviation – Variance Measure of Shape – skewness Descriptive Statistics (for numerical data) Measures of Location – Sample Mean – Sample Median – Sample Mode Measures of Variability – Sample StDev – Sample Variance Measure of Shape – Sample skewness

A positively-skewed pdf Mode is the most likely value P(X median) = 0.5 Mean is the probability- weighted average Skewness > 0

A negatively-skewed pdf Skewness < 0

An exhibit at MOMA invites visitors to mark their heights on a wall. A normal distribution results:exhibit Well, not quite. The distribution is actually slightly negatively skewed by the confounding presence of children, who are obviously shorter than adults - you can see this in the great number of names well below the central band which are not mirrored by names higher up. Rest assured, however, that the ex- children distribution is itself Gaussian.

The Normal pdf tes06.html Mean = μ median = μ mode = μ Skewness = 0

Measures of Variability w=1226&bih=866&tbm=isch&tbnid=pppxDi8aC37y8M:&imgrefurl= comfsm.fm/~dleeling/statistics/notes06.html&docid=Hu1RM- siu0MevM&imgurl= iff_sx.gif&w=401&h=322&ei=9qAqT8KXAcPptgfC3uX0Dw&zoom=1&iact=hc&vpx =748&vpy=508&dur=1013&hovh=201&hovw=251&tx=142&ty=111&sig= &page=1&tbnh=149&tbnw=186&start=0&ndsp=20&ved=1t:4 29,r:13,s:0 σ = 0.7 σ = 1.0 σ = 1.5

Skewed pdfs can also have different standard deviations Which pdf has the largest σ?

Pdfs Can have different means, but identical standard deviations Which pdf has the largest σ? Which pdf has the largest μ?

Characteristics of probability distributions Measures of Location – Mean – Median – Mode Measures of Variability – Standard Deviation – Variance Measure of Shape – skewness Descriptive Statistics (for numerical data) Measures of Location – Sample Mean – Sample Median – Sample Mode Measures of Variability – Sample StDev – Sample Variance Measure of Shape – Sample skewness Probability weighted average 50% point Most likely Expected squared distance from mean Neg if skewed left, 0 if symmetric, pos if skewed right.

Characteristics of probability distributions Measures of Location – Mean – Median – Mode Measures of Variability – Standard Deviation – Variance Measure of Shape – skewness Descriptive Statistics (for numerical data) Measures of Location – Sample Mean – Sample Median – Sample Mode Measures of Variability – Sample StDev – Sample Variance Measure of Shape – Sample skewness Probability weighted average 50% point Most likely Expected squared distance from mean Neg if skewed left, 0 if symmetric, pos if skewed right. =average() =median() =mode() =stdev() =var() =skew()

Characteristics of probability distributions Measures of Location – Mean – Median – Mode Measures of Variability – Standard Deviation – Variance Measure of Shape – skewness Descriptive Statistics (for numerical data) Measures of Location – Sample Mean – Sample Median – Sample Mode Measures of Variability – Sample StDev – Sample Variance Measure of Shape – Sample skewness Probability weighted average 50% point Most likely Expected squared distance from mean Neg if skewed left, 0 if symmetric, pos if skewed right. GET THEM ALL USING DATA ANALYSIS, DESCRIPTIVE STATISTICS, SUMMARY STATISTCS

Characteristics of probability distributions Measures of Location – Mean – Median – Mode Measures of Variability – Standard Deviation – Variance Measure of Shape – skewness Descriptive Statistics (for numerical data) Measures of Location – Sample Mean – Sample Median – Sample Mode Measures of Variability – Sample StDev – Sample Variance Measure of Shape – Sample skewness Probability weighted average 50% point Most likely Expected squared distance from mean Neg if skewed left, 0 if symmetric, pos if skewed right. RANGE COUNT

The sample standard deviation

Understanding sample standard deviation stdev stdev stdev8.07 It measures variability about the mean. All the data contribute to the measure. It measures variability …. In either direction. XXXXX XXXXX

Our Data Section ND IDGender (M=1)HS Stat? HtValue

Data/DataAnalysis/DescriptiveStatistics SummaryStatistics Section ND ID Gender (M=1) HS Stat? Ht Value Mean4.493Mean Mean0.609Mean0.217Mean69.351Mean0.185 Standard Error0.061 Standard Error Standard Error0.059 Standard Error0.050 Standard Error0.477 Standard Error0.056 Median Median1 0 70Median0 Mode4 #N/AMode1 0 71Mode0 Standard Deviation0.504 Standard Deviation Standard Deviation0.492 Standard Deviation0.415 Standard Deviation3.959 Standard Deviation0.465 Sample Variance0.254 Sample Variance Sample Variance0.242 Sample Variance0.173 Sample Variance Sample Variance0.216 Kurtosis-2.060Kurtosis0.555Kurtosis-1.847Kurtosis-0.039Kurtosis-0.793Kurtosis6.385 Skewness0.030Skewness-0.581Skewness-0.455Skewness1.401Skewness-0.307Skewness2.706 Range Range1 1 16Range2 Minimum Minimum0 0 60Minimum0 Maximum Maximum1 1 76Maximum2 Sum310Sum Sum42Sum15Sum Sum12.75 Count69Count69Count69Count69Count69Count69

Data/DataAnalysis/DescriptiveStatistics SummaryStatistics Section ND ID Mean4.493Mean Standard Error0.061Standard Error Median Mode4 #N/A Standard Deviation0.504 Standard Deviation Sample Variance0.254Sample Variance Kurtosis-2.060Kurtosis0.555 Skewness0.030Skewness Range Minimum Maximum Sum310Sum Count69Count69

Data/DataAnalysis/DescriptiveStatistics SummaryStatistics Gender (M=1) HS Stat? Mean0.609Mean0.217 Standard Error0.059Standard Error0.050 Median1 0 Mode1 0 Standard Deviation0.492Standard Deviation0.415 Sample Variance0.242Sample Variance0.173 Kurtosis-1.847Kurtosis Skewness-0.455Skewness1.401 Range1 1 Minimum0 0 Maximum1 1 Sum42Sum15 Count69Count69

Data/DataAnalysis/DescriptiveStatistics SummaryStatistics Ht Value Mean69.351Mean0.185 Standard Error0.477Standard Error0.056 Median70Median0 Mode71Mode0 Standard Deviation3.959Standard Deviation0.465 Sample Variance15.673Sample Variance0.216 Kurtosis-0.793Kurtosis6.385 Skewness-0.307Skewness2.706 Range16Range2 Minimum60Minimum0 Maximum76Maximum2 Sum Sum12.75 Count69Count69

Fill Test Data Normal(10.2,0.16)? EXHIBIT 2 LOREX PHARMACEUTICALS Filling Line Test Results with Target =

Fill Test Data Descriptive Statistics Summary Statistics Amount Mean Standard Error0.014 Median Mode#N/A Standard Deviation0.163 Sample Variance0.026 Kurtosis0.771 Skewness0.245 Range0.997 Minimum9.758 Maximum Sum Count144

Fill Test Data Histogram BinFrequency More1 1 data point was < data points were between and was above Data Data Analysis Histogram Check chart output

Preview of Coming Attractions Class 07 – Find out how to use these counts to test H0: these data came from N(10.2,.16) – Find out how to use the Denmark family counts to test H0: those data came from Binomial(4,.5)