Last Time Normal Distribution –Density Curve (Mound Shaped) –Family Indexed by mean and s. d. –Fit to data, using sample mean and s.d. Computation of Normal.

Last Time Normal Distribution –Density Curve (Mound Shaped) –Family Indexed by mean and s. d. –Fit to data, using sample mean and s.d. Computation of Normal Probabilities –Using Excel function, NORMDIST –And Big Rules of Probability

Reading In Textbook Approximate Reading for Today’s Material: Pages 61-62, 66-70, 59-61, 322-326 Approximate Reading for Next Class: Pages 337-344, 488-498

Normal Density Fitting Idea: Choose μ and σ to fit normal density to histogram of data, Approach: IF the distribution is “mound shaped” & outliers are negligible THEN a “good” choice of normal model is:

Normal Density Fitting Melbourne Average Temperature Data

Computation of Normal Probs EXCEL Computation: probs given by “lower areas” E.g. for X ~ N(1,0.5) P{X ≤ 1.3} = 0.726

Computation of Normal Probs Computation of upper areas: (use “1 –”, i.e. “not” formula) = 1 -

Computation of Normal Probs Computation of areas over intervals: (use subtraction) = -

Z-score view of populations Idea: Reproducible view of “where data point lies in population”

Z-score view of populations Idea: Reproducible view of “where data point lies in population” Context 1: List of Numbers Context 2: Probability distribution

Z-score view of Lists of #s Idea: Reproducible view of “where data point lies in population”

Z-score view of Lists of #s Idea: Reproducible view of “where data point lies in population” Thought model: population is Normal

Z-score view of Lists of #s Idea: Reproducible view of “where data point lies in population” Thought model: population is Normal Population mean: μ

Z-score view of Lists of #s Idea: Reproducible view of “where data point lies in population” Thought model: population is Normal Population mean: μ Population standard deviation: σ

Z-score view of Lists of #s Idea: Reproducible view of “where data point lies in population” Thought model: population is Normal Population mean: μ Population standard deviation: σ Interpret data as “s.d.s away from mean”

Z-score view of Lists of #s Approach: Transform data

Z-score view of Lists of #s Approach: Transform data By subtracting mean & dividing by s.d

Z-score view of Lists of #s Approach: Transform data By subtracting mean & dividing by s.d. To get

Z-score view of Lists of #s Approach: Transform data By subtracting mean & dividing by s.d. To get (gives mean 0, s.d. 1)

Z-score view of Lists of #s Approach: Transform data By subtracting mean & dividing by s.d. To get (gives mean 0, s.d. 1) Interpret as

Z-score view of Lists of #s Approach: Transform data By subtracting mean & dividing by s.d. To get (gives mean 0, s.d. 1) Interpret as I.e. “ is sd’s above the mean”

Z-score view of Normal Dist. Approach: For

Z-score view of Normal Dist. Approach: For Subtract mean & divide by s.d

Z-score view of Normal Dist. Approach: For Subtract mean & divide by s.d. To get

Z-score view of Normal Dist. Approach: For Subtract mean & divide by s.d. To get (gives mean 0, s.d. 1, i.e. Standard Normal)

Z-score view of Normal Dist. Approach: For Subtract mean & divide by s.d. To get (gives mean 0, s.d. 1, i.e. Standard Normal) Interpret as

Z-score view of Normal Dist. Approach: For Subtract mean & divide by s.d. To get (gives mean 0, s.d. 1, i.e. Standard Normal) Interpret as I.e. “ is sd’s above the mean”

Z-score view of Normal Dist. HW: 1.117

Interpretation of Z-scores Z-scores

Interpretation of Z-scores Z-scores are on N(0,1) scale,

Interpretation of Z-scores Z-scores are on N(0,1) scale, so use areas to interpret them

Interpretation of Z-scores Z-scores are on N(0,1) scale, so use areas to interpret them Important Areas:

Interpretation of Z-scores Z-scores are on N(0,1) scale, so use areas to interpret them Important Areas: 1.Within 1 sd of mean

Interpretation of Z-scores Z-scores are on N(0,1) scale, so use areas to interpret them Important Areas: 1.Within 1 sd of mean “the majority”

Interpretation of Z-scores Z-scores are on N(0,1) scale, so use areas to interpret them Important Areas: 1.Within 1 sd of mean “the majority” ≈ 68%

Interpretation of Z-scores Z-scores are on N(0,1) scale, so use areas to interpret them Important Areas: 2.Within 2 sd of mean “really most” ≈ 95%

Interpretation of Z-scores Z-scores are on N(0,1) scale, so use areas to interpret them Important Areas: 3.Within 3 sd of mean “almost all” ≈ 99.7%

Interpretation of Z-scores Summary: these are called the “68 - 95 - 99.7 % Rule”

Interpretation of Z-scores Summary: these are called the “68 - 95 - 99.7 % Rule” Mean +- 1 - 2 – 3 sd’s

Interpretation of Z-scores Summary: “68 - 95 - 99.7 % Rule” Excel Calculation From Class Example 9: http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg9.xls

Interpretation of Z-scores Summary: “68 - 95 - 99.7 % Rule” Excel Calculation

Interpretation of Z-scores HW: 1.115, 1.116 (50%, 2.5%, 0.18-0.22) 1.119

Inverse Normal Probs Idea, for a given cutoff value, x

Inverse Normal Probs Idea, for a given cutoff value, x Calculated P{X < x}

Inverse Normal Probs Idea, for a given cutoff value, x Calculated P{X < x} as Area under normal density

Inverse Normal Probs Idea, for a given cutoff value, x Calculated P{X < x} as Area under normal density Using Excel function: NORMDIST

Inverse Normal Probs Now for a given P{X < x}, i.e. Area

Inverse Normal Probs Now for a given P{X < x}, i.e. Area Find corresponding cutoff x

Inverse Normal Probs Now for a given P{X < x}, i.e. Area Find corresponding cutoff x Terminology:

Inverse Normal Probs Now for a given P{X < x}, i.e. Area Find corresponding cutoff x Terminology: Quantile

Inverse Normal Probs Now for a given P{X < x}, i.e. Area Find corresponding cutoff x Terminology: Quantile Percentile

Inverse Normal Probs E.g. Given area = 80%

Inverse Normal Probs E.g. Given area = 80% This x is the

Inverse Normal Probs E.g. Given area = 80% This x is the 0.8-quantile

Inverse Normal Probs E.g. Given area = 80% This x is the 0.8-quantile 80-th percentile

Inverse Normal Probs Now for a given P{X < x}, i.e. Area Find: Quantile Percentile

Inverse Normal Probs Now for a given P{X < x}, i.e. Area Find: Quantile Percentile Excel Computation: NORMINV

Inverse Normal Probs Excel Computation: NORMINV

Inverse Normal Probs Excel Computation: NORMINV (very similar to other Excel functions)

Inverse Normal Probs Excel Computation: NORMINV (very similar to other Excel functions) (and reasonably well organized)

Inverse Normal Probs Excel Computation: NORMINV Examples in: http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg9.xls

Inverse Normal Probs Excel Computation: NORMINV

Inverse Normal Probs Excel Computation: NORMINV Set: Mean = 0

Inverse Normal Probs Excel Computation: NORMINV Set: Mean = 0 s.d. = 1 prob = 0.8

Inverse Normal Probs Excel Computation: NORMINV Set: Mean = 0 s.d. = 1 prob = 0.8 Get answer

Inverse Normal Probs Excel Computation: NORMINV or can just type in formula

Inverse Normal Probs Excel Computation: NORMINV or can just type in formula Get answer

Inverse Normal Probs Now for a given P{X < x}, i.e. Area Find: Quantile Percentile = 0.84

Inverse Normal Probs Excel Computation: NORMINV Another example: for X ~ N(100,20)

Inverse Normal Probs Excel Computation: NORMINV Another example: for X ~ N(100,20) Find x, so that 30% = P{X < x}

Inverse Normal Probs Excel Computation: NORMINV Another example: for X ~ N(100,20) Find x, so that 30% = P{X < x} i.e. the 30-th percentile

Inverse Normal Probs Excel Computation: NORMINV Another example: for X ~ N(100,20) Find x, so that 30% = P{X < x} i.e. the 30-th percentile Answer: slightly less than mean

Inverse Normal Probs Example: Quality Control

Inverse Normal Probs When a machine works normally, it fills bottles with mean = 25 oz, and SD = 0.2 oz.

Inverse Normal Probs When a machine works normally, it fills bottles with mean = 25 oz, and SD = 0.2 oz. The machine is “out of control” when it overfills.

Inverse Normal Probs When a machine works normally, it fills bottles with mean = 25 oz, and SD = 0.2 oz. The machine is “out of control” when it overfills. Choose an “alarm level”, which will give only 1 % false alarms.

Inverse Normal Probs When a machine works normally, it fills bottles with mean = 25 oz, and SD = 0.2 oz. The machine is “out of control” when it overfills. Choose an “alarm level”, which will give only 1 % false alarms. Want: cutoff, x, so that Area above = 1%

Inverse Normal Probs When a machine works normally, it fills bottles with mean = 25 oz, and SD = 0.2 oz. The machine is “out of control” when it overfills. Choose an “alarm level”, which will give only 1 % false alarms. Want: cutoff, x, so that Area above = 1% Note: Area below = 100% - Area above = 99%

Inverse Normal Probs When a machine works normally, it fills bottles with mean = 25 oz, and SD = 0.2 oz. Want: cutoff, x, so that Area above = 1% Note: Area below = 100% - Area above = 99%

Inverse Normal Probs When a machine works normally, it fills bottles with mean = 25 oz, and SD = 0.2 oz. Want: cutoff, x, so that Area above = 1% Note: Area below = 100% - Area above = 99% So set alarm threshold to 25.47

Inverse Normal Probs HW: 1.122 (-0.675, 0.385) 1.123 1.132 (1294) 1.133 1.139

And Now for Something Completely Different A fun idea. Can you read this?

And Now for Something Completely Different A fun idea. Can you read this? Olny srmat poelpe can raed this.

And Now for Something Completely Different A fun idea. Can you read this? Olny srmat poelpe can raed this. I cdnuolt blveiee that I cluod aulaclty uesdnatnrd what I was rdanieg.

And Now for Something Completely Different A fun idea. Can you read this? Olny srmat poelpe can raed this. I cdnuolt blveiee that I cluod aulaclty uesdnatnrd what I was rdanieg. The phaonmneal pweor of the hmuan mnid, aoccdrnig to rscheearch at Cmabrigde Uinervtisy.

And Now for Something Completely Different The phaonmneal pweor of the hmuan mnid, aoccdrnig to rscheearch at Cmabrigde Uinervtisy.

And Now for Something Completely Different The phaonmneal pweor of the hmuan mnid, aoccdrnig to rscheearch at Cmabrigde Uinervtisy. It deosn't mttaer in what oredr the ltteers in a word are, the olny iprmoatnt tihng is that the first and last ltteer be in the rghit pclae.

And Now for Something Completely Different The phaonmneal pweor of the hmuan mnid, aoccdrnig to rscheearch at Cmabrigde Uinervtisy. It deosn't mttaer in what oredr the ltteers in a word are, the olny iprmoatnt tihng is that the first and last ltteer be in the rghit pclae. The rset can be a taotl mses and you can still raed it wouthit a porbelm.

And Now for Something Completely Different The rset can be a taotl mses and you can still raed it wouthit a porbelm.

And Now for Something Completely Different The rset can be a taotl mses and you can still raed it wouthit a porbelm. Tihs is bcuseae the huamn mnid deos not raed ervey lteter by istlef, but the word as a wlohe.

And Now for Something Completely Different The rset can be a taotl mses and you can still raed it wouthit a porbelm. Tihs is bcuseae the huamn mnid deos not raed ervey lteter by istlef, but the word as a wlohe. Amzanig huh?

And Now for Something Completely Different The rset can be a taotl mses and you can still raed it wouthit a porbelm. Tihs is bcuseae the huamn mnid deos not raed ervey lteter by istlef, but the word as a wlohe. Amzanig huh? Yaeh and I awlyas tghuhot slpeling was ipmorantt!

Checking Normality Idea: For which data sets, will the normal distribution be a good model?

Checking Normality Idea: For which data sets, will the normal distribution be a good model? Recall fitting normal density to data:

Normal Density Fitting Idea: Choose μ and σ to fit normal density to histogram of data, Approach: IF the distribution is “mound shaped” & outliers are negligible THEN a “good” choice of normal model is:

Normal Density Fitting Melbourne Average Temperature Data

Checking Normality Idea: For which data sets, will the normal distribution be a good model? Useful graphical device to check: IF the distribution is “mound shaped” & outliers are negligible

Checking Normality Useful graphical device:

Checking Normality Useful graphical device: Quantile – Quantile plot

Checking Normality Useful graphical device: Quantile – Quantile plot Varying Terminology:

Checking Normality Useful graphical device: Quantile – Quantile plot Varying Terminology: Q-Q plot

Checking Normality Useful graphical device: Quantile – Quantile plot Varying Terminology: Q-Q plot Normal Quantile plot (text book)

Checking Normality Q-Q plot

Checking Normality Q-Q plot Idea: graphical comparison

Checking Normality Q-Q plot Idea: graphical comparison of data distribution

Checking Normality Q-Q plot Idea: graphical comparison of data distribution vs. normal distribution

Checking Normality Q-Q plot Idea: graphical comparison of data distribution vs. normal distribution as data quantiles vs. normal quantiles

Checking Normality Q-Q plot, implementation:

Checking Normality Q-Q plot, implementation: Sort data, to find data quantiles

Checking Normality Q-Q plot, implementation: Sort data, to find data quantiles Assign corresponding probabilities:

Checking Normality Q-Q plot, implementation: Sort data, to find data quantiles Assign corresponding probabilities: (equally spaced, strictly between 0 and 1)

Checking Normality Q-Q plot, implementation: Sort data, to find data quantiles Assign corresponding probabilities: Compute corresponding normal quantiles

Checking Normality Q-Q plot, implementation: Sort data, to find data quantiles Assign corresponding probabilities: Compute corresponding normal quantiles (using NORMINV)

Checking Normality Q-Q plot, implementation: Sort data, to find data quantiles Assign corresponding probabilities: Compute corresponding normal quantiles (using NORMINV) Make plot with x-axis

Checking Normality Q-Q plot, implementation: Sort data, to find data quantiles Assign corresponding probabilities: Compute corresponding normal quantiles (using NORMINV) Make plot with x-axis & y-axis

Checking Normality Q-Q plot, interpretation:

Checking Normality Q-Q plot, interpretation: When distribution is normal:

Checking Normality Q-Q plot, interpretation: When distribution is normal: –Points lie close to a line

Checking Normality Q-Q plot, interpretation: When distribution is normal: –Points lie close to a line –For standard normal quantiles

Checking Normality Q-Q plot, interpretation: When distribution is normal: –Points lie close to a line –For standard normal quantiles Y-intercept of line is mean Slope of line is s.d.

Checking Normality Q-Q plot, interpretation: When distribution is normal: –Points lie close to a line –For standard normal quantiles Y-intercept of line is mean Slope of line is s.d. For non-normal distribution:

Checking Normality Q-Q plot, interpretation: When distribution is normal: –Points lie close to a line –For standard normal quantiles Y-intercept of line is mean Slope of line is s.d. For non-normal distribution: –Q-Q plot will curve away from line

Checking Normality Q-Q plot, e.g.

Checking Normality Q-Q plot, e.g. Excel analyses available in: http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg10.xls

Checking Normality Q-Q plot, e.g. n = 1000 from N(0,1) Data simulated as:

Checking Normality Q-Q plot, e.g. n = 1000 from N(0,1) Data simulated as:  Data Tab

Checking Normality Q-Q plot, e.g. n = 1000 from N(0,1) Data simulated as:  Data Tab  Data Analysis

Checking Normality Q-Q plot, e.g. n = 1000 from N(0,1) Data simulated as:  Data Tab  Data Analysis  Random Number Generation

Checking Normality Q-Q plot, e.g. n = 1000 from N(0,1) Data simulated as:  Data Tab  Data Analysis  Random Number Generation  Set parameters

Checking Normality Q-Q plot, e.g. n = 1000 from N(0,1) Next sort data  Copy to another column

Checking Normality Q-Q plot, e.g. n = 1000 from N(0,1) Next sort data  Copy to another column  Highlight

Checking Normality Q-Q plot, e.g. n = 1000 from N(0,1) Next sort data  Copy to another column  Highlight  Data Tab

Checking Normality Q-Q plot, e.g. n = 1000 from N(0,1) Next sort data  Copy to another column  Highlight  Data Tab  Sort Button

Checking Normality Q-Q plot, e.g. n = 1000 from N(0,1) Next sort data  Copy to another column  Highlight  Data Tab  Sort Button Gives Data Quantiles

Checking Normality Q-Q plot, e.g. n = 1000 from N(0,1) Next compute Normal Quantiles

Checking Normality Q-Q plot, e.g. n = 1000 from N(0,1) Next compute Normal Quantiles  1 st type indices  Range of probs i / (n+1)

Checking Normality Q-Q plot, e.g. n = 1000 from N(0,1) Next compute Normal Quantiles  1 st type indices  Range of probs i / (n+1)  Normal quantiles

Checking Normality Q-Q plot, e.g. n = 1000 from N(0,1) Now plot Data Quantiles vs. Normal Quantiles

Checking Normality Q-Q plot, e.g. n = 1000 from N(0,1) Now plot Data Quantiles vs. Normal Quantiles  Insert Tab

Checking Normality Q-Q plot, e.g. n = 1000 from N(0,1) Now plot Data Quantiles vs. Normal Quantiles  Insert Tab  Scatter Button

Checking Normality Q-Q plot, e.g. n = 1000 from N(0,1) Now plot Data Quantiles vs. Normal Quantiles  Insert Tab  Scatter Button  Fill out menu (as before)

Checking Normality Q-Q plot, e.g. n = 1000 from N(0,1) Results: Looks very linear

Checking Normality Q-Q plot, e.g. n = 1000 from N(0,1) Results: Looks very linear As expected

Checking Normality Q-Q plot, e.g. n = 1000 from N(0,1) Results: Looks very linear As expected Y-intercept = 0 (= mean)

Checking Normality Q-Q plot, e.g. n = 1000 from N(0,1) Results: Looks very linear As expected Y-intercept = 0 (= mean) Slope = 1 (= s.d.)

Checking Normality Q-Q plot, e.g. Buffalo Snowfalls

Checking Normality Q-Q plot, e.g. Buffalo Snowfalls Recall Histogram

Checking Normality Q-Q plot, e.g. Buffalo Snowfalls Recall Histogram - Roughly symmetric

Checking Normality Q-Q plot, e.g. Buffalo Snowfalls Recall Histogram - Roughly symmetric - Mound shaped

Checking Normality Q-Q plot, e.g. Buffalo Snowfalls Recall Histogram - Roughly symmetric - Mound shaped - Does Normal Curve fit the data?

Checking Normality Q-Q plot, e.g. Buffalo Snowfalls Approximately linear

Checking Normality Q-Q plot, e.g. Buffalo Snowfalls Approximately linear Suggests normal

Checking Normality Q-Q plot, e.g. Buffalo Snowfalls Approximately linear Suggests normal But some wiggles?

Checking Normality Q-Q plot, e.g. Buffalo Snowfalls Approximately linear Suggests normal But some wiggles? Due to natural sampling variation?

Checking Normality Q-Q plot, e.g. Buffalo Snowfalls Approximately linear Suggests normal But some wiggles? Due to natural sampling variation? Study with smaller simulation

Checking Normality Q-Q plot, e.g. n = 100 from N(0,1)

Checking Normality Q-Q plot, e.g. n = 100 from N(0,1) Approximately linear

Checking Normality Q-Q plot, e.g. n = 100 from N(0,1) Approximately linear Some wiggliness

Checking Normality Q-Q plot, e.g. n = 100 from N(0,1) Approximately linear Some wiggliness Suggests Buffalo variation is usual

Checking Normality Q-Q plot, e.g. n = 100 from N(0,1) Approximately linear Some wiggliness Suggests Buffalo variation is usual Make this more precise?

Checking Normality Q-Q plot, e.g. British Suicides

Checking Normality Q-Q plot, e.g. British Suicides Recall Histogram

Checking Normality Q-Q plot, e.g. British Suicides Recall Histogram  Strong right skewness

Checking Normality Q-Q plot, e.g. British Suicides Recall Histogram  Strong right skewness  So mean >> median

Checking Normality Q-Q plot, e.g. British Suicides Recall Histogram  Strong right skewness  So mean >> median  Not mound shaped

Checking Normality Q-Q plot, e.g. British Suicides

Checking Normality Q-Q plot, e.g. British Suicides Distinct non-linearity (curvature)

Checking Normality Q-Q plot, e.g. British Suicides Distinct non-linearity (curvature) Conclude data not normal

Checking Normality Q-Q plot, e.g. British Suicides Distinct non-linearity (curvature) Conclude data not normal Characteristic of right skewness

Checking Normality Q-Q plot, e.g. Log10 British Suicides Recall: log10 transformation resulted in mound shape

Checking Normality Q-Q plot, e.g. Log10 British Suicides Recall Histogram

Checking Normality Q-Q plot, e.g. Log10 British Suicides Recall Histogram: o Much more mound shaped

Checking Normality Q-Q plot, e.g. Log10 British Suicides Recall Histogram: o Much more mound shaped o Check for normality with Q-Q plot

Checking Normality Q-Q plot, e.g. Log10 British Suicides

Checking Normality Q-Q plot, e.g. Log10 British Suicides Looks very linear

Checking Normality Q-Q plot, e.g. Log10 British Suicides Looks very linear Indicates normal distribution is good fit

Checking Normality Q-Q plot, e.g. Log10 British Suicides Looks very linear Indicates normal distribution is good fit I.e. transformation worked!

Checking Normality HW: 1.143 1.145 1.146 (a. approx. normal + big outlier; b. close to normal; c. right skew + one big outlier; d. Non-normal with several clusters

Last Time Normal Distribution –Density Curve (Mound Shaped) –Family Indexed by mean and s. d. –Fit to data, using sample mean and s.d. Computation of Normal.

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Last Time Normal Distribution –Density Curve (Mound Shaped) –Family Indexed by mean and s. d. –Fit to data, using sample mean and s.d. Computation of Normal.

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