Section 2.5 The Normal Distribution

 68% of values lie within 1 SD of the mean.  Including to the right and left  90% of the values lie with SDs of the mean.  95% lie within about 2 SDs (actually 1.96 SDs) of the mean.  99.7% of the data lie within 3 SDs of the mean.

 Think of a case variable that when measured will give results that are normally distributed.  Determine how you would measure the variable:  Measure a time  Measure length or distance  Survey people  Actually run an experiment or get survey results and collect data sufficient enough to determine if the data is roughly normally distributed.  Plot the data with a histogram or dot plot and create a smooth curve over it to show its basic shape.  Find a measure a center and spread for the data.  If it is roughly normal, check to see if approximately 68% of the data is within 1 SD of the mean, 95% within 2 SDs and 99.7% within 3 SDs.  If it is not normal, find the quartiles and create a box plot of the data.

 Very common distribution of data throughout many disciplines.  SAT / ACT scores  Measure of diameter of tennis balls  Heights / weights of people  Once we know a distribution is Normal, there is a tremendous amount of information we can determine or predict about it.

 All normal distributions have the same basic shape.  The difference: tall and thin vs. short and fat  However, we could easily stretch the scale of the tall thin curve to make it identical to the short fat one.  The area under the curve can be thought of in terms of proportions or percentage of data.  The total area under the curve is 1.0 (100%)

 We can standardize any normal curve to be identical.  We do this by treating the mean as Zero and the SD as One.  The variable along the x-axis becomes what we call a z score.  The z score is the number of SDs away from the mean.

 Practice problems:  1) Normal distribution with:  mean = 45 and SD = 5  Find the z score for a data value of 19  Find the z score for a data value of 52  2) Normal distribution with:  Mean = 212 and SD = 24  Find the z score for a data value of 236

 We can use the standard normal curve to find proportion of data in a range of values.  Normal Curve example: SAT I Math scores  Mean = 500 SD = 40  Find the proportion of data in the score range 575 or less.  Using z tables: Table A very back of book  Find the proportion of data above 575.  Find the proportion of data between 490 and 550.

 Read all of 2.5.  Be prepared for quiz on Tuesday.

 You have collected data regarding the weights of boys in a local middle school. The distribution is roughly normal. The mean is 113 lbs and the SD is 10 lbs.  A) What proportion of boys are below 100 lbs?  B) What proportion are above 120 lbs?  C) What proportion are in between 90 & 120 lbs?

 You can also use the TI-83 or higher to find these same proportions:  2 nd, Distr, normalcdf(low, high,mean,SD)  When using z scores you can leave mean,SD blank. normalcdf(low, high) it will default to mean=0 and SD=1.  This will give you the same area under the curve (proportion of data) as the z table.

 If you know the percent of data covered under a normal distribution, you can find the z-score.  Simply look up the percent (proportion) in the z table and relate it to the corresponding z score.  Find the value that is closest to the percent given  Another method is with the calculator.  2 nd,Distr, invNorm(proportion, mean, SD)

 Find the z-score that has the given percent of values below it in a standard normal distribution:  a) 32% b) 41% (use the z-table)  c) 87% d) 94%(use your calculator)

 If you know how many SDs a value is from the mean, you can use this (z-score) to find the actual data value:  x = mean + (z SD)  Example: The mean weight of the boys at a middle school is 113 lbs, with a SD of 10 lbs. One boy is determined to be 2.2 SDs above the mean. How much did the boy weigh?

 So now, if you know the percentage of data above or below a data value and you know the mean and SD, you can figure out that data value:  Use z-table to find the z-score, then use the z score with mean and SD to find the data value.  Or you can use the invNorm function on your calc. ▪ invNorm(proportion, mean, SD)

 The heights of U.S yr old females is roughly normally distributed with a mean of 64.8 in. and a SD of 2.5 in.  Estimate the percent of women above 5’8”  What height would a US female be if she was 1.5 SDs below the mean? Give your answer in ft & in.  What height would a US female be if she was considered to be in the 80 th percentile?

 What percentage of US females is above 5’7”?  What percent are between 5’7” and 5’0”?

 The cars in Clunkerville have a mean age of 12 years and a SD of 8 years. What percentage of cars are more than 4 years old?  Why is this a trick question?

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