Stats: Chapter 6- What's Normal?

the first three rules for working with normal models

1) make a picture 2) make a picture 3) make a picture histograms will help describe the models - ex) the normal curve is bell- shaped and symmetric around its mean the place where the bell shape changes from curving downward to cuving back up- the inflection point- is exactly one standard deviation away from the mean

what happens when we find the z- score?

1) we shift the data by subtracting the mean- shift the mean to 0 (does not change sd) 2) rescale the values by dividing by the standard deviation when we divide all of the shifted values by the standard deviation, the sd should be divided by s as well- new standard deviation becomes 1 affects on the distribution: - standardixing into z-scores does not change the shape of the distribution of a variable - standardizing into z- scores changes the center by making the mean 0 - standardizing into z scores changes the spread by making the standard deviation 1

the range, standard deviation, and IQR...

do not change even when the values are changed (as long as they are changed consistently)

use the mean or standard devaitation (the difference between the data and the mean) to describe the distribution when...

the data is symmetric

standardizing

we standardize to eliminate units. standardized values can be compared and combined even if the original variables had different units and magnitudes uses the standard deviation as a ruler to measure distance from the mean, creating z- scores

finding normal percentages by hand

when data does not fall exactly 1,2, or 3 standard deviations from the mean, we can look it up in a table of standard normal percentages: correspond the value on the z- table to find an area value: multiple by 100 to get a percentage (area)- shows that all of the data to the left of the value is __% of the whole data the whole model is 100%: context

when are values unusual in terms of the 68-95-99.7 rule?

when they are above or below 3 standard deviations away from the mean

standard normal model

N(u, o)- Normal model with a mean (u) of 0 and standard deviation (o) of 1- N(0,1) aka the standard normal distribution unimodal and symmetric, no outliers (typically)

formula to find z scores

[y (value) - y- bar (mean)]/ s (standard deviation) has no values measures the distance of a value from the mean negative z- score: left of the mean positive z- score: to the right of the mean

nearly normal condition

a distribution is nearly Normal if it is unimodal and symmetric- usually free of outliers determined by a histogram

parameter

a numerically valued attributed of a model- ex) u, o the mean and standard deviation are not numerical summarizes of the data but rather, they are part of the model: they are numbers that we choose to help specify the model that are not part of the given

normal model

a useful family of models for unimodal, symmetric distributions- "bell- shapes" curves doesn't have every rivets cannot be used when the distribution is not unimodal and summetric + don't use mean and sd when outliers are present

statistic

a value calculated from data to summarize aspects of the data ex) mean (y-, s) and standard deviation

standardized value

a value found by subtracting the mean (from the given) and then dividing by the standard deviation also known as the z- score (denoted with the letter z) have no units compare original values to the mean have been converted from their original units to the unit of standard deviation from the mean- allows us to compare values that are measured on different scales, w/ different units, or in different populations

shifting data in terms of z- scores

add or subtract a constant amount to/ from each value- adds or subtracts the same constant from the mean, median, etc leaves measures of spread the same leaves measures of percentiles and center shifted by the constant value

re-scaling data in terms of z- scores

change the shape of the data multiply or divide all values by a constant value measures of spread (range, IQR, mean, med, mode)- divided and multiplied by that same value

68-95-99.7 rule

in a normal model, about 68% of values fall within 1 standard of the mean, about 95% fall within 2 standard deviations of the mean, and about 99.7% fall within 3 standard deviations of the mean

standardizing units into z- scores: shifting and rescaling

shifts the data by subtracting the mean and rescales the data by diving by their standard deviation consequences of standardizing: does not change the shape changes the center by making the mean zero changes the spread by making the standard deviation 1

the percentiles to scores: z in scores

start with the area (percentages): find the corresponding z score using the chart

z- score

tells how many standard deviations a value is from the mean- meaurement: negatives simply mean in the opp direction (below mean), positive= above mean-- the farther the value is from the mean, the more unusual it is have a mean of 0 and a standard deviation of 1 when working with data, use to statistics mean and standard deviation: (value - mean)/ (standard deviation) allow us to compare values from different distributions or values based on different units can identify unusual or surprising values among data

standard deviation

tells us how much the whole collection of values varies- a natural ruler for comparing an individual value to the group

normal percentile

the normal percentile corresponding to a z-score gives the percentage of values in a standard Normal distribution found at that z-score or below

normality assumption

we somtimes work with Normal tables- are based on the Normal modelp- we cannot assume that every model will be Normal