The Standard Deviation as a Ruler & The Normal Model

Expressing the distance from the mean of a set of data in standard deviations ___ the values

Standardizes (can compare 2 or more sets of data)

Z-score

measures the distance of a value from the mean in standard deviations (does NOT have units)

Z-score formula

z-score = (individual data point - mean)/standard deviation

2 things done by standardizing data to get a z-score

1. Shifting data by subtracting the mean

2. Rescale the values by dividing by their standard deviation

Shifting Data

Adding or subtracting a constant to every single data point in a set (changes measure of center by that constant)

Rescaling Data

Multiply every number in the data set by a constant (changes measures of center and spread by that constant)

Standardizing into z-scores doesn’t change the ___ of the distribution of a variable

shape (shifting and rescaling didnt change it)

Standardizing changes the ___ by making the mean zero

center (Because of the shift)

Standardizing changes the ___ by making the Standard Deviation one

spread (because its being divided by the standard deviation)

How far away from 0 does a z-score have to be to be unusual or interesting

No universal standard but the farther the z-score is from 0 (positive or negative) the more unusual it is

The Normal Model are appropriate for many distributions whose shapes are ___ and ___

UNIMODAL & SYMMETRIC

Normal Model distributions…

provide a measure of how extreme a z-score is

N (μ,σ)

symbol that represents the Normal model with mean of μ and standard deviation of σ

Parameters

numbers used to specify the model (written usually with Greek Letters) that arent numerical summaries of the data (doesnt come from the data)

Statistics

summaries of the data that are usually written with Latin Letters

Z-score formula of a Normal Model

z= (x-μ)/σ

Standard Normal Model/ Standard Normal Distribution

Normal Model with a mean of 0 and Standard Deviation of 1 (N(0,1)) that can be used to standardize any normal model

Empirical Rule/ 68,95,97.7 rule

• About 68% of the data falls within 1 Standard Deviation of the mean

• 95% fall within 2 Standard Deviations

• 97.7% falls within 3 Standard Deviations

Inflection point

The place where the bell shape changes its curvature and is exactly 1 standard deviation away from the mean

Figuring out the percentile (percentage of data that falls below) with a z-score

1. find the z-score

2. Use a normal percentile table or technology

To figure out the percentage ABOVE the given z-score

subtract the percentile by 1

Normal Probability Plot

used to find if data is normal. If the distribution of the data is roughly normal, the plot is roughly a diagonal line. Deviations from a straight line indicate that the distribution isnt normal

It’s reasonable to use a normal model when…

the distribution is symmetrical and unimodal or ROUGHLY symmetric and unimodal

It’s NOT reasonable to use a normal model when…

The data is definitely skewed

What can go wrong?

• Dont use a normal model when the data isnt unimodal & symmetric

• Dont use the mean and standard deviation when outliers are present

• Dont round your results in the middle of your calculations

• Do what we say, not what we do

• Dont worry about minor differences in results