2.1 - Normal Distributions & Z-Scores

Standard Normal Curve

A specific normal distribution with a mean of 0 and a standard deviation of 1, often used in statistics to compute probabilities and z-scores.

• Gaussian Distribution

Important Properties of the Normal Curve

• defined by mean & SD

• mean splits into = upper & lower halves

• mean, median, mode are the same

Mean & SD of Standard Normal Curve

mean = 0

SD = 1

• X much meaning outside of this

How is the Normal Curve defined by the mean & SD?

The normal curve is defined by its mean and standard deviation, where the mean indicates the center of the distribution and the standard deviation determines the width and height of the curve. Together, they determine the shape and spread of the distribution.

• required to render a normal distrib

• changing either makes a new distrib

• wide = bigger SD, narrow = smaller SD } different curves

• used to determine analyses, if other data meets mean & SD

Z-Score

way to compare SDs

• standardized, shows where indiv data point falls on curve relative to the mean of the distribution.

Goal of Z-Scores

• allow us to convert a datum/observation into something standard

• can use the standard to compare different data points across distributions. The goal of Z-scores is to express how many standard deviations an individual data point is from the mean.

Calculating Z-Scores

To calculate Z-scores, subtract the mean from the individual data point and divide the result by the standard deviation. This standardized score indicates how many standard deviations an observation is above or below the mean of the distribution.

Z Score of 1

always 1 SD above the mean, in any relevant units

• -1 is 1 less, +2 is 2 above, etc

Z Scores for Normality of Values

can be used to assess how unusual or weird a value is

• rule of thumb = more than ± SD from mean counts as unusual

• large z score = unusual, X impossible

Individual Observations & Z Scores

can be used to find out where an indiv observation fits on a curve or to calc percentiles

How are SDs used to find proportions?

Standard deviations help determine the proportion of values that lie within certain ranges under the normal distribution curve. By converting raw scores to z-scores, one can use the z-table to find the area under the curve, which corresponds to probabilities or proportions.

How can distributions be skewed?

positive or negative skew

Kurtosis

peaky or tailsy distrubtions

• platykurtic

• leptokurtic

• mesokurtic

Platykurtic Distributions

very flat, have long tails

Leptokurtic Distributions

very pointy/peaky

Mesokurtic Distributions

just right, not too flat or long, or pointy

How can data be determined as normal?

whether data is normal enough → depends on what you do w/ it

• need to justify what you’re doing

Kolmogorov-Smirnov & Shapiro-Wilk Tests

tests of normality, both very sensitive to n

• will show significant deviation even when data is usable

• use only on n < 50, SW better to use

Q-Q Plot

a graphical tool to compare the quantiles of a dataset against the quantiles of a theoretical normal distribution to assess normality.

• good visual method for double checking data, especially for larg n

• X a test → eyeballing data/graphical tool

Interpreting a K-S/S-W Test

p </= 0.05 → consider data non-normal

• generally best for small datasets (n < 50)

• very likely to give a misleading (+) the larger the dataset gets

Q-Q Plot Normal Distribution

values mostly sticking to expected line

• few deviations of low value

Assessing Skewness/Kurtosis

more than 3 times standard error = non-normal