STATS ex 2 - Normality of Distribution

Normality of a Distribution

display the frequency of a continuous variable

normal curve has a symmetrical or equal distribution of scores around the mean with a small number of outliers in the two tails

can be displayed in a table or a figure

frequency table

method of organizing data by listing every possible value in the first column of numbers and the frequency of each value as the second column

steps to normality of distribution is on slide deck

okkk

Normal Curve

theoretical normal curve is an expression of statistical theory

• it is a theoretical frequency distribution of all possible scores

• however, no real distribution exactly fits the normal curve

• the theoretical curve is symmetrical, unimodal and has continuous values (mean, median and mode are equal)

Skewness

any frequency distribution that is not symmetrical is referred to as skewed or asymmetrical

in skewed distribution mean, median, and mode are not equal

skewness interferes with validity of many statistical analyses

Positively Skewed

largest portion of data is below the mean

mean is greater than the median which is greater than the mode

Negatively skewed

largest portion of data is above the mean

the mean is less than the median which is less than the mode

Kurtosis

explains the degree of peakedness of the frequency distribution, which is related to the spread of variance of scores

extreme kurtosis can affect the validity of statistical analysis because the scores have little variance

Leptokurtic

extremely peaked distribution

Mesokurtic

intermediate degree of kurtosis

Platykurtic

relatively flat distribution

kurtosis indications

0 - indicates curve is mesokurtic

value above 0 - indicate curve is leptokurtic

values below 0 (negative) - indicate platykurtic curve

skewness and kurtosis should be assessed when

prior to statistical analysis - importance of such non-normality needs to be determined by both the researcher and the statistician

Skewness and Kurtosis statistic values of ≥+1 or ≤ -1 are..

fairly severe and could impact outcomes form parametric analysis techniques

if kurtosis and skewness is severe nonparametric testing should be considered

Shapiro-Wilk’s W test

is a formal test of normality that assesses whether a variable’s distribution is skewed and or kurtotic

this test has the ability to calculate both skewness and kurtosis by comparing the shape of the variable’s frequency distribution to that of a perfect normal curve

Looking at result for “Number of Times Fired from Job” what can you conclude

1. positively skewed (>1)

2. Kurtosis = 0.576 (slightly leptokurtic)

3. Significant deviation from normal (p=.001)

Looking at the results for “Age of 1st arrest” what can you conclude

1. positively skewed (>1)

2. Kurtosis = 2.043 (Leptokurtic)

3. Shapiro Wilk = 0.017 (Significant deviation from normal)

Looking at age enrollment what did you conclude ?

appeared to be negatively skewed but did not yield skewness nor kurtosis or Shapiro-Wilk values that indicated deviations from normality (p=.373)

What is skewness

measure of asymmetry of the distribution

What is kurtosis

the degree of peakedness of the frequency distribution

(≥+1 or ≤ -1 are fairly severe)

How would you characterize the kurtosis of this distribution

leptokurtic - small variation.

How would you characterize the skewness of the distribution

negatively skewed (tail on left)

how would you characterize the kurtosis of the distribution

platykurtic

does not have extreme kurtosis (=-.575)

positively skewed