Characteristics of Distributions

random variable

is a numeric function of the outcomes of an experiment. (how many heads in a certain number of coin tosses)
A random variable is discrete if it can only assume a countable number of possible values

A discrete probability function

describes how to calculate probabilities about a discrete random variable

Characteristics of a distribution

Measures of central tendency (typical value)

– Mean

– Median

mode?

• Measures of spread/variability, degree of dispersion

– Variance

– Standard deviation

• Types of shapes

– Symmetric, how values are apportioned throughout range of distribution

– Positively skewed (right skew)

– Negatively skewed (left skew)

measures of central tendency

mean

median

Types of shapes

Symmetric

Positively skewed (right skew)

Negatively skewed (left skew)

Mean

mu is true mean of the distribution, expected value/mean of random variable weighted by porbabilities
x is the possible value for the random variable

P(X = x) is the probability of seeing x accodging to discrete probability function

Median

X = is random variable

median can be a single value in the dataset or an interval

Measures of spread

variance

standard deviation

Variance

Standard deviation

average deviation of random variable from mean of distribution

Types of shapes

mean is a possible median!

Mean vs. median

• When is the mean a better measure of a “typical” value?

– Symmetric distribution

• Mean and median are the same

• Mean has nicer mathematical properties

• When is the median a better measure of a typical value?

– Skewed distribution or presence of extreme values

• Median better represents a “typical” value

• Mean more sensitive to extreme values