bio stat (exam 2) lecture 9a + 9b

the area under the curve must equal

equal to 1 AKA TOTAL Probability

aera =

the proportion of random variable values between a and b

normal distribution

Are symmetric, single-peaked, and bell-shaped

Each normal distribution is determined by its mean and standard deviation (spread)

facts about normal distribution

• Most widely used distribution in statistics/data/analysis/machine learning

• Often abused and misused

• Also known as bell-shaped or Gaussian

• Defined by mean and standard deviation

z-scores

The number of standard deviations the observation (data point) is from the mean

The formula is Z = x (standard value given) - mean / standard deviation

mean

average, center of distribution

standard deviation

measures how spread out the data are

unimodal

curve has one peak

larger standard deviation means

wider, flatter curve

smaller standard deviation means

narrower, taller curve

empirical rule

what percentage of data falls within 1,2, and 3 standard deviations of the mean

1 standard deviation

68%

2 standard deviation

95%

3 standard deviation

99.7%

standard normal distribution

mean = 0

sd = 1

is a normal distribution, but instead of working with the orginal values, we convert them into standardized scores called z-scores

z-score

tells us how many standard deviations a value is above or below them

ex. mean = 80

standard deviation = 5

what percentage of students scored between 75 and 85

80 - 5 = 75

80 + 5 = 85

apply empirical rule

answer = 68%

when z score = 0

exactly at the mean

when z-score = 1

exactly one SD above the mean

ex. z score

mean = 80

sd = 5

your score = 90

z = (90 - 80) / 5

z = 10 / 5

z = 2

meaning your score is 2 SD above the mean

percentile

tells you the percentage of observations that fall at or below a particular value

ex. 50th percentile = median

75th percentile = 75% of observations are at or below that value

distributional assumption

where we assume the data comes from a specific probability model

• normal

• binomial

• exponential

distributional assumption visual representations

• histogram

• qq plot

histograms

used to visualize if data is adequately large

qq plots

• A great visual to check a distributional assumption (does not need to be normal)

• Shows the relationship between observed (sampled) data and what we would expect in theory

• If the relationship = linear, then assumption is valid