Ch 13: The Normal Distribution

density curves

shows the frequency of observations at each point

shape of a normal density curve

1. single peak

2. bell shape

3. approximately symmetrical

central unit theorem

more observations w/ SRS → distribution looks “normal”

what’s the difference between histograms and density curves?

histograms show the shape of the sample, but the density curves “smooth” it out and create the more ideal shape that we would likely see in the population

what is variability represented by?

standard deviation

how can you identify standard deviation by looking at a graph?

by looking at the slope of the curve or the inflection point

what happens if we change the mean of the distribution?

the shape of the distribution will be unchanged; the curve will simply shift to the left or right

what happens if we change the standard deviation of the distribution?

• smaller SD → less variability and more sharply peaked

• larger SD → higher variability and less sharply peaked

68% rule

68% of observations fall within ±1 standard deviation on either said of the mean

95%

95% of observations fall within ±2 standard deviations on either side of the mean

99.7%

99.7% of observations fall within ±3 standard deviations on either side of the mean

z-score

standard score/standard deviation units

positive z-score

the observations is above the mean

negative z-score

below the mean

how can we use standard score

to compare how well each individual score relative to the norms for their respective tests

the c^th percentile

a value such that c% of the observations lie below it and the rest lie above it

each standard score has a specific percentile

this percentile will be the same no matter what the mean and standard deviation of the original distribution are