Unit 2 - Modeling Distributions of Data

pth percentile

percentage of observations at or below a given observation 

_{^{^include values below or = to the chosen value, but do not include the chosen value itself! (ex: value is 22, if there are two 22s only include one of them)}}

cumulative relative frequency graph

displays the cumulative relative frequency of each class of a frequency distribution

_{^{^think of y-axis as percentile}}

_{^{^go to next class → cumulates, add up percentage for that class and the ones below it}}

z-score

how many standard deviations from the mean an observation is

^ z = (observation - mean)/standard deviation

_{^{*aka standardized score}}

_{^{^higher - more above avg}}

*allows comparison of different data sets (ex: SAT vs. ACT scores, who did better on their respective test)

*higher z-score means more above the mean/did ‘more’ better than others, but does not mean higher # of smth (b/c z-score says nothing abt sample size!)

adding/subtracting to transform data

• add ‘a’ to/subtract ‘a’ from measures of center & location (mean, median, quartiles, percentiles) 

• shape/spread do not change _{^{(the rest change)}}

multiplying/dividing to transform data

• multiply/divide measures of center & location (mean, median, quartiles, percentiles) by ‘b’

• mult/divide measures of spread by |b|

• shape does not change _{^{(unless b is negative) (the rest change)}}

!!! more unusual if percentile is farther from median (50th percentile)

Q₁ 25th percentile, Q₃ 75th percentile

write units

cumulative rel freq graph, what will be the shape of histogram -> look at where the median is, if it is more left prob right-skewed, if more right it is prob left-skewed

make histogram from cumulative rel freq graph:

• x-axis same, make the gaps the bins

• y-axis is percent, make bars as tall as the change in cumulative rel freq from the graph (if looking at btwn 10-20 and 10 is 8 and 20 is 30, then the change is 22, so that is the height of the bar on the histogram!!)

density curve

a curve that is always on or above the horizontal axis and has a total area of 1 below the curve

^mean μ = at ‘balance point’

^standard deviation σ

^median = point where ½ data is above and ½ is below

_{^{*describes overall pattern of a distribution; ideal description of a distribution of data}}

_{^{*excludes outliers; not perfect}}

normal curve

• describes a normal distribution

• always same shape (symmetric, unimodal/1 peak, bell-shaped)

• completely described by its mean/standard deviation

• mean and median in the center of the normal curve

normal distribution

• described by a normal density curve

• notation: N(μ, σ)

• _{^{standard deviation is the distance from the mean to the change-of-curvature points on either side}}

68-95-99.7 Rule

68% of observations fall within 1 standard deviation (σ)

95% of observations fall within 2 σs

99.7% of observations fall within 3 σs _{^{(0.3% left, 0.15% on both sides)}}

standard normal distribution

the normal distribution has a mean of 0 and standard deviation of 1 (same units!)

standard normal table A

table of areas under the standard normal curve

^only used for z-scores

^rows are ones and tenths place, columns are hundredths

^using it helps you find the proportion of observations BELOW the given # (like z < [given])

• If you want to find above (z > [given]), do 1 minus the # you get

• if a < z < b, find the difference (do the value you get for b (what's on the table) minus the value you get for a, no doing any '1 - [value you get]')

• working backwards - given percentile -> convert to decimal (if '#% of all observations are GREATER than z,' do 1 - [the decimal]), find on table, look at what z it is!

!!!

normal curve → mean = median

• if a curve is skewed, the mean is pulled towards the tail of the data

peak of the curve is the mode