Ch 2 - Modeling Distributions of Data

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32 Terms

1
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percentile definition

the value with p percent of the observations less than or equal to it (“at” not “in)

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percentile sentence template

  • ___ is at the pth percentile

  • p% of data points (units) were less than/greater than or equal to____

(general = less than, specific = greater than)

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cumulative relative frequency

a graph showing the accumulation of frequencies up to each category in a dataset, allowing comparison of different percentiles.

<p>a graph showing the accumulation of frequencies up to each category in a dataset, allowing comparison of different percentiles. </p>
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z score

how many standard deviations from the mean & observation falls & whether it is above (pos.) or below (neg.) the mean

  • also known as the standardized score

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z score formula

data item - mean / standard deviation

<p>data item - mean / standard deviation</p>
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z score sentence template

The “data item” is “absolute value of z-score” standard deviations above/below the mean.

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measures of center & location

  • mean, median, quartiles

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measures of spread

  • standard deviation, range, IQR

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transforming data with addition or subtraction (of constant a)

  • center & location (mean, median, quartiles, max, min) change by a

  • spread (standard deviation, range, IQR) and shape stay the same

  • Z SCORE DOESNT CHANGE (bc total stand devs)

bc the pts are still the same relative to each other (horizontal/vertical shift)

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transforming data w/ a multiplier (of constant b)

  • center & location (mean, median, quartiles) shrink/stretch by b

  • spread (standard deviation, range, IQR) shrink/stretch by b

  • shape does not change

  • Z SCORE DOESNT CHANGE (bc total stand devs)

bc this is a stretch/shrink it zooms in/out the graph

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μ

mean :)

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M

population mean (to do w/ normal curve)

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x bar

sample mean

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σ

standard deviation :)

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a density curve

the smooth curve drawn thru the tops of HISTOGRAM bars (2 characteristics)

  • is always on or above the horizontal axis

  • has TOTAL area exactly 1 underneath it

(no set of real data can be exactly described by density curve, curve is an approximation)

  • does not matter on a continuous x axis if x=# bc if it equals its a vertical line ~ 0 

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median of a density curve

equal area point where half the area under the curve lies to the left and half to the right, representing the middle value of the distribution.

<p>equal area point where half the area under the curve lies to the left and half to the right, representing the middle value of the distribution. </p>
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mean of a density curve

balance point of a density curve (the center of gravity of the distribution, not necessarily where the areas are equal.

  • Skew pulls the mean toward the longer tail.

<p>balance point of a density curve (the&nbsp;<em>center of gravity</em> of the distribution, not necessarily where the areas are equal.</p><ul><li><p>Skew pulls the mean toward the longer tail.</p></li></ul><p></p>
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proportion probability

4 things

  1. [0,1] bc def of density curve

  2. P(x>#) = x(height) = .decimal area

  3. =%

  4. UNITS

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P(point = x)

is about 0 bc is a vertical line

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Normal Distributions

“special” type of density curve

  • notation: Normal

  • same overall shape: symmetric, unimodal, and bell-shaped

(not all symmetric curves are normal, ie bimodal)

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Describing normal distribution

Must identify 2 things:

  • mean μ→ the center (bc symmetric)

  • standard deviation σ → is the dis. from the center to the change-of curvature points on either side (points of inflection)

    • 1 standard dev. above mean, 1 standard dev. below

  • larger stand dev = more spread

<p>Must identify 2 things:</p><ul><li><p>mean μ→ the center (bc symmetric)</p></li><li><p>standard deviation σ → is the dis. from the center to the change-of curvature points on either side (points of inflection)</p><ul><li><p>1 standard dev. above mean, 1 standard dev. below</p></li></ul></li><li><p>larger stand dev = more spread</p></li></ul><p></p>
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normal distribution notation

uses N(µ, σ), where µ is the mean and σ is the standard deviation

  • X ~ N (70,2)

  • “x is distributed” on normal curve mean of 70, standard deviation of 2

<p>uses N(µ, σ), where µ is the mean and σ is the standard deviation</p><ul><li><p>X ~ N (70,2)</p></li><li><p>“x is distributed” on normal curve mean of 70, standard deviation of 2</p></li></ul><p></p>
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any Normal curve “standardized” =

N(0,1)

converting this to z score

  • mean: 0

  • standard deviation: 1

STANDARDIZING DOES NOT CHANGE the overall shape of the distribution.

<p>N(0,1)</p><p>converting this to z score</p><ul><li><p>mean: 0 </p></li><li><p>standard deviation: 1</p></li></ul><p>STANDARDIZING DOES NOT CHANGE the overall shape of the distribution. </p><p></p>
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Empirical Rule

68-95-99.7 rule

  • appx. 68% of area falls within σ of the mean µ

    • if the curve is symmetric & normal

    • (1 stand below mean - 1 stand dev above mean)

  • appx. 95% of area falls within 2σ of the mean µ

    • (2 stand below mean - 2 stand dev above mean)

  • appx. 99.7% of the area falls within 3σ of the mean µ

    • (3 stand below mean - 3 stand dev above mean)

.15%, 2.35%, 13.5%, 34%, 34%, 13.5%, 2.35%, .15%

<p>68-95-99.7 rule</p><ul><li><p>appx. 68% of area falls within σ of the mean µ</p><ul><li><p>if the curve is symmetric &amp; normal</p></li><li><p>(1 stand below mean - 1 stand dev above mean)</p></li></ul></li><li><p>appx. 95% of area falls within 2σ of the mean µ</p><ul><li><p>(2 stand below mean - 2 stand dev above mean)</p></li></ul></li><li><p>appx. 99.7% of the area falls within 3σ of the mean µ</p><ul><li><p>(3 stand below mean - 3 stand dev above mean)</p></li></ul></li></ul><p>.15%, 2.35%, 13.5%, 34%, 34%, 13.5%, 2.35%, .15% </p><p></p>
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f(μ) of a normal curve

<p></p>
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steps to proportion when z score is not integer

  1. draw normal curve and shade area of interest

  2. use proper notation to write proportion of interest p(z < #) = proportion of area to left of z score

  3. (if u use a diff variable u must define it and x~N(115,6) before converting to z score

<ol><li><p>draw normal curve and shade area of interest</p></li><li><p>use proper notation to write proportion of interest p(z &lt; #) = proportion of area to left of z score</p></li><li><p>(if u use a diff variable u must define it and x~N(115,6) before converting to z score</p></li></ol><p></p>
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Is the data normal?

Make a Normal Probability Plot (which assumes data is normal)

  • enter data

  • → stat plot

  • → select graph

  • → zoom 9

“The Normal Probability Plot is roughly linear so the data is approximately Normal”

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if its asks abt a percentile, proportion or percent its asking for?

area!

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if it asks abt a data item, mean, or standard deviation?

find z score! go backwards

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<p>normal probability plot ∩</p>

normal probability plot ∩

majority of data points falls on the right side (of the linear line), which indicates a tail on the left, so the data is skewed left

<p>majority of data points falls on the right side (of the linear line), which indicates a tail on the left, so the data is skewed left </p>
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<p>normal probability plot&nbsp;∪</p>

normal probability plot ∪

majority of data points falls on the left side (of the linear line), which indicates a tail on the right, so the data is skewed right

<p>majority of data points falls on the left side (of the linear line), which indicates a tail on the right, so the data is skewed right</p>
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when changing the paramenters of a Normal curve?

restate the distribution