Download Density Curves

Alexis, Sommy, Elizabeth, Charlie
Density Curves
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Always on or above the horizontal axis
Has an area of exactly one underneath it
Describes the overall pattern of the
distribution and shows the proportion of all
observations within a certain interval
Median: “equal areas point”, divides the area
of the curve in half
Mean: “balance point”, the point at which the
curve would balance if made of sold material
Solving problems
with density curves
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Plot your data, make a graph, usually a histogram or
stem plot
Look for a pattern (SOCS)
Calculate a numerical summary to describe center
and spread
Empirical Rule
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In a normal distribution with mean µ
and standard deviation σ
68% of the observations fall within σ
of the mean µ
95% of the observations fall within 2σ
of the mean µ
99.7% of the observations fall within
3σ of the mean µ
Percentile
Pth percentile is the value such
that p percent of the observations
fall at or below it
 Often used for test scores in which
the data is normally distributed
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Comparing Distributions
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Z-scores
Percentiles: p percent of values that fall at or
below the given number
 Can use either z-scores or percentiles to
compare data across two different distributions
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Standardizing Data
Z score: tells how many standard deviations
away from the mean and the original
observation falls and in which direction
 Observations larger than the mean are positive
when standardized
 Observations smaller than the mean are
negative when standardized
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Standard Normal Table
Gives the area under the standard normal curve
 The table entry value z is the area under the curve left of z
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Normal Proportions Calculations
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State the problem in terms of the observed
variable x
Draw a picture of the distribution and
shade the area of interest under the curve
Standardize x to restate the problem in
terms of the normal variable z. Draw a
picture to show the area of interest under
the standard normal curve
Use the table to find the required area
under the standard Normal curve
Write your conclusion in context
Normal Probability Plot
Used to see if a normal model is adequate for the
data
 If the points lie close to a straight line the plot
indicates that the data are Normal
 Systematic deviations from a straight line indicate
a Non-Normal distribution
 Outliers appear as points that are far from the
overall pattern of the plot
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Calculator Keystrokes
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Histogram
STAT > 1. Edit > ENTER > L1>ENTER> (enter data) > 2nd Y = > 1. Plot
> ENTER > On> ENTER> arrow over to histogram drawing> ENTER>
arrow down to X list> L1 (second 1)> GRAPH
Z-score
2nd VARS> 2. Normalcdf( > lower (lowest score) >ENTER> upper(
maximum score) > ENTER> mean of data> ENTER > standard
deviation of data > ENTER> paste> ENTER> answer is z score from
table
Score needed
2nd VARS> 3. invNorm( > area (z-score) >ENTER> mean of data>
ENTER > standard deviation of data > ENTER> paste> ENTER>
answer is score needed to lie in that area of distribution
Five Number Summary
STAT > 1. Edit > ENTER > L1>ENTER> (enter values) > STAT> arrow
right to CALC> 1. 1-VAR Stats> ENTER> List (L1)> arrow down to
calculate> ENTER
(mean, standard deviation, min, Q1, Median, Q3, Max)