stats 201: The normal distributions

density curve and key properties, z scores,6895 and 99.7

exploring a distribution:

plot data, look for overall aptter(shape, center and variability), calculate numerical summaryand now we add one more step—smooth curve

for exploring data on a single numerical variable… what is step 1

ploatting data with histogram or a box plot

for exploring data on a single numerical variable… what is step 2

looking for the overall apttern—shape,center and varibaility and outliers and choose either five number summary or the mean and stdev to briefly describe the center and varibaility

for exploring data on a single numerical variable… what is step 3

calulating numerical summary to briefly descirb the cneter and varibaiilty

for exploring data on a single numerical variable… what is step 4

sometimes the overall pattern of a large number of observations is so regular that we can describe it by a smooth curve

if we have large sample sizea histirgram can be descrieb by a

smooth curve aka density curve ( approximate)

what are the two IMPORTANT DISTINCTIONS between histogram and density curve

-a density curve is intended to reflect the idealized shape of the population distribution

-the total area under the density curve is =1 or 100%

Whta s the area of a line?

0

Two important properties of a density curve

A density curve is a curve that

- is always on or above the horizontal axis.

- has an area of exactly 1=100% underneath it.

density curve describes the overall pattern of a distribution and is

is intended to reflect the idealized shape of the population distribution.

The median of a density curve is the

equal-areas point— which divides the area under the curve in half

The mean of the density curve is the

balance point or center of gravity at which that curve would balance if it were made of solid material

for a symmmertic density curve..

the mean and the median are the same and both lie at the center of the curve

the mean of a skewed density curve is 

pulled away from the median in the direction of the long tail

Special density curves—normal curves descirbe

nromal distribution

what are normal curves?

symmetric, single-peaked, annd bell shaped

normal curve is completely described by 

mean and standard deviation—> mu and standard deviation

The mean is located at the center of the symmetric curve and is the

same as median

What does the std control

-the variability or shape of a nromal curve

when sigma(STDEV) is larger the curve

spreads out further and the area under the normal curve is less concentrated about the mean

In the special normal distribution —what are values of the mew and stdev

mew is zero and sigma is 1 N(0,1)

mapping ibservatiosn from N(mew, sigma) to N(0,1) is called standardizing

yes true

Z-score ::we standardize using z= x-mew/sigma

z=observation-mean/stdev—-ZSCORE

The standard score for any observation is called a 

z score

A positive z-score indicates that

the observation is above the mean

A negative z-score indicates that the observation

is below the mean

the cumulative proportion for a value x^0 in a distrivution is the proportion of observations that are

less than or equal to x^0

because all normal distributions are the same once we standardize..

we can find the areas under any normal curve from a single table—called standard normal table

The table entry for each value z is the?

area under the curve to the left of the z of the z

c-th percentile of a distribution is a

value such that c percent of the observations lie below it.

The 68 -95-99.1 rule for normal distributions explain

68%: of the observations fall within one standard dev of the mean (u-o,u+o)

95%: of the observations fall within two standard deviations of the mean: (u-2o, u+2o)

99.7%: of the observations fall within three standard deviations of the mean (u-3o,u+3o)