Database Management Exam > Database Management Videos > Mastering R Programming: For Data Science and Analytics > Normal Distribution; Z Scores; and Normal Probabilities in R (R Tutorial 3.3)
Normal Distribution; Z Scores; and Normal Probabilities in R (R Tutorial 3.3) Video Lecture | Mastering R Programming: For Data Science and Analytics - Database Management
FAQs on Normal Distribution; Z Scores; and Normal Probabilities in R (R Tutorial 3.3) Video Lecture - Mastering R Programming: For Data Science and Analytics - Database Management
| 1. What is a normal distribution and why is it important in statistics? |  |
Ans. A normal distribution, also known as a Gaussian distribution, is a probability distribution that is symmetric and bell-shaped. It is important in statistics because many natural phenomena follow a normal distribution, allowing for easier analysis and prediction. Additionally, many statistical techniques and tests are based on the assumption of normality.
| 2. What are Z-scores and how are they used in statistics? | |
Ans. Z-scores, also known as standard scores, measure the number of standard deviations an observation or data point is from the mean of a normal distribution. They are calculated by subtracting the mean from the observation and dividing the result by the standard deviation. Z-scores are used to standardize data and compare observations from different distributions. They also help in determining the probability associated with a particular observation.
| 3. How can R be used to calculate probabilities in a normal distribution? | |
Ans. R provides various functions to calculate probabilities in a normal distribution. The function "pnorm(x, mean, sd)" can be used to calculate the cumulative probability up to a given value "x" in a normal distribution with a specific mean and standard deviation. Similarly, the function "qnorm(p, mean, sd)" can be used to calculate the quantile (value) corresponding to a given cumulative probability "p". These functions are handy in determining the likelihood of observing a particular value or range in a normal distribution.
| 4. What is the significance of the mean and standard deviation in a normal distribution? | |
Ans. The mean (μ) is the central tendency of a normal distribution and represents the average value of the data. It determines the location of the peak of the distribution. The standard deviation (σ) measures the spread or variability of the data. It determines the width of the bell-shaped curve. The mean and standard deviation together characterize the entire distribution and provide valuable information about the data's central tendency and dispersion.
| 5. How can R be used to generate random numbers from a normal distribution? | |
Ans. In R, the function "rnorm(n, mean, sd)" can be used to generate 'n' random numbers from a normal distribution with a specific mean and standard deviation. This function is useful for simulating data or creating random samples that follow a normal distribution. By specifying appropriate mean and standard deviation values, one can generate random numbers that mimic real-world data distributions.
Text Transcript from Video
hi I'm Mike Marin and in this video
we'll talk about calculating
probabilities percentiles and taking
random samples from a normally
distributed variable throughout this
video we will use an example with a
variable X which is known to be normally
distributed with a mean of 75 and a
standard deviation of 5 we can calculate
probabilities for a normal distribution
using the P norm command to access the
help menu type help and in the brackets
the command you would like help for or
simply throw a question mark in front of
the name of the command that is
mentioned we can use the P norm command
to calculate probabilities for a normal
random variable let's go ahead and
calculate the probability that X is less
than or equal to 70 for our example to
do so we would like the probability for
a normal variable the probability
associated with a value of 70 we have a
mean of 75 a standard deviation of 5 and
we would like a lower tail probability
so we will set this equal to true by
lower tail we mean less than 70
in our the default for the lower tail
argument is true so it's not necessary
to include this if we would like a lower
tail probability let's go ahead and
calculate the probability that X is
greater or equal to 85 to do so again we
can use the P norm command we would like
the probability associated with 85 and
here we would like the upper tail so we
would set the lower tail to false the P
norm command can also be used to
calculate probabilities for Z the
standard normal let's go ahead and
calculate the probability that Z is
greater or equal to 1 to do so again we
want the probability for a normal here
we're on a z value of 1 this has a mean
of 0 and a standard deviation of 1 again
we want to set the lower tail to false
or have an upper tail probability we can
now see the area above a z-score of 1 is
approximately 16% the Q norm function
can be used to calculate quantiles or
percentiles for a normal random variable
let's go ahead and calculate the first
quartile for this variable to do so we
will use the Q norm command here we
would like to find the 25th percentile
or the first quartile this variable has
a mean of 75 a standard deviation of 5
and again lower tail equal to true we
would like a lower tail probability of
25% seventy one point six is approximate
value for the first quartile we may also
want to find and/or plot the probability
density function to do so we may use the
d norm function let's go ahead and
create a plot
the probability density function for
this example we've been discussing a
normal variable with a mean of 75
standard deviation of 5 to do so first
we will create a sequence of x-values
that run from 55 up to 95 in increments
of 0.25 we will save this in an object
called X here we can see the sequence
we've created we can then go ahead and
find the value of the probability
density function for each of these X
values using the D norm command we will
save this in an object called de NS here
we would like to find the values of the
probability density for the normal
associated with X a mean of 75 and a
standard deviation of 5 we can then go
ahead and plot the values of X versus
the densities to see the probability
density function if we like we may have
all the points in this plot attached
using a line to do so we can include the
type argument and set this equal to L
for a line
if desired one may add a title to this
plot and label axes using arguments
covered earlier in this series of videos
I've just copied and pasted this command
I entered earlier to produce a nice plot
with labels and titles
if one would like to add a vertical line
at the mean of 75 this can be done using
the a/b line argument here we would like
to include a vertical line at the value
x equals 75 finally one may wish to have
our draw a random sample from a normally
distributed population to do so we can
use the our norm command let's go ahead
and draw a sample of N equals 40
observations from this normally
distributed population with a mean of 75
and a standard deviation of 5 again I
will go ahead and save this random
sample in an object labeled R a and D
here we can see the random sample we've
selected we can also take a quick look
at a histogram for this random sample
it's worth noting here that even though
this sample was taken from a normally
distributed population the sample
histogram itself may not necessarily
look very normal thanks for watching
this video and make sure to check out my
other instructional videos
we'll talk about calculating
probabilities percentiles and taking
random samples from a normally
distributed variable throughout this
video we will use an example with a
variable X which is known to be normally
distributed with a mean of 75 and a
standard deviation of 5 we can calculate
probabilities for a normal distribution
using the P norm command to access the
help menu type help and in the brackets
the command you would like help for or
simply throw a question mark in front of
the name of the command that is
mentioned we can use the P norm command
to calculate probabilities for a normal
random variable let's go ahead and
calculate the probability that X is less
than or equal to 70 for our example to
do so we would like the probability for
a normal variable the probability
associated with a value of 70 we have a
mean of 75 a standard deviation of 5 and
we would like a lower tail probability
so we will set this equal to true by
lower tail we mean less than 70
in our the default for the lower tail
argument is true so it's not necessary
to include this if we would like a lower
tail probability let's go ahead and
calculate the probability that X is
greater or equal to 85 to do so again we
can use the P norm command we would like
the probability associated with 85 and
here we would like the upper tail so we
would set the lower tail to false the P
norm command can also be used to
calculate probabilities for Z the
standard normal let's go ahead and
calculate the probability that Z is
greater or equal to 1 to do so again we
want the probability for a normal here
we're on a z value of 1 this has a mean
of 0 and a standard deviation of 1 again
we want to set the lower tail to false
or have an upper tail probability we can
now see the area above a z-score of 1 is
approximately 16% the Q norm function
can be used to calculate quantiles or
percentiles for a normal random variable
let's go ahead and calculate the first
quartile for this variable to do so we
will use the Q norm command here we
would like to find the 25th percentile
or the first quartile this variable has
a mean of 75 a standard deviation of 5
and again lower tail equal to true we
would like a lower tail probability of
25% seventy one point six is approximate
value for the first quartile we may also
want to find and/or plot the probability
density function to do so we may use the
d norm function let's go ahead and
create a plot
the probability density function for
this example we've been discussing a
normal variable with a mean of 75
standard deviation of 5 to do so first
we will create a sequence of x-values
that run from 55 up to 95 in increments
of 0.25 we will save this in an object
called X here we can see the sequence
we've created we can then go ahead and
find the value of the probability
density function for each of these X
values using the D norm command we will
save this in an object called de NS here
we would like to find the values of the
probability density for the normal
associated with X a mean of 75 and a
standard deviation of 5 we can then go
ahead and plot the values of X versus
the densities to see the probability
density function if we like we may have
all the points in this plot attached
using a line to do so we can include the
type argument and set this equal to L
for a line
if desired one may add a title to this
plot and label axes using arguments
covered earlier in this series of videos
I've just copied and pasted this command
I entered earlier to produce a nice plot
with labels and titles
if one would like to add a vertical line
at the mean of 75 this can be done using
the a/b line argument here we would like
to include a vertical line at the value
x equals 75 finally one may wish to have
our draw a random sample from a normally
distributed population to do so we can
use the our norm command let's go ahead
and draw a sample of N equals 40
observations from this normally
distributed population with a mean of 75
and a standard deviation of 5 again I
will go ahead and save this random
sample in an object labeled R a and D
here we can see the random sample we've
selected we can also take a quick look
at a histogram for this random sample
it's worth noting here that even though
this sample was taken from a normally
distributed population the sample
histogram itself may not necessarily
look very normal thanks for watching
this video and make sure to check out my
other instructional videos
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Nov 24, 2024 Last updated |
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