The Normal Distribution and the 68-95-99.7 Rule (5.2)
Summary
TLDRThis video script explores the concept of the normal distribution, also known as the bell curve, and its characteristics, including the role of population parameters like mean (μ) and standard deviation (σ). It explains the 68-95-99.7 rule, which approximates the distribution's area within one, two, or three standard deviations from the mean. The script uses practical examples like exam scores and heights to illustrate these principles, offering viewers a clear understanding of how data clusters around the central value and the distribution's spread.
Takeaways
- 📚 A parameter is a number that describes a population, while a statistic describes a sample. For example, the population mean is denoted by the Greek letter mu (μ), and the sample mean by x-bar (𝑥̄).
- 📊 The normal distribution, also known as the bell curve, is a symmetrical density curve that shows data clustering around a central value, the population mean.
- 🌟 The position of the normal distribution on the number line is determined by the population mean (μ), while the spread is determined by the population standard deviation (Σ).
- 📉 The larger the standard deviation (Σ), the more spread out the normal distribution becomes, and the flatter the curve. Conversely, a smaller standard deviation results in a less spread out, taller curve.
- 🔍 The normal distribution is unimodal and symmetric, meaning it has a single peak and can be divided into two equal halves around the mean.
- 📈 The 68-95-99.7 rule states that approximately 68% of data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations.
- 🚫 A normal distribution never touches the x-axis, extending infinitely in both directions, but the area beyond three standard deviations becomes very small.
- 📐 The normal distribution's shape and spread are characterized by two parameters: the mean (μ) and the standard deviation (Σ), and it can be denoted as X ~ N(μ, Σ²).
- 📝 The 68-95-99.7 rule can be applied to any normal distribution to approximate the areas under the curve, regardless of its specific shape or size.
- 📑 The script provides examples and practice questions to illustrate the application of the normal distribution and the 68-95-99.7 rule in calculating areas under the curve.
- 💻 For further learning, the video suggests visiting the website simpleearningpower.com for study guides and practice questions related to the normal distribution.
Q & A
What is the difference between a parameter and a statistic?
-A parameter is a number that describes data from a population, while a statistic is a number that describes data from a sample.
What symbols are used to represent the sample mean and sample standard deviation?
-The sample mean is represented by x-bar (x̄) and the sample standard deviation is represented by s.
What symbols are used to represent the population mean and population standard deviation?
-The population mean is represented by the Greek letter mu (μ) and the population standard deviation is represented by the Greek letter sigma (σ).
What is a normal distribution and why is it sometimes called the bell curve?
-A normal distribution is a special type of density curve that is bell-shaped. It is called the bell curve because of its shape.
How does the population mean (mu) affect the position of the normal distribution?
-The population mean (mu) determines the position of the normal distribution. If the mean increases, the curve shifts to the right; if the mean decreases, the curve shifts to the left.
How does the population standard deviation (sigma) affect the spread of the normal distribution?
-The population standard deviation (sigma) determines the spread of the normal distribution. A larger standard deviation results in a more spread-out distribution, while a smaller standard deviation results in a less spread-out distribution.
What does the notation N(μ, σ) mean in the context of a normal distribution?
-The notation N(μ, σ) indicates that the variable X follows a normal distribution with a mean of μ and a standard deviation of σ.
What does the 68-95-99.7 rule state?
-The 68-95-99.7 rule states that in a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, 95% falls within two standard deviations, and 99.7% falls within three standard deviations.
If the mean height of students at a university is 5.5 feet with a standard deviation of 0.5 feet, what percentage of students are between 5 and 6 feet tall?
-Approximately 68% of students are between 5 and 6 feet tall, according to the 68-95-99.7 rule.
In a normal distribution with a mean of 70 and a standard deviation of 10, what is the approximate area contained between 70 and 90?
-The approximate area contained between 70 and 90 is 47.5%, as it represents half of the area within two standard deviations (95%).
For a normal distribution with a mean of 0 and a standard deviation of 1, what is the approximate area contained between -2 and 1?
-The approximate area contained between -2 and 1 is 81.5%, calculated by adding 47.5% (area from -2 to 0) and 34% (area from 0 to 1).
Outlines
📚 Introduction to Normal Distribution and Parameters vs. Statistics
This paragraph introduces the concept of the normal distribution, also known as the bell curve, which is a symmetrical density curve that represents data clustering around a central value, the population mean (μ). It distinguishes between parameters, which are characteristics of a population (e.g., μ and population standard deviation, Σ), and statistics, which describe sample data (e.g., sample mean, x-bar, and sample standard deviation, s). The paragraph also explains the significance of the mean and standard deviation in determining the position and spread of the normal distribution curve.
📊 Understanding the 68-95-99.7 Rule in Normal Distribution
The second paragraph delves into the 68-95-99.7 rule, which is a statistical principle used to approximate the distribution of data in a normal distribution. It states that approximately 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. The paragraph uses the example of student heights to illustrate this rule, explaining how the areas under the normal distribution curve correspond to percentages of the population. It also addresses the infinite nature of the normal distribution, which never touches the x-axis and extends indefinitely, and the diminishing areas beyond three standard deviations from the mean.
Mindmap
Keywords
💡Normal Distribution
💡68-95-99.7 Rule
💡Parameter
💡Statistic
💡Bell Curve
💡Population Mean (mu)
💡Population Standard Deviation (Sigma)
💡Density Curve
💡Symmetric
💡Unimodal
💡Empirical Rule
Highlights
The video explains the concept of the normal distribution and the 68-95-99.7 rule.
Distinguishes between a parameter and a statistic, with examples of mean and standard deviation.
Clarifies the use of symbols x-bar for sample mean and s for sample standard deviation.
Describes the Greek letters mu and Sigma as symbols for population mean and standard deviation.
Explains the bell-shaped curve of the normal distribution and its relation to data clustering around the mean.
Discusses the natural occurrence of the normal distribution in variables such as height, weight, and blood pressure.
Illustrates how the population mean (mu) determines the position of the normal distribution.
Describes the role of the population standard deviation (Sigma) in the spread of the normal distribution.
Explains the effect of standard deviation on the shape of the normal distribution curve.
States that the normal distribution is unimodal and symmetric about its mean.
Details the notation for a normally distributed population with mean mu and standard deviation Sigma.
Introduces the 68-95-99.7 rule for approximating the areas under the normal distribution curve.
Demonstrates the application of the 68-95-99.7 rule using an example of students' heights.
Clarifies that the normal distribution extends to infinity and does not touch the x-axis.
Provides practice questions to apply understanding of the normal distribution and the 68-95-99.7 rule.
Shows how to calculate the area between specific values using the 68-95-99.7 rule.
Encourages viewers to support the channel for more educational content.
Directs viewers to the website for additional study guides and practice questions.
Transcripts
in this video we'll be learning about
the normal distribution and the 6895
99.7 rule when we talk about normal
distributions we refer to data we get
from a population or sample so before we
actually talk about the normal
distribution we need to first
distinguish the difference between a
parameter and a statistic a parameter is
a number that describes the data from a
population whereas a statistic is a
number that describes the data from a
sample examples of parameters and
statistics are the mean and standard
deviation but because of the definitions
we just talked about we have to be very
careful with what symbols we use to
represent these numbers when we are
dealing with a sample we use the symbol
x-bar to represent the sample mean and
we use the letter s to represent the
sample standard deviation these are
statistics when we are dealing with a
population we use the Greek letter mu to
represent the population mean and we use
the Greek letter Sigma to represent the
population standard deviation these are
parameters the population parameters mu
and Sigma are very important when we
talk about normally distributed
populations so what is a normal
distribution anyways a normal
distribution is a special type of
density curve that is bell-shaped for
this reason the normal distribution is
sometimes called the bell curve or the
normal curve the normal distribution
describes the tendency for data to
cluster around a central value in fact
this central value is the population
mean mu which is always located in the
middle of the curve so for any normal
distribution we can say that some data
points will fall below the mean other
data points will fall above the mean but
most of the data values are located near
the mean the normal distribution and its
shape actually arises from many
different variables found in nature such
as weight height volume blood pressure
and many more this is why the normal
distribution is commonly studied for
example exam scores are known to follow
a normal distribution
some people do great on exams some
people do poorly on exams but a large
majority of people score near the
average or the mean in this example the
average exam score is 50
because it is located in the middle of
the curve now that you know what a
normal distribution looks like we need
to talk about the population mean meal
and the population standard deviation
Sigma
both of these tell us important
information about how the normal
distribution looks we all talk about the
population mean mu first the population
mean mu characterizes the position of
the normal distribution if you increase
the mean the curve will follow and move
towards the right and if you decrease
the mean the curve will still follow and
move towards the left this happens
because the data will always cluster
around the mean in normally distributed
populations as a result the value of the
mean determines the position of the
normal distribution on the other hand
the population standard deviation Sigma
characterizes the spread of the normal
distribution the larger the standard
deviation the more spread out the
distribution will be and the smaller the
standard deviation the less spread order
will be notice that when the spread
increases the curve gets much flatter
and when the spread decreases the curve
gets taller the reason for this is
because the normal distribution is a
density curve and the total area of any
density curve must remain equal to one
or a hundred percent so changes in the
width of the curve must be compensated
for by changes in the height of the
curve and vice versa overall here are
some points about the normal
distribution the normal distribution is
unimodal this means that the
distribution has a single peak the
normal curve is symmetric about its mean
so you can clearly see that the
distribution can be cut into two equal
halves the parameters mu and Sigma
completely characterized the normal
distribution the population mean mu
determines the location of the
distribution and where the data tends to
cluster around the population standard
deviation Sigma determines how spread
out the distribution will be the
notation given to a population that
follows a normal distribution can be
written like this although it looks
scary it means what it says for the
variable X it follows a normal
distribution and has the mean mu with a
standard deviation of Sigma now that
you've been introduced to the normal
distribution
we can talk about the 6895 99.7 rule if
we were measuring the heights of all
students at a local university and found
that it was normally distributed with a
mean height of 5.5 feet and a standard
deviation of half a foot or 0.5 we can
construct a normal distribution as
follows
from here we can create intervals that
increase by the standard deviation so
we'll have six six point five and seven
and on the other side we'll have five
four point five and four so what the 68
95 99 point seven rule says is that
within one standard deviation away from
the mean it contains a total area of
zero point six eight or 68% because of
this we can say that 68% of the
population are between five and six feet
tall and if he go to standard deviations
away from the mean it contains an area
of 95 percent this means that 95 percent
of the people in the population have a
height between four point five and six
point five feet and finally within three
standard deviations away from the mean
it contains a total area of ninety-nine
point seven percent this means that for
the population we are studying
ninety-nine point seven percent of the
people are between four and seven feet
tall now you might be wondering what
happens if we go four standard
deviations away from the mean or five or
six standard deviations away from the
mean and to answer that you actually can
a normal distribution actually never
touches the x-axis it continues on to
infinity so you can go as many standard
deviations away from the mean as you
want but the area contained within these
regions will be very very small
the 6895 99.7 rule is a great way for
approximating the areas of a normal
distribution and this works for any
normal distribution no matter what shape
and size so let's do some practice
questions feel free to pause the video
at any point so you can try these
questions for yourself
question number one the normal
distribution below has a standard
deviation of 10 approximately what area
is contained between 70 and 90
in this question we know that the
population mean is equal to 70 because
it's in the center of the distribution
we also know from the question that one
standard deviation is equal to 10 and we
can see this because each interval goes
up by 10 according to the 6895 99.7 rule
we know that there is an area of 95
percent contained within two standard
deviations of the mean
two standard deviations to the right
gets us to 90 and two standard
deviations to the left gets us to 50
according to the 68 95 99 point 7 rule
this means that there is an area of 95
percent contained within this interval
however we are only interested in the
area from 70 to 90 so dividing this area
by two gives us our area of interest
95 percent divided by two gives us an
area of forty seven point five percent
and that is our answer question number
two for the normal distribution below
approximately what area is contained
between negative two and one in this
example we know that we have am u of
zero because zero is in the center of
the distribution and we know that we
have a sigma of one because each
interval goes up by one to approximate
the area between negative two and one we
use the 6895 99.7 rule we can
strategically divide this area into two
parts so that we can easily incorporate
this rule we'll start with the right
half which goes from zero to one we know
that one standard deviation away from
the mean gives us 68% and half of this
is 34% giving us our area from zero to
one the next half goes from zero to
negative two but we know that within two
standard deviations from the mean we
have an area of 95% dividing this by two
gives us the area from zero to negative
two which is equal to forty seven point
five percent and finally to get that
total area contained between negative
two and one
all we have to do is add these two areas
together and when we do we get a total
area of 81.5% if you found this video
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