The Total Area Always Equals 100%
Normal distributions are a family of distributions that have the
same general shape. They are symmetric with scores more concentrated in the middle than in
the tails. Normal distributions are sometimes described as bell shaped.
Here are some examples. Notice that they differ in how spread out they are. But even though
the shapes are different, the area under each curve is the same. The total area accounted for
under any curve is 100%. This never changes and is critical to our understanding of how to
apply the Normal distribution. Because we know that the area under the curve is always 1
or 100%, we can understand a lot about individual scores and groups of scores to which the
Normal distribution is applied. We'll need a Normal curve table for this. More on that one,
shortly. Click images to enlarge.
The Normal curve has other characteristics that are always true. Once again, the fact that we
can always count on these characteristics provides a good model for understanding numeric
trends in data. The following are other important characteristics of the Normal curve:
All Normal curves are symmetric around the mean of the distribution. In other words, the left
half of the Normal curve is a mirror image of the right half.
All Normal curves are unimodal. Because Normal curves are symmetric, the most frequently
observed score in a Normal distribution— the mode— is the same as the mean.
Since the Normal curves are unimodal and symmetric, the mean, median, and mode of all
Normal distributions are equal.
All Normal curves are asymptotic to the horizontal axis of the distribution. Scores in a
Normal distribution descend rapidly as one moves along the horizontal axis from the center
of the distribution toward the extreme ends of the distribution, but they never actually touch
it. This is because scores on the Normal curve are continuous and held to describe an infinity
of observations.
All Normal curves have the same proportions of scores under the curve relative to particular
locations on the horizontal axis when scores are expressed as areas, percentiles, probabilities,
etc.
Hey, It's As Natural As Big Feet!
As we discussed in a previous chapter, research in any field must deal with variability. We
know that too much variability probably means that we have more error in our methods and
data, whereas less variability is one indication that our methods and data comprise less error.
So, less variability is good, but there will always be some. Why? Everyone doesn't respond
the same way to the same medication; different people have different memory abilities; some
people are taller and some people are shorter. Turns out that variability is natural, as is the
Normal distribution. In other words, organisms inherit physical and derivative psychological
traits...well..."Normally."
We take this Normality as a common pattern or "process" of nature and our "observations"
of it. And eventhough we may not be able to identify all of the factors (we never will, by the
way) that make up the thing we like to call "intelligence," when we measure this thing in
large numbers and with proper research methods, we get that nice Normal curve. Go figure!
Consider the phenomenon of Bigfoot or Sasquatch. I know...I know. A fairytale. Right? Like
there could really be populations of half-human/half-ape creatures that exist in various remote
locations and are only detectable through their forensic remains. Before we dismiss it too
quickly, let's try the hard thing. Let's try to argue FOR the existence of Sasquatch based on
Normality. How could we do this?
As you may know, footprints are the standard stock in
trade of Sasquatch research, and their sometimes inhuman length assures almost immediate
measurement, even by first-time witnesses. The process here consists of foot lengths and
the observations are the measurements of footprints. Foot lengths are going to be affected by
a lot of factors: Gender of the creature. Family genetics. Nutrition. Surface from which the
foot lengths were measured--snow, mud, grass, etc. Length of time between the creation of
the footprint and its measurement. Amount of alcohol consumed by everyone involved. It's
complicated!
Nonetheless, as can be seen here, a sample of 410 independently collected footprints
(ostensibly left by a Bigfoot) forms a fairly Normal curve (with frequency plotted on
the y axis and foot length plotted on the x axis). The Normal distribution overall argues
compellingly for the existence of Sasquatch as a genuine species, in that production of
fictitious data over 40 years by hundreds of people independently of each other would likely
have generated a distribution with many peaks. A further factor that supports the authenticity
of the data is the fact that foot length, foot width, heel width, and gait are interrelated in a
logical and cohesive fashion, a congruence not plausible by pure chance.
Hmmmm....very interesting. Are you a true believer, yet? If you want to learn a little more
about forensic research on the big fella, you can read this research paper.
Why don't we frame this in less cryptozoological
terms. Let's look at the SAT. The process here consists of the students taking the test, and
the observations are the students’ scores. Now, my score, for example, is going to be due to a
whole set of different factors: my IQ, what I had for breakfast, how much I studied the night
before, how good my teachers are, which butterflies were flapping their wings in Beijing this
morning, and so on. In short, my score is the result of a whole set of hard-to-predict factors.
The same with my fellow students. And yet, even though all these factors are hard to predict,
if you take the scores of a large number of students from a single population, the scores will
be Normally distributed as you see here. Once again, when we see such a Normal curve in
our data, we're inclined to think that we're on the right track.
The 68-95-99 Rule
The standard normal curve is a special example of the normal distribution. The height
of a Normal distribution can be specified mathematically in terms of two population
parameters: the mean (μ) and the standard deviation (σ). Instead of calculating our curve
parameters in painstaking, mathematic long hand, we will simply use sample statistics (s
and x-bar) to estimate the properties or distribution shape of our actual population. In other
words, we can do some short cutting.
Every time you look at a group of scores (sample of data), you want to be thinking
about those scores as comprising a shape. Even though you will see data listed in groups
and columns, underneath every data set is a shape. Whenever we perform statistical analyses,
we're hoping that this shape comes as close as possible to bell-shaped or Normal. As we
move along with our discussion, this idea of "shape" will become more concrete.
The distances along the horizontal axis of our curve, when divided into standard deviations,
will always include the same proportion of the total area: Between -1 and +1 standard
deviation units lies about 68% of the area. Between -2 and +2 standard deviation units lies
about 95% of the area. Between -3 and +3 standard deviation units lies about 99% of the area.
This is true of a standard normal curve whether it is perfectly bell-shaped, a little narrower or
a little wider. This graphic depicts the approximate 68-95-99 breakdown for a bell-shaped,
standard normal curve.
This conception of the normal curve starts to become powerful when we "map" it onto
normally distributed variables. One example of a variable that forms a normal curve is I.Q.
In this case, we can tell what percentage of people are in any area of the curve. A normal
distribution of 1000 cases will have 683 (about 68%) people between +/-1 standard deviation,
about 954 (about 95%) people between +/-2 standard deviations, and 997 (about 99%) people
between +/-3 standard deviations. Only 3 people will be outside 3 standard deviations from
the mean, if the sample size is 1000. In other words, in a perfectly normal distribution based
on such data, we would expect only about three people to have I.Q. scores above and below
the I.Q. scores associated with z scores of +3 and -3.
source:-http://www.mesacc.edu/~derdy61351/230_web_book/module3/normal/
Normal distributions are a family of distributions that have the
same general shape. They are symmetric with scores more concentrated in the middle than in
the tails. Normal distributions are sometimes described as bell shaped.
Here are some examples. Notice that they differ in how spread out they are. But even though
the shapes are different, the area under each curve is the same. The total area accounted for
under any curve is 100%. This never changes and is critical to our understanding of how to
apply the Normal distribution. Because we know that the area under the curve is always 1
or 100%, we can understand a lot about individual scores and groups of scores to which the
Normal distribution is applied. We'll need a Normal curve table for this. More on that one,
shortly. Click images to enlarge.
The Normal curve has other characteristics that are always true. Once again, the fact that we
can always count on these characteristics provides a good model for understanding numeric
trends in data. The following are other important characteristics of the Normal curve:
All Normal curves are symmetric around the mean of the distribution. In other words, the left
half of the Normal curve is a mirror image of the right half.
All Normal curves are unimodal. Because Normal curves are symmetric, the most frequently
observed score in a Normal distribution— the mode— is the same as the mean.
Since the Normal curves are unimodal and symmetric, the mean, median, and mode of all
Normal distributions are equal.
All Normal curves are asymptotic to the horizontal axis of the distribution. Scores in a
Normal distribution descend rapidly as one moves along the horizontal axis from the center
of the distribution toward the extreme ends of the distribution, but they never actually touch
it. This is because scores on the Normal curve are continuous and held to describe an infinity
of observations.
All Normal curves have the same proportions of scores under the curve relative to particular
locations on the horizontal axis when scores are expressed as areas, percentiles, probabilities,
etc.
Hey, It's As Natural As Big Feet!
As we discussed in a previous chapter, research in any field must deal with variability. We
know that too much variability probably means that we have more error in our methods and
data, whereas less variability is one indication that our methods and data comprise less error.
So, less variability is good, but there will always be some. Why? Everyone doesn't respond
the same way to the same medication; different people have different memory abilities; some
people are taller and some people are shorter. Turns out that variability is natural, as is the
Normal distribution. In other words, organisms inherit physical and derivative psychological
traits...well..."Normally."
We take this Normality as a common pattern or "process" of nature and our "observations"
of it. And eventhough we may not be able to identify all of the factors (we never will, by the
way) that make up the thing we like to call "intelligence," when we measure this thing in
large numbers and with proper research methods, we get that nice Normal curve. Go figure!
Consider the phenomenon of Bigfoot or Sasquatch. I know...I know. A fairytale. Right? Like
there could really be populations of half-human/half-ape creatures that exist in various remote
locations and are only detectable through their forensic remains. Before we dismiss it too
quickly, let's try the hard thing. Let's try to argue FOR the existence of Sasquatch based on
Normality. How could we do this?
As you may know, footprints are the standard stock in
trade of Sasquatch research, and their sometimes inhuman length assures almost immediate
measurement, even by first-time witnesses. The process here consists of foot lengths and
the observations are the measurements of footprints. Foot lengths are going to be affected by
a lot of factors: Gender of the creature. Family genetics. Nutrition. Surface from which the
foot lengths were measured--snow, mud, grass, etc. Length of time between the creation of
the footprint and its measurement. Amount of alcohol consumed by everyone involved. It's
complicated!
Nonetheless, as can be seen here, a sample of 410 independently collected footprints
(ostensibly left by a Bigfoot) forms a fairly Normal curve (with frequency plotted on
the y axis and foot length plotted on the x axis). The Normal distribution overall argues
compellingly for the existence of Sasquatch as a genuine species, in that production of
fictitious data over 40 years by hundreds of people independently of each other would likely
have generated a distribution with many peaks. A further factor that supports the authenticity
of the data is the fact that foot length, foot width, heel width, and gait are interrelated in a
logical and cohesive fashion, a congruence not plausible by pure chance.
Hmmmm....very interesting. Are you a true believer, yet? If you want to learn a little more
about forensic research on the big fella, you can read this research paper.
Why don't we frame this in less cryptozoological
terms. Let's look at the SAT. The process here consists of the students taking the test, and
the observations are the students’ scores. Now, my score, for example, is going to be due to a
whole set of different factors: my IQ, what I had for breakfast, how much I studied the night
before, how good my teachers are, which butterflies were flapping their wings in Beijing this
morning, and so on. In short, my score is the result of a whole set of hard-to-predict factors.
The same with my fellow students. And yet, even though all these factors are hard to predict,
if you take the scores of a large number of students from a single population, the scores will
be Normally distributed as you see here. Once again, when we see such a Normal curve in
our data, we're inclined to think that we're on the right track.
The 68-95-99 Rule
The standard normal curve is a special example of the normal distribution. The height
of a Normal distribution can be specified mathematically in terms of two population
parameters: the mean (μ) and the standard deviation (σ). Instead of calculating our curve
parameters in painstaking, mathematic long hand, we will simply use sample statistics (s
and x-bar) to estimate the properties or distribution shape of our actual population. In other
words, we can do some short cutting.
Every time you look at a group of scores (sample of data), you want to be thinking
about those scores as comprising a shape. Even though you will see data listed in groups
and columns, underneath every data set is a shape. Whenever we perform statistical analyses,
we're hoping that this shape comes as close as possible to bell-shaped or Normal. As we
move along with our discussion, this idea of "shape" will become more concrete.
The distances along the horizontal axis of our curve, when divided into standard deviations,
will always include the same proportion of the total area: Between -1 and +1 standard
deviation units lies about 68% of the area. Between -2 and +2 standard deviation units lies
about 95% of the area. Between -3 and +3 standard deviation units lies about 99% of the area.
This is true of a standard normal curve whether it is perfectly bell-shaped, a little narrower or
a little wider. This graphic depicts the approximate 68-95-99 breakdown for a bell-shaped,
standard normal curve.
This conception of the normal curve starts to become powerful when we "map" it onto
normally distributed variables. One example of a variable that forms a normal curve is I.Q.
In this case, we can tell what percentage of people are in any area of the curve. A normal
distribution of 1000 cases will have 683 (about 68%) people between +/-1 standard deviation,
about 954 (about 95%) people between +/-2 standard deviations, and 997 (about 99%) people
between +/-3 standard deviations. Only 3 people will be outside 3 standard deviations from
the mean, if the sample size is 1000. In other words, in a perfectly normal distribution based
on such data, we would expect only about three people to have I.Q. scores above and below
the I.Q. scores associated with z scores of +3 and -3.
source:-http://www.mesacc.edu/~derdy61351/230_web_book/module3/normal/
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