Psychologist Daniel Gilbert wrote, "The secret of happiness is variety, but the
secret of variety, like the secret of all spices, is knowing when to use it."
Not only is variety--or variation--the secret of happiness, it is also one of the
underlying principles of statistics. As researchers, we are interested in
IF the data vary, HOW the data vary, and, ultimately, WHY the data vary.
The first step to determining IF the data vary is to summarize the data, and
you're probably familiar with measures of central tendency: the mean, median, and mode.
Now let's look at measures of dispersion, which help us understand the
range and variability of scores in a dataset.
The focus of this video is to
provide a brief overview of the most common measures of dispersion and
provide more detail on two of those measures: Variance and Standard Deviation.
Fair warning: We will focus on understanding the concepts of Variance
and Standard Deviation, not how to calculate these statistics.
That will be covered in a later video.
Let's start with a brief discussion of Measures of Dispersion,
which deal with the variability--or spread, sometimes called the "spatter"--of scores.
Measures of dispersion help us understand the range and variability of scores.
For our example, let's turn to a popular web site
in my household called Rotten Tomatoes, which reviews movies on a scale of 0 to 100,
or what they call a film's "freshness level."
Movies rated on the low end of the scale are considered "rotten"--
actually Rotten Tomatoes establishes the cut-off point at 59% -
if a movie is rated 59%or lower, it's classified as "rotten"--
while those on the higher end of the scale, rated at 60% or higher,
are classified as "fresh."
Unfortunately, I can't show you their cute images due to copyright issues.
Here is the list of movies that Rotten Tomatoes identified as
the 25 best Christmas movies, along with their ratings.
Now some of these movies I've never heard of,
and, some of them, I personally wouldn't classify as a Christmas movie.
Notice the scores:
The lowest score listed for this dataset
is 80, for Batman Returns, which came out in 1992, while the highest scores listed is
100 for the 1940 movie, The Shop Around the Corner,
and the 1942 movie, Holiday Inn.
This relates to the first three measures of dispersion:
minimum--the lowest value in a dataset, which, in this case, is 80--
maximum, or the highest score in the dataset, 100, and
the range, which can be thought of in two different ways: first as a quantity, as in
the difference between the highest and lowest scores in a distribution: "The
range of scores is 20," and, second, as an interval, with the lowest and highest
scores as the range: "The range is 80 to 100," or "The scores range between 80 to 100," which
which could be written as 80, 100.
That covers the first three measures of dispersion.
The next three are interrelated: Variance, Standard Deviation, and Z-score.
You calculate Variance on the way to calculating Standard Deviation,
and you need Standard Deviation to calculate the Z-score.
The first two, Variance and Standard Deviation, deal with how the numbers are spread out,
or how they vary, in relation to the mean. These statistics relate to the entire dataset.
Z-scores, however are used to standardize individual scores or data points,
so that we can compare them with other data points in the dataset.
You can think of these scores as measuring a particular scores relationship to the mean
in a group of scores.
The rest of this video will focus on Variance and Standard Deviation.
First, let's learn the symbols for the two statistical measures,
as that will help us in understanding how one can be used to calculate the other,
And we'll start with Standard Deviation.
If you are reporting the Standard Deviation of a population,
you designate that with the lowercase Sigma character,
the one that looks like a little o with the tail on the top (σ).
If you are reporting the Standard Deviation of a sample, it simple: a lowercase s.
In some cases, however, you'll just see Standard Deviation denoted with an uppercase SD.
Here's the connection between the two:
Standard Deviation is the square root of Variance,
so the symbol for Variance is the same symbol for Standard Deviation, but squared.
Variance describes the extent to which values in a dataset vary;
it measures how far a set of numbers is spread out in relation to the mean.
Are the scores spread out a lot? Or are they all clustered together around the mean?
The larger the Variance statistic you calculate, the more spread out the data are.
The smaller the Variance statistic, the more the scores are clustered around the mean.
A Variance of 0 means that all the scores in a dataset are the same--
they are the mean.
Standard Deviation is the most frequently used measure of variability.
Consider word choice: a deviation from something that is standard;
in this case, the mean is the standard.
Standard Deviation explains the average distance that the individual points are away from the mean.
Not surprisingly, as Standard Deviation is the square root of Variance,
the larger the spread of numbers, the larger the Standard Deviation statistic.
Let's look at Standard Deviation in more detail to understand why it's helpful.
Assuming you have a normal distribution, the mean would, of course, be in the center
As the whole curve encompasses 100% of the scores, this means that 50% of the scores
in the distribution are higher than the mean and 50% of the scores are lower.
Another way to say it is that the probability of a scorer being above the mean is 50%,
while the probability of a scorer being below the mean is 50%.
At the mean, the standard deviation is zero.
In a normal distribution the probability is
that 34% of the scores are within one standard deviation above the mean and
about 34% of the scores are within one standard deviation below the mean.
Added together, this means that there is a 68% probability that a
score is within one standard deviation of the mean.
The same principle applies to two standard deviations from the mean.
There is about a 95% probability that a score lies within two standard deviations of the mean.
Well, actually, to be precise,
the 95% probability is that a score is within 1.96 standard deviations of the mean.
And the same principle for three standard deviations,
which would encompass virtually all of the scores in a normal distribution.
Now, again, to be precise,
there is a 99% probability that a score lies within 2.58 standard deviations of the mean.
All of the standard deviations to the right of the mean, as they are higher than the mean, are
while all of the standard deviations below the mean, or to the left, are negative.
Let's see how this works:
Assume we have a mean of 42 and a standard deviation calculated at 12.
The mean, of course, goes right here, in the very center of the distribution curve.
Now we have to do a little bit of addition and subtraction.
If one standard deviation is 12,
then two standard deviations would be twice that: 12 times 2, or 24.
And three standard deviations would be 3 times 12, or 36.
So we can plot all of these, starting with the mean.
The mean is 42, and, when adding one standard deviation score of 12 is 54.
So a score between 42, the mean, and 54 would be plotted at one standard deviation above the mean.
We can do the same for one standard deviation below the mean:
42, minus the one standard deviation score of 12, is 30.
So 30 gets plotted at one standard deviation below the mean.
This means that, assuming the scores are normally distributed,
68% of the scores are between 30 and 42--
or plus or minus one standard deviation of the mean.
Let's continue for two standard deviations:
The mean of 42 + 24 (two standard deviations) is 66,
so we can plot the score of 66 at plus two standard deviations.
And the mean of 42 minus 24 (or two standard deviations) is 18,
so we can plot the score of 18 at negative two standard deviations.
Now, assuming the scores were normally distributed about 95% of the scores are
between 18 and 66--within plus or minus two standard deviations of the mean.
One last time for three standard deviations:
the mean of 42 plus 36 (the score for three standard deviations) is 78,
which is plotted at plus three standard deviations.
And the mean of 42 minus 36 is 6 which is plotted at minus three standard
Remember we said earlier that, assuming a normal distribution, 99% of the scores
are within plus or minus 2.58 standard deviation of the mean?
If we use this is the multiplier, we would take the standard deviation of 12 and multiply it
by 2.58 to get 30.96. That means that 99% of our scores would be between 11.04 and 72.96.
Let's see what happens with the same mean but a smaller standard deviation, such as 1.5.
The mean is still in the very center of the distribution curve.
Let's figure out all the positive standard deviations first.
The mean of 42 plus one standard deviation, or, in this case, 1.5 is 43.5.
The mean plus two standard deviations, so 42 plus 3, is 45, and the mean plus three
And the mean plus three standard deviations, so 42 + 4.5, is 46.5.
And we can plot these numbers at plus one, plus two and plus three standard deviations, respectively.
We can do the same for the negative standard deviations and plot those now.
You can see that the scores are much closer together--there is less variation--
and we can see what the likelihood is that a score will fall within certain parameters.
This, then, is the value of standard deviation.
Okay, let's add an example that will hopefully take all this statistical stuff and make it more relevant.
Let's say that you were considering going to a movie and are split between two possibilities.
Thirty of your friends have seen both movies so
you ask them to rate each movie on a scale of 1 to 10, 10 being high.
Knowing what you do about measures of central tendency,
you calculate the mean on the scores for both movies.
Wow. the mean is the same at 7.33.
Kinda hard to make a decision here based solely on the mean.
So now you look at the measures of dispersion, in particular
the minimum, the maximum, and the range--
--which are also the same for both movies
ranging from the minimum score of 1 to the maximum score of 10.
So, no help here.
Now it's time to calculate the standard deviation for the two datasets.
Okay, now we see a difference.
The standard deviation for the first movie is higher than the standard deviation for the second movie:
3.570 vs 1.456.
This tells you that there is more variation in the scores for the first movie --
the scores are all spread out--than in the second movie
where the scores are much closer together.
Indeed ,if you graph the data, you'll see that there is much more consistency in
the scores for the second movie:L The majority of your friends rated movie at
a 6 or better and only two were unhappy with it. For the first movie, however. it
looks like your friends either loved it or hated it. So if you're only going to
go see one movie, I'd recommend the second one as it looks like it's a safer
bet that you will like that second movie.
Another example: Assume an airline is
designing a new aircraft and is looking at seat size.
At this time, the average economy seat width is 17.8 inches.
They know that they can't provide seats that
all 100% of their passengers will fit in, but they want to be able to accommodate
most of their potential air travelers. As airlines want to make a profit,
their mission is to answer the question: What is the most cost effective seat size?
If they know the mean size of air travelers, likely determined from a sample
of air travelers, and the standard deviation of the sample of air travelers,
then they can design their seats accordingly.
if the airlines want to accommodate about 68 percent of their passengers,
then they would design their seats to accommodate people
based upon the average size plus one standard deviation.
If, however, they want to accommodate about 95% of their potential passengers,
they would design the seat based upon two standard deviations of the mean.
Look: A real purpose for statistics!
Processing time!
What do measures of dispersion do?
Well, they display the range and variability of scores.
What are the most common measures of dispersion?
well the easy ones are the range, the minimum, and the maximum,
but there's also Variance, Standard Deviation, and Z-score.
What do Variance and Standard Deviation do?
They tell us how scores are spread out from the mean.
The higher the number, the more spread out the distribution is,
While the lower the number, the more compressed the scores are.
With your basic understanding of measures of dispersion,
it's time for you to pull out your calculator and start calculating them.